Recent zbMATH articles in MSC 53https://www.zbmath.org/atom/cc/532022-05-16T20:40:13.078697ZWerkzeugProfessor Oldřich Kowalski passed awayhttps://www.zbmath.org/1483.010152022-05-16T20:40:13.078697Z"Abbassi, M. T. K."https://www.zbmath.org/authors/?q=ai:abbassi.mohamed-tahar-kadaoui"Mikeš, J."https://www.zbmath.org/authors/?q=ai:mikes.josef"Vanžurová, A."https://www.zbmath.org/authors/?q=ai:vanzurova.alena"Bejan, C. L."https://www.zbmath.org/authors/?q=ai:bejan.cornelia-livia"Belova, O. O."https://www.zbmath.org/authors/?q=ai:belova.olga-o(no abstract)A homology vanishing theorem for graphs with positive curvaturehttps://www.zbmath.org/1483.050692022-05-16T20:40:13.078697Z"Kempton, Mark"https://www.zbmath.org/authors/?q=ai:kempton.mark"Münch, Florentin"https://www.zbmath.org/authors/?q=ai:munch.florentin"Yau, Shing-Tung"https://www.zbmath.org/authors/?q=ai:yau.shing-tungSummary: We prove a homology vanishing theorem for graphs with positive Bakry-Émery curvature, analogous to a classic result of Bochner on manifolds [\textit{S. Bochner}, Ann. Math. (2) 49, 379--390 (1948; Zbl 0038.34401)]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by \textit{A. Grigor'yan} et al. [``Homologies of path
complexes and digraphs'', Preprint, \url{arXiv:1207.2834}]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-Émery curvature, analogous to a classic result of Myers on manifolds [\textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; Zbl 0025.22704)]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Émery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by \textit{A. Grigor'yan} et al. [Pure Appl. Math. Q. 10, No. 4, 619--674 (2014; Zbl 1312.05063)], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.Corrigendum to: ``On two conjectures concerning convex curves''https://www.zbmath.org/1483.140572022-05-16T20:40:13.078697Z"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Shapiro, Michael"https://www.zbmath.org/authors/?q=ai:shapiro.michael-zThe Hodge numbers of O'Grady 10 via Ngô stringshttps://www.zbmath.org/1483.140712022-05-16T20:40:13.078697Z"de Cataldo, Mark Andrea A."https://www.zbmath.org/authors/?q=ai:de-cataldo.mark-andrea-a"Rapagnetta, Antonio"https://www.zbmath.org/authors/?q=ai:rapagnetta.antonio"Saccà, Giulia"https://www.zbmath.org/authors/?q=ai:sacca.giuliaIn complex dimension 10 there is a sporadic class of irreducible holomorphic symplectic manifolds introduced by O'Grady, its deformation class is denoted by \(OG10\). The first main result of the paper is the computation of the Betti numbers and Hodge numbers of the deformation class \(OG10\). The main idea of the proof is to consider two Lagrangian fibered irreducible holomorphic symplectic manifolds \(\tilde{M}\) and \(\tilde{N}\) with base space \(\mathbb{P}^5\): one is of \(OG10\)-type and the other one is of \(K3^{[5]}\)-type. The key ingredient is the Ngô's Support Theorem which allows one to compare their cohomologies. The second main Theorem determines the pure Hodge structure of the manifolds \(\tilde{M}\) in terms of the pure Hodge structure of a general polarized \(K3\) surface of genus 2.
Reviewer: Andrea Galasso (Taipei)Lagrangian constant cycle subvarieties in Lagrangian fibrationshttps://www.zbmath.org/1483.140732022-05-16T20:40:13.078697Z"Lin, Hsueh-Yung"https://www.zbmath.org/authors/?q=ai:lin.hsueh-yungSummary: We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold \(X\) whose general fiber is of dimension strictly between 0 and \(\dim X\) is rationally connected. Using this result, we construct for any hyper-Kähler manifold \(X\) admitting a Lagrangian fibration a Lagrangian constant cycle subvariety \(\Sigma_H\) in \(X\) which depends on a divisor class \(H\) whose restriction to some smooth Lagrangian fiber is ample. If \(\dim X=4\), we also show that up to a scalar multiple, the class of a zero-cycle supported on \(\Sigma_H\) in \(CH_0(X)\) depend neither on \(H\) nor on the Lagrangian fibration (provided \(b_2(X)\geq 8)\).Elliptic curves in hyper-Kähler varietieshttps://www.zbmath.org/1483.140742022-05-16T20:40:13.078697Z"Nesterov, Denis"https://www.zbmath.org/authors/?q=ai:nesterov.denis"Oberdieck, Georg"https://www.zbmath.org/authors/?q=ai:oberdieck.georgLet \(X\) be a hyper-Kähler manifold of dimension \(2n\) and assume that the Picard group of \(X\) is generated by an ample divisor \(H\). Then the space of cohomology classes of curves is one-dimensional and we denote by \(\beta\) its effective generator, i.e. \(H_2(X, \mathbb{Z})_{\text{alg}}=\mathbb{Z} \cdot \beta\). Let \(M_{g,n}(X, \beta )\) be the moduli space of stable genus \(g\) curves in \(X\) with \(n\) distinct markings and cohomology class equals to \(\beta\). By looking at the dimension of the tangent space of the Hilbert scheme of the family of such curves, the expected dimension of this family of curves is \(vd := (\dim (X) - 3)(1 - g) + n + 1.\) The authors conjecture that \(M_{1,0}(X, \beta)\) is pure of dimension 1. This conjecture is well-known in dimension two by Bogomolov-Mumford Theorem, where the family of genus one curves is always non-trivial and one-dimensional. In the general case, we should allow the family to be empty. Indeed the authors proves that \(M_{1,0}(X, \beta)=\emptyset \) for a very general generalised Kummer fourfold. The authors provide some Gromov--Witten calculations for \(K3\) surface and for the Hilbert scheme of two points on a \(K3\) surface. Then they focus on the Fano variety of lines on a very general cubic fourfold that we will denote by \(X\). This variety is a projective hyper-Kähler manifold of dimension four that is deformation equivalent to the Hilbert scheme of two points on a \(K3\) surface. The aim of the paper is to understand the genus one curves contained in \(X\) with minimal class, i.e. \(M_{1,0}(X, \beta )\). In order to describe this locus, they show that the space \(\operatorname{CH}_{\beta}(X)\) of effective 1-cycles represented by \(\beta\) can be decomposed as \(\operatorname{CH}_{\beta}(X)=S\cup \Sigma\) where \(S\) parametrizes the rational curves and \(\Sigma\) the genus-one curves. The intersection \(S \cap \Sigma\) consists of at most 3780 points corresponding precisely to the nodal rational curves. Moreover, \(\Sigma\) is not isotrivial and generically there are 3780 curves with a fixed \(j\)-invariant.
Reviewer: Fabrizio Anella (Roma)Optimal destabilizing centers and equivariant K-stabilityhttps://www.zbmath.org/1483.140762022-05-16T20:40:13.078697Z"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanThe notion of K-stability is introduced as an algebraic condition for the existence of Kähler-Einstein metrics on Fano varieties. K-stability of a Fano variety \(X\) is defined by positivity of Futaki invariant of test configurations of \(X\), which are \(\mathbb{G}_m\)-equivariant polarized one parameter degenerations of \(X\). One of the main challenges in the theory of K-stability is to find an effective way of checking K-stability for an explicit Fano variety. If the Fano variety \(X\) is equipped with an action of an algebraic group \(G\), then it is expected that to test K-stability it suffices to check those test configurations that are \(G\)-equivariant, known as the \(G\)-equivariant K-stability. This expectation was proved in cases when \(X\) is a smooth Fano manifold and \(G\) reductive by \textit{V. Datar} and \textit{G. Székelyhidi} [Geom. Funct. Anal. 26, No. 4, 975--1010 (2016; Zbl 1359.32019)], when \(G\) is an algebraic torus by \textit{C. Li} et al. [J. Am. Math. Soc. 34, No. 4, 1175--1214 (2021; Zbl 1475.14062)], or when \(G\) is a finite group by \textit{Y. Liu} and \textit{Z. Zhu} [Int. J. Math. 33, No. 1, Article ID 2250007, 21 p. (2022; Zbl 07488259)]. Note that the approaches of Datar-Székelyhidi and Liu-Zhu heavily relies on analytic methods from solutions of the Yau-Tian-Donaldson conjecture.
The main theorem of this paper verifies this expectation in full generality using purely algebraic methods, showing the following result.
Theorem 1.1. Let \(X\) be a Fano variety with an action of an algebraic group \(G\). Then
\begin{itemize}
\item[1.] If \(X\) is \(G\)-equivariantly K-semistable, then \(X\) is K-semistable.
\item[2.] If \(X\) is \(G\)-equivariantly K-polystable and \(G\) is reductive, then \(X\) is K-polystable.
\end{itemize}
To prove Theorem 1.1, the author proceed in two steps. The first step addresses the uniqueness of the minimal optimal destabilizing center. The stability threshold \(\delta(X)\) of a Fano variety \(X\), introduced by \textit{K. Fujita} and \textit{Y. Odaka} [Tohoku Math. J. (2) 70, No. 4, 511--521 (2018; Zbl 1422.14047)], measures the singularity of an average anti-canonical \(\mathbb{Q}\)-divisor on \(X\). An equivalent definition by \textit{H. Blum} and \textit{M. Jonsson} [Adv. Math. 365, Article ID 107062, 57 p. (2020; Zbl 1441.14137)] gives that
\[
\delta(X)=\min_{v} \frac{A_X(v)}{S(v)},
\]
where \(v\) runs over all real valuations of \(K(X)\). According to Fujita-Odaka and Blum-Jonsson, \(X\) is K-semistable if and only if \(\delta(X)\geq 1\). If \(\delta(X)<1\), an optimal destabilizing center \(Z\subset X\) is the center of a valuation \(v\) achieving the minimum in the above equality. In Theorem 1.5, the author shows that if \(X\) is K-unstable, then there exists a unique minimal optimal destabilizing center of \(X\). Then in the second step, the author shows that the stability threshold around the minimal optimal destabilizing center can be computed asympotically by a \(G\)-equivariant prime divisor. The proofs of these results cleverly combine the connectedness theorem of Kollár-Shokurov, compatible divisors explored in an earlier joint work of the author with \textit{H. Ahmadinezhad} [``K-stability of Fano varieties via admissible flags'', Preprint, \url{arXiv:2003.13788}], and induction on dimension from inversion of adjunction.
Reviewer: Yuchen Liu (Evanston)Special cases of the orbifold version of Zvonkine's \(r\)-ELSV formulahttps://www.zbmath.org/1483.140922022-05-16T20:40:13.078697Z"Borot, Gaëtan"https://www.zbmath.org/authors/?q=ai:borot.gaetan"Kramer, Reinier"https://www.zbmath.org/authors/?q=ai:kramer.reinier"Lewanski, Danilo"https://www.zbmath.org/authors/?q=ai:lewanski.danilo"Popolitov, Alexandr"https://www.zbmath.org/authors/?q=ai:popolitov.aleksandr"Shadrin, Sergey"https://www.zbmath.org/authors/?q=ai:shadrin.sergeySummary: We prove the orbifold version of Zvonkine's \(r\)-ELSV formula in two special cases: the case of \(r=2\) (completed 3-cycles) for any genus \(g\geq 0\) and the case of any \(r\geq 1\) for genus \(g=0\).Quasimaps to GIT fibre bundles and applicationshttps://www.zbmath.org/1483.141012022-05-16T20:40:13.078697Z"Oh, Jeongseok"https://www.zbmath.org/authors/?q=ai:oh.jeongseokThe main result of this paper is a Givental style mirror theorem for fiber bundles \(X\) over a smooth projective variety, whose fiber is a partial flag variety (and hence a GIT quotient). The theorem is phrased as a relation between the \(J\)-function, the first derivative of genus \(0\) Gromov-Witten invariants, and its mirror counterpart, the \(I\)-function. Namely, the \(I\)-function is shown to lie on a shifted Lagrangian cone related to the \(J\)-function. The latter can then be recovered from the former via Birkhoff factorization and a change of variables.
When \(X\) is a GIT quotient, \textit{I. Ciocan-Fontanine} et al. [J. Geom. Phys. 75, 17--47 (2014; Zbl 1282.14022)] defined an analog of \(I\)-function in terms of invariants of quasimaps, morphisms taking the generic point to \(X\), and proved the mirror theorem when \(X\) is a complete intersection in a toric variety. The author generalizes this construction to flag variety bundles by introducing maps that project to prestable maps to the base and quasimaps from contracted domain curves to the fiber. The mirror theorem is then derived from an equivariant version for the natural fiberwise torus action on the bundle by taking the limit over one of the equivariant parameters.
The equivariant proof is somewhat similar to \textit{J. Brown}'s proof [Int. Math. Res. Not. 2014, No. 19, 5437--5482 (2014; Zbl 1307.14077)] of Elezi's conjecture for fiber bundles whose fiber is a toric variety. Namely, the elements of the Lagrangian cone that are generated by the \(J\)-function are characterized, and the \(I\)-function is shown to meet this characterization. However, while Brown used asymptotic analysis to verify his characterization, the author uses the geometric interpretation in terms of the moduli spaces of maps instead. The characterization, in addition to the recursion relation and the polynomiality condition of Givental and Ciocan-Fontanine-Kim, involves an initial condition for the recursion in terms of the base of the bundle. Moreover, the polynomiality condition has to be extended to any derivative along a vector at the origin of the cohomology of the base. As an application, the author shows how the Gromov-Witten invariants of the base and the total space are related to each other when \(X\) is Fano or Calabi-Yau.
Reviewer: Sergiy Koshkin (Houston)The genus two G-function for the cubic elliptic singularityhttps://www.zbmath.org/1483.141022022-05-16T20:40:13.078697Z"Wang, Xin"https://www.zbmath.org/authors/?q=ai:wang.xin.3|wang.xin.4|wang.xin.6|wang.xin.8|wang.xin.7|wang.xin.5|wang.xin.2|wang.xin.12|wang.xin|wang.xin.13|wang.xin.10|wang.xin.1|wang.xin.11|wang.xin.9In 2001 \textit{B. Dubrovin} and \textit{Y. Zhang} [``Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants'', Preprint, \url{arXiv:math/0108160}] gave a complicated explicit formula for the genus \(2\) generating function for semisimple Frobenius manifolds. In 2012, in collaboration with \textit{S.-Q. Liu} [Russ. J. Math. Phys. 19, No. 3, 273--298 (2012; Zbl 1325.53114)], they split it into contributions from genus \(2\) dual graphs and the so called genus \(2\) \(G\)-function \(G^{(2)}\). In 2015 \textit{X. Liu} and \textit{X. Wang} [Adv. Math. 274, 631--650 (2015; Zbl 1368.53058)] showed that \(G^{(2)}\) vanishes for simple singularities and \(\mathbb{P}^1\) orbifolds of Fano type. In this paper the author studies properties of \(G^{(2)}\) for the cubic elliptic singularity and derives closed form formulas for its derivatives along the mirror map.
The derivation is based on the Saito-Givental theory for isolated polynomial singularities. First, Frobenius manifold structure is put on the deformation space of the singularity using Saito's relative differential form, which is semisimple at the generic point. Then Gromov-Witten invariants and the \(I\)-function are defined using Givental's formalism. From the \(I\)-function two series, \(X\) and \(L\), are defined, which are quasi-modular forms with the modular group \(\Gamma_0(3)\) up to analytic continuation and symplectic transformation. Finally, the derivative of \(G^{(2)}\) is expressed as an explicit polynomial in \(X\) and \(L\). The modular properties are naturally interpretable in terms of Gromov-Witten invariants of the corresponding \((3,3,3)\) type elliptic orbifold.
Reviewer: Sergiy Koshkin (Houston)From Schouten to Mackenzie: notes on bracketshttps://www.zbmath.org/1483.170012022-05-16T20:40:13.078697Z"Kosmann-Schwarzbach, Yvette"https://www.zbmath.org/authors/?q=ai:kosmann-schwarzbach.yvetteSummary: In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher-Nijenhuis, then in the work of Gerstenhaber and Nijenhuis-Richardson in cohomology theory.A construction by deformation of unitary irreducible representations of \(\mathrm{SU}(1, n)\) and \(S\mathrm{SU}(n + 1)\)https://www.zbmath.org/1483.170082022-05-16T20:40:13.078697Z"Cahen, Benjamin"https://www.zbmath.org/authors/?q=ai:cahen.benjaminIn the paper under review, the author constructs holomorphic discrete series representations of \(\mathrm{SU}(1, n)\) and some unitary irreducible representations of \(\mathrm{SU}(n)\) by deforming a minimal realization of \(\mathfrak{g}=\mathfrak{sl}(n+ 1,\mathbb C)\). The minimal realization refers to a representation \(\rho_0\) of \(\mathfrak{g}\) in the space of complex polynomials with \(n\) variables, which is given by the classical Weyl correspondence. The term ``minimal'' indicates that the construction of \(\rho_0\) is closely related to the minimal nilpotent coadjoint orbit of \(\mathfrak{g}\). The deformation of \(\rho_0\) is given over the space \(M\) of the complex polynomials with \(2n\) variables and is controlled by the first Chevalley-Eilenberg cohomology space \(H^1(\mathfrak{g},M)\).
Reviewer: Husileng Xiao (Harbin)On the growth of \(L^2\)-invariants of locally symmetric spaces. II: Exotic invariant random subgroups in rank onehttps://www.zbmath.org/1483.220072022-05-16T20:40:13.078697Z"Abert, Miklos"https://www.zbmath.org/authors/?q=ai:abert.miklos"Bergeron, Nicolas"https://www.zbmath.org/authors/?q=ai:bergeron.nicolas"Biringer, Ian"https://www.zbmath.org/authors/?q=ai:biringer.ian"Gelander, Tsachik"https://www.zbmath.org/authors/?q=ai:gelander.tsachik"Nikolov, Nikolay"https://www.zbmath.org/authors/?q=ai:nikolov.nikolay"Raimbault, Jean"https://www.zbmath.org/authors/?q=ai:raimbault.jean"Samet, Iddo"https://www.zbmath.org/authors/?q=ai:samet.iddoThis paper is a sequel to [Ann. Math. (2) 185, No. 3, 711--790 (2017; Zbl 1379.22006)] by the same authors. The former paper had extended the validity of the Lück approximation theorem to the setting of Benjamini-Schramm convergence for uniformly discrete sequences of lattices \(\Gamma_n\) in a higher-rank symmetric space of noncompact type \(X=G/K\). That is, it proved \(\lim_{n\to\infty}\frac{b_k(\Gamma_n\backslash X)}{\mathrm{Vol}(\Gamma_n\backslash X)}=\beta_k^{(2)}(X)\) under the assumption that \(\Gamma_n\backslash X\) BS-converges to \(X\).
Their approach towards this theorem had been to introduce the notion of invariant random subgroups (IRSs), that is, conjugation-invariant Borel probability measures on \(\mathrm{Sub}(G)\). (\(\mathrm{Sub}(G)\) is the space of closed subgroups with the Chabauty topology.) For a lattice \(\Gamma\subset G\) one has a map \(\Gamma\backslash G\to \mathrm{Sub}(G)\) sending \(\Gamma g\) to \(g^{-1} \Gamma g\), and one can use the finite measure on \(\Gamma\backslash G\) to define an IRS on \(\mathrm{Sub}(G)\). BS-convergence of \(\Gamma_n\backslash X\) to \(X\) is equivalent to convergence of \(\mu_{\Gamma_n}\) to \(\mu_{\mathrm{id}}\) for the weak-*-topology on \(\mathrm{IRS}(G)\). (The latter is compact, so sequences converge up to extraction.)
By ergodic decomposition it suffices to study ergodic IRSs. If \(\mathrm{rank}_{\mathbb R}(G)\ge 2\), then the Nevo-Stuck-Zimmer theorem implies that the only ergodic IRSs are \(\mu_G,\mu_{\mathrm{id}}\) and \(\mu_\Gamma\) for some lattice \(\Gamma\). Moreover, for every sequence of pairwise non-conjugate lattices, \(\mu_{\Gamma_n}\) converges to \(\mu_{\mathrm{id}}\). This was a main ingredient in the proof by the authors of the improved Lück approximation theorem.
If \(\mathrm{rank}_{\mathbb R}(G)=1\), there are much more possibilities for IRSs. First, in this case for a lattice \(\Gamma\subset G\) the Margulis normal subgroup theorem does not apply. There are many normal subgroups of infinite index which yield an IRS. Next, lattices \(\Gamma\) may have epimorphisms to the free group \(F_2\), and by the work of \textit{L. Bowen} [Groups Geom. Dyn. 9, No. 3, 891--916 (2015; Zbl 1358.37011)] there are many exotic IRSs on free groups. Using the epimorphism one obtains then exotic IRSs supported on \(\Gamma\).
The paper under review is devoted to the construction of other uncountable families of IRSs in \(\mathrm{SO}(n,1)=\mathrm{Isom}^+({\mathbb H}^n)\). It follows from the Borel density theorem, that an ergodic IRS \(\mu\not=\mu_G\) is almost-surely discrete. Thus it can be seen as a probability measure on the set of discrete subgroups or equivalently on the set of (framed) hyperbolic manifolds. The authors describe several constructions of random hyperbolic manifolds, which frequently can not be induced by lattices.
One such construction takes two hyperbolic \(n\)-manifolds \(N_0\) and \(N_1\), whose totally geodesic boundaries consist both of the same two copies of some hyperbolic \((n-1)\)-manifold. To each \(\alpha\in\left\{0,1\right\}^{\mathbb Z}\) one obtains a hyperbolic \(n\)-manifold \(N_\alpha\) by glueing copies of \(N_0\) and \(N_1\) according to the pattern prescribed by \(\alpha\). Each shift-invariant measure on \(\left\{0,1\right\}^{\mathbb Z}\) yields a random hyperbolic \(n\)-manifold. This IRS is not induced by a lattice if \(N_0\) and \(N_1\) are not embedded in non-commensurable compact arithmetic \(n\)-manifolds and \(\alpha\) is not supported on a shift-periodic orbit.
Another construction takes a topological surface \(S\) glued from infinitely many pairs of pants along the pattern of an infinite \(3\)-regular tree. Hyperbolic metrics on \(S\) are described by Fenchel-Nielsen coordinates. Choosing Fenchel-Nielsen coordinates randomly from \(\left(0,\infty\right)\times S^1\) one obtains a random hyperbolic surface. For appropriately measures on \(\left(0,\infty\right)\) and the Lebesgue measure on \(S^1\) one obtains IRSs not induced by a lattice.
A further construction takes a subgroup of the mapping class group \(\mathrm{Mod}(\Sigma)\) freely generated by pseudo-Anosov \(\phi_1,\ldots,\phi_n\), such that orbits on Teichmüller space are quasi-convex. For a sequence of words with \(\vert w_i\vert\to\infty\) let \(\Gamma_i\backslash{\mathbb H}^3\) be hyperbolic \(3\)-manifold fibering over \(S^1\) with monodromy \(w_i\). The sequence \(\mu_{\Gamma_i}\) converges (up to extraction) to an IRS \(\mu\). If the words \(w_i\) are chosen appropriately, then \(\mu\) is not induced by a lattice.
All these constructions yield weak-*-limits of sequences \(\mu_{\Gamma_n}\) for lattices \(\Gamma_n\) and the authors ask whether this must be the case for every ergodic IRS \(\mu\not=\mu_G\).
Reviewer: Thilo Kuessner (Augsburg)The fundamental theorem of algebra and Liouville's theorem geometrically revisitedhttps://www.zbmath.org/1483.300272022-05-16T20:40:13.078697Z"Almira, Jose Maria"https://www.zbmath.org/authors/?q=ai:almira.jose-maria"Romero, Alfonso"https://www.zbmath.org/authors/?q=ai:romero.alfonsoSummary: If \(f(z)\) is either a polynomial with no zeroes or a bounded entire function, then a Riemannian metric \(g_f\) is constructed on the complex plane \(\mathbb{C}\). This metric \(g_f\) is shown to be flat and geodesically complete. Therefore, the Riemannian manifold \((\mathbb{C}, g_f)\) must be isometric to \((\mathbb{C}, |dz|^2)\), which implies that \(f(z)\) is a constant.Uniformization of branched surfaces and Higgs bundleshttps://www.zbmath.org/1483.300782022-05-16T20:40:13.078697Z"Biswas, Indranil"https://www.zbmath.org/authors/?q=ai:biswas.indranil"Bradlow, Steven"https://www.zbmath.org/authors/?q=ai:bradlow.steven-b"Dumitrescu, Sorin"https://www.zbmath.org/authors/?q=ai:dumitrescu.sorin"Heller, Sebastian"https://www.zbmath.org/authors/?q=ai:heller.sebastian-gregorThe induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometryhttps://www.zbmath.org/1483.300842022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Danciger, Jeffrey"https://www.zbmath.org/authors/?q=ai:danciger.jeffrey"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Schlenker, Jean-Marc"https://www.zbmath.org/authors/?q=ai:schlenker.jean-marcA theorem by Alexandrov and Pogorelov says that any smooth Riemannian metric on the 2-sphere with curvature \(K>-1\) coincides with the induced metric on the boundary of some compact convex subset of hyperbolic 3-space with smooth boundary and, furthermore, that this compact convex subset is unique up to a global isometry of hyperbolic 3-space. In the paper under review, the authors study a generalization of this result to unbounded convex subsets of hyperbolic 3-space, more especially to convex subsets bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, they supplement the notion of induced metric on the boundary of the convex set so that it includes a gluing map at infinity which records how the asymptotic geometries of the two surfaces fit together near the limiting Jordan curve. They restrict their study to the case where the induced metrics on the two bounding surfaces have constant curvature \(K \in [-1, 0)\) and were the Jordan curve at infinity is a quasicircle. In this case the gluing map becomes a quasisymmetric homeomorphism of the circle and the authors prove that for \(K\) in the given interval, any quasisymmetric map can be obtained as the gluing map at infinity along some quasicircle. They also obtain Lorentzian analogous of these results, in which hyperbolic 3-space is replaced by the 3-dimensional anti-de Sitter space \(\mathbb{A}d\mathbb{S}^3\), whose natural boundary is the Einstein space \(\mathrm{Ein}^{1,1}\), a conformal Lorentzian analogue of the Riemannian sphere. The authors say that their results may be viewed as universal versions of a conjecture of Thurston about the realization of metrics on boundaries of convex cores of quasifuchsian hyperbolic manifolds and of an analogue of this conjecture, due to Mess, in the setting of globally hyperbolic anti-de Sitter spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)From hierarchical to relative hyperbolicityhttps://www.zbmath.org/1483.300852022-05-16T20:40:13.078697Z"Russell, Jacob"https://www.zbmath.org/authors/?q=ai:russell.jacobSummary: We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmüller space for surfaces with complexity three.On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in 3-dimensional hyperbolic spacehttps://www.zbmath.org/1483.300872022-05-16T20:40:13.078697Z"Smith, Graham"https://www.zbmath.org/authors/?q=ai:smith.graham-a|smith.graham-mSummary: For all \(k\in ]0,1[\), we construct a canonical bijection between the space of ramified coverings of the sphere of hyperbolic type and the space of complete immersed surfaces in 3-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to \(k\). We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.
For the entire collection see [Zbl 1473.53006].Sharp bounds for the anisotropic \(p\)-capacity of Euclidean compact setshttps://www.zbmath.org/1483.310242022-05-16T20:40:13.078697Z"Li, Ruixuan"https://www.zbmath.org/authors/?q=ai:li.ruixuan"Xiong, Changwei"https://www.zbmath.org/authors/?q=ai:xiong.changwei.1|xiong.changweiSummary: We prove various sharp bounds for the anisotropic \(p\)-capacity \(\mathrm{Cap}_{F , p}(K)\) (\(1 < p < n\)) of compact sets \(K\) in the Euclidean space \(\mathbb{R}^n\) (\(n \geq 2\)). Our results are mainly the anisotropic generalizations of some isotropic ones in [\textit{M. Ludwig} et al., Math. Ann. 350, No. 1, 169--197 (2011; Zbl 1220.26020); \textit{J. Xiao}, Ann. Henri Poincaré 17, No. 8, 2265--2283 (2016; Zbl 1345.83014); Adv. Math. 308, 1318--1336 (2017; Zbl 1361.31008); Adv. Geom. 17, No. 4, 483--496 (2017; Zbl 1387.53024)]. Key ingredients in the proofs include the inverse anisotropic mean curvature flow (IAMCF), the anisotropic Hawking mass and its monotonicity property along IAMCF for certain surfaces, and the anisotropic isocapacitary inequality.A weak notion of visibility, a family of examples, and Wolff-Denjoy theoremshttps://www.zbmath.org/1483.320062022-05-16T20:40:13.078697Z"Bharali, Gautam"https://www.zbmath.org/authors/?q=ai:bharali.gautam"Maitra, Anwoy"https://www.zbmath.org/authors/?q=ai:maitra.anwoyThe authors study the form of visibility introduced recently by the first author and \textit{A. Zimmer} [Adv. Math. 310, 377--425 (2017; Zbl 1366.32005)]. They show that some of the theorems alluded there follow merely from the latter notion of visibility (domains that possess this property are called visibility domains with respect to the Kobayashi distance).
In the paper the authors provide a sufficient condition for a domain in \(\mathbb C^n\) to be a visibility domain. They also construct a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains.
Finally, they also establish two new Wolff-Denjoy-type theorems.
Reviewer: Paweł Zapałowski (Kraków)Finite time blow-up for the heat flow of \(H\)-surface with constant mean curvaturehttps://www.zbmath.org/1483.350482022-05-16T20:40:13.078697Z"Li, Haixia"https://www.zbmath.org/authors/?q=ai:li.haixiaSummary: We consider an initial boundary value problem for the heat flow of the equation of surfaces with constant mean curvature which was investigated in [\textit{T. Huang} et al., Manuscr. Math. 134, No. 1--2, 259--271 (2011; Zbl 1210.53012)], where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.Chiti-type reverse Hölder inequality and torsional rigidity under integral Ricci curvature conditionhttps://www.zbmath.org/1483.350512022-05-16T20:40:13.078697Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of \textit{N. Gamara} et al. [Open Math. 13, 557--570 (2015; Zbl 06632233)] and \textit{D. Colladay} et al. [J. Geom. Anal. 28, No. 4, 3906--3927 (2018; Zbl 1410.58016)] from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to \(p\)-Laplacian.Li-Yau multipLier set and optimal Li-Yau gradient estimate on hyperboLic spaceshttps://www.zbmath.org/1483.350592022-05-16T20:40:13.078697Z"Yu, Chengjie"https://www.zbmath.org/authors/?q=ai:yu.chengjie"Zhao, Feifei"https://www.zbmath.org/authors/?q=ai:zhao.feifeiThis paper is motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound. To reach this aim, the authors first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold, then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Lastly, as an application of the previous results, a sharp and interesting Harnack inequalities on hyperbolic spaces is shown.
Reviewer: Vincenzo Vespri (Firenze)Parabolic quaternionic Monge-Ampère equation on compact manifolds with a flat hyperKähler metrichttps://www.zbmath.org/1483.351332022-05-16T20:40:13.078697Z"Zhang, Jiaogen"https://www.zbmath.org/authors/?q=ai:zhang.jiaogenSummary: The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the Kähler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperKähler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Ampère equation. Our goal is to prove the long time existence and \(C^{\infty}\) convergence for normalized solutions as \(t\rightarrow\infty \). As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Ampère equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.Inverse source problems in transport equations with external forceshttps://www.zbmath.org/1483.351742022-05-16T20:40:13.078697Z"Lai, Ru-Yu"https://www.zbmath.org/authors/?q=ai:lai.ru-yu"Zhou, Hanming"https://www.zbmath.org/authors/?q=ai:zhou.hanmingSummary: This paper is concerned with the inverse source problem for the transport equation with external force. We show that both direct and inverse problems are uniquely solvable for generic absorption and scattering coefficients. In particular, for inverse problems, generic injectivity and a stability estimate of the source are derived. The analysis employs the Fredholm theorem and the Santalo's formula.The translating soliton equationhttps://www.zbmath.org/1483.351782022-05-16T20:40:13.078697Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez.rafael-beltran|lopez-camino.rafaelSummary: We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.
For the entire collection see [Zbl 1473.53006].A density theorem for asymptotically hyperbolic initial datahttps://www.zbmath.org/1483.352612022-05-16T20:40:13.078697Z"Dahl, Mattias"https://www.zbmath.org/authors/?q=ai:dahl.mattias"Sakovich, Anna"https://www.zbmath.org/authors/?q=ai:sakovich.annaSummary: When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, we show that an asymptotically hyperbolic initial data set with nonnegative local energy density can be approximated by an initial data set with strictly positive local energy density and a simple structure at infinity, while changing the mass arbitrarily little. This is achieved by suitably modifying the argument used by \textit{M. Eichmair} et al. [J. Eur. Math. Soc. (JEMS) 18, No. 1, 83--121 (2016; Zbl 1341.53067)] in the asymptotically Euclidean case.Tracking the critical points of curves evolving under planar curvature flowshttps://www.zbmath.org/1483.352802022-05-16T20:40:13.078697Z"Fehér, Eszter"https://www.zbmath.org/authors/?q=ai:feher.eszter"Domokos, Gábor"https://www.zbmath.org/authors/?q=ai:domokos.gabor"Krauskopf, Bernd"https://www.zbmath.org/authors/?q=ai:krauskopf.berndSummary: We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function \(r(\varphi)\) measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function \(r(\varphi)\) and of the curvature \(\kappa (\varphi)\) (characterized by \(dr/d\varphi = 0\) and \(d\kappa /d\varphi = 0\), respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.
We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically -- in the spirit of experimental mathematics -- we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.Koszul information geometry and Souriau Lie group thermodynamicshttps://www.zbmath.org/1483.370422022-05-16T20:40:13.078697Z"Barbaresco, Frédéric"https://www.zbmath.org/authors/?q=ai:barbaresco.fredericSummary: The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from ``Characteristic Functions'', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of ``Information Geometry'' theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean ``Moment map'' by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author.
For the entire collection see [Zbl 1470.00021].On the relation between action and linkinghttps://www.zbmath.org/1483.370522022-05-16T20:40:13.078697Z"Senior, David Bechara"https://www.zbmath.org/authors/?q=ai:senior.david-bechara"Hryniewicz, Umberto L."https://www.zbmath.org/authors/?q=ai:hryniewicz.umberto-l"Salomão, Pedro A. S."https://www.zbmath.org/authors/?q=ai:salomao.pedro-a-sThe authors focus on numerical invariants of contact forms in dimension three.
Some questions related to the application of these invariants in geometry of contact forms (periodic orbits, Reeb flows, contact volume, systolic norm, Reinhardt domains, Seifert surfaces, etc.) are formulated. The main result is so-called Action-Linking Lemma that contains an important identity for every adapted Seifert surface in a contact manifold.
Reviewer: Mihail Banaru (Smolensk)A note on the commutator of Hamiltonian vector fieldshttps://www.zbmath.org/1483.370672022-05-16T20:40:13.078697Z"Żołądek, Henryk"https://www.zbmath.org/authors/?q=ai:zoladek.henrykSummary: We present two proofs of the Jacobi identity for the Poisson bracket on a symplectic manifold.
For the entire collection see [Zbl 1456.14003].Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torushttps://www.zbmath.org/1483.370712022-05-16T20:40:13.078697Z"Agapov, S. V."https://www.zbmath.org/authors/?q=ai:agapov.sergei-vadimovich"Valyuzhenich, A. A."https://www.zbmath.org/authors/?q=ai:valyuzhenich.aleksandr-andreevich"Shubin, V. V."https://www.zbmath.org/authors/?q=ai:shubin.v-vThis paper aims at fixing a gap in the proof of the main result from [the first two authors, Discrete Contin. Dyn. Syst. 39, No. 11, 6565--6583 (2019; Zbl 1432.37082)].
Consider conformal coordinates \((x,y)\) on the \(2\)-torus with metric \(ds^2=\Lambda(x,y)(dx^2+dy^2)\), and let \[H=\frac{p_x^2+p_y^2}{2 \Lambda(x,y)}.\] The magnetic gedesic flow is the Hamiltonian flow of \(H\) with respect to the magnetic Poisson bracket \[\{-,-\}_{\text{mg}} = \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial p_x}+ \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial p_y} + \Omega(x,y) \frac{\partial}{\partial p_x} \wedge \frac{\partial}{\partial p_y}\,,\] where \(\Omega(x,y)\, dx\wedge dy\) is a closed \(2\)-form defining a fixed nonzero magnetic field.
The main result of [loc. cit.] states that if the magnetic geodesic flow has a first integral \(F\) which is polynomial of degree \(N\) in the momenta \((p_x,p_y)\) and which has periodic analytic coefficients at \(N+2\) energy levels \(\{H=E_j\}\), \(E_1,\ldots, E_{N+2}\in \mathbb{R}\) pairwise distinct, then \(\Omega\) and \(\Lambda\) are functions of a unique variable. Furthermore \(H\) admits a first integral \(F_1\) which is linear in momenta. In Section 2 the authors outline the main steps of the proof given in [loc. cit.] and identify an argument which can fail. It is explained in Section 3 how to fix this issue without altering the statement of the important result that was presented.
Reviewer: Maxime Fairon (Glasgow)Superintegrable systems on moduli spaces of flat connectionshttps://www.zbmath.org/1483.370722022-05-16T20:40:13.078697Z"Arthamonov, S."https://www.zbmath.org/authors/?q=ai:arthamonov.semeon"Reshetikhin, N."https://www.zbmath.org/authors/?q=ai:reshetikhin.nikolai-yuLet \(G\) be a connected simple linear algebraic group over \(\mathbb{C}\). The aim of this paper is to construct superintegrable Hamiltonian systems on moduli spaces of flat \(G\)-connections over any oriented surface with nonempty boundary. Hamiltonians of such systems are traces of holonomies along non-intersecting non-self-intersecting curves. The construction naturally works in the same way for various real forms of \(G\), for example for compact simple Lie groups or for split real forms. The moduli space is a Poisson variety with Atiyah-Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars-Schneider models.
This paper is organized as follows. In Section 1 the authors recall the definition of superintegrable (degenerately integrable) systems. There they also define the notion of affine superintegrable systems in the algebro-geometrical setting. Section 2 is an overview of basic notions about moduli spaces of flat connections on a surface. In this section they recall the definition of graph functions and the description of Poisson brackets between two such functions. In Section 3 they describe the main result, the construction of a family of Hamiltonian systems defined by a choice of non-intersecting, non-self-intersecting cycles and prove their superintegrability. At the end of this section they introduce the notion of a partial order on such systems. In Section 4 they explain how solutions to equations of motion of these superintegrable systems can be solved using the projection method. Section 5 has some genus one examples. In the conclusion the authors define a system on the space of chord diagrams which is conjectured to be superintegrable. They discuss the case of non-generic conjugation orbits and some further directions.
Reviewer: Ahmed Lesfari (El Jadida)Bi-Hamiltonian structure of spin Sutherland models: the holomorphic casehttps://www.zbmath.org/1483.370742022-05-16T20:40:13.078697Z"Fehér, L."https://www.zbmath.org/authors/?q=ai:feher.laszlo-gyIn the previous work [Lett. Math. Phys. 110, No. 5, 1057--1079 (2020; Zbl 1445.37042)] the author developed a bi-Hamiltonian interpretation for a family of Sutherland spin models having hyperbolic and trigonometric form. The current paper expands this investigation to what the author calls the holomorphic spin Sutherland hierarchy. This consists of the holomorphic evolution equations of the form \(\dot{Q} = (L^k)_0 Q\), \(\dot{L} = [\mathcal{R}(Q)(L^k), L]\) for all \(k \in \mathbb{N}\). Here \(Q\) is an invertible \(n \times n\) complex diagonal matrix, \(L\) is an arbitrary \(n \times n\) complex matrix, and the subscript \(0\) means diagonal part. It is assumed that the eigenvalues of \(Q\) are distinct, so that the operator \(\mathcal{R}(Q) = 1/2 (\mathrm{Ad}_Q + \mathrm{id})( \mathrm{Ad}_Q - \mathrm{id})^{-1}\) (where \(\mathrm{Ad}_Q(X) = QXQ^{-1}\)) is a well-defined linear operator on the off-diagonal subspace of gl(\(n, \mathbb{C}).\)
The holomorphic spin Sutherland hierarchy is known to be a reduction of the natural integrable system on the cotangent bundle \(\mathcal{M} = T^* \mathrm{GL}(n, \mathbb{C})\). The author's main result is that the unreduced integrable system on \(\mathcal{M}\) leads to a bi-Hamiltonian structure for the spin Sutherland hierarchy after Poisson reduction. One of the two compatible Poisson structures is associated with the canonical cotangent bundle symplectic form on \(\mathcal{M}\), and the second one is constructed from the Semenov-Tian-Shansky Poisson bracket of the Heisenberg double of \(T^* \mathrm{GL}(n, \mathbb{C})\) with its standard Poisson-Lie group structure.
The author also shows that the bi-Hamiltonian structures of the hyperbolic and trigonometric real forms described in his previous work can be recovered from the holomorphic form.
Reviewer: William J. Satzer Jr. (St. Paul)Poincaré inequalities and uniform rectifiabilityhttps://www.zbmath.org/1483.460282022-05-16T20:40:13.078697Z"Azzam, Jonas"https://www.zbmath.org/authors/?q=ai:azzam.jonasSummary: We show that any \(d\)-Ahlfors regular subset of \(\mathbb{R}^n\) supporting a weak \((1, d)\)-Poincaré inequality with respect to surface measure is uniformly rectifiable.Sharp Sobolev inequalities involving boundary terms revisitedhttps://www.zbmath.org/1483.460332022-05-16T20:40:13.078697Z"Tang, Zhongwei"https://www.zbmath.org/authors/?q=ai:tang.zhongwei"Xiong, Jingang"https://www.zbmath.org/authors/?q=ai:xiong.jingang"Zhou, Ning"https://www.zbmath.org/authors/?q=ai:zhou.ningIt is known that there is a constant \(S\) such that
\[ S^{-1} = \inf \biggl\{ \frac{\| \nabla u \|^2_{L^2(\mathbb R^n)} } { \| u \|^2_{L^{2^*}(\mathbb R^n)} } : u \in L^{2^*}(\mathbb R^n) \setminus \{ 0 \} ,\ |\nabla u | \in L^2(\mathbb R^n) \biggr\} \] and the extremals are known. The authors study similar Sobolev inequalities on compact Riemannian manifolds.
Let \((M, g)\) be a smooth compact \(n\)-dimensional Riemannian manifold with smooth boundary \(\partial M\) and \(n \geq 3, 2^*= 2n/(n - 2)\). If the manifold \((M, g)\) supports the inequality
\[
\Bigl(\int_M |u|^{2*} dv_g \Bigr)^{{2}/{2^*} } \leq 2^{\frac2n}S \int_M |\nabla_g u |^2 dv_g
\]
for any \(u \in H^1_0 (M)\setminus \{ 0 \}\), Li-Zhu proved that there exists a constant \(A(M, g) > 0\) such that
\[
\Bigl(\int_M |u|^{2*} dv_g \Bigr)^{{2}/{2^*} } \leq 2^{\frac2n}S \int_M |\nabla_g u |^2 dv_g + A(M,g) \int_{\partial M} |u|^2 d s_g.
\]
where \(dv_g\) is the induced volume form on \(M\) and \(ds_g\) is the induced volume form on \(\partial M\). They also proved that the first inequality is necessary for the second. If the inequality is not supported, \textit{Y. Y. Li} and \textit{M. Zhu} [Geom. Funct. Anal. 8, No.~1, 59--87 (1998; Zbl 0901.58066)] proved that
\[
\Bigl(\int_M |u|^{2*} dv_g \Bigr)^{{2}/{2^*} }) \leq 2^{\frac2n}S \int_M |\nabla_g u |^2 dv_g + A^{\prime}(M,g) \Bigl( \int_{M} |u|^2 d v_g + \int_{\partial M} |u|^2 d s_g \Bigr).
\]
The authors extend this by giving a version involving the mean curvature of the manifold. Suppose \((M, g)\) is a smooth compact \(n\)-dimensional Riemannian manfiold with \(n \geq 7\) which supports the inequality
\[
\Bigl(\int_M |u|^{2*} dv_g \Bigr)^{{2}/{2^*} } \leq 2^{\frac2n}S \int_M |\nabla_g u |^2 dv_g ;
\]
then there is a constant \(A_1(M,g) >0\) such that
\[
\Bigl(\int_M |u|^{2*} dv_g \Bigr)^{{2}/{2^*} } \leq 2^{2/n}S \Bigl(\int_M |\nabla_g u |^2 dv_g + \frac{n - 2}{2} \int_{\partial M} h_g |u|^2 d s_g \Bigr) + A_1 \| u \|^2_{L^r(\partial M)}, \] for \(u \in H^1(M)\), where \(h_g\) is the mean curvature of \(\partial M\) and \(r = \frac{2(n - 1)}{n}\).
If the inequality is not supported, the inequality still holds but with two final terms, the squares of the \(L^r\) norm over \(\partial M\) and the square of the \(L^{r_1}\) norm over \(M\), where \( r_1 = \frac{n}{n + 2}\). I refer to the paper for the precise form.
The constants are sharp as before and in addition \(h_g\) cannot be replaced by any function that is strictly less than \(h_g\) at any point of \(\partial M\), nor can the exponents in the \(L^r\) norms be replaced by any smaller number.
Reviewer: Raymond Johnson (Columbia)Topological algebras -- geometry -- physics; some interactionshttps://www.zbmath.org/1483.460532022-05-16T20:40:13.078697Z"Fragoulopoulou, Maria"https://www.zbmath.org/authors/?q=ai:fragoulopoulou.mariaSummary: This is a survey account on some interactions among topological algebras, geometry and physics.
For the entire collection see [Zbl 1466.46001].On weakly reflective PF submanifolds in Hilbert spaceshttps://www.zbmath.org/1483.460762022-05-16T20:40:13.078697Z"Morimoto, Masahiro"https://www.zbmath.org/authors/?q=ai:morimoto.masahiroSummary: A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.Contributions of abstract differential geometry (ADG) in physicshttps://www.zbmath.org/1483.530012022-05-16T20:40:13.078697Z"Zafiris, Elias"https://www.zbmath.org/authors/?q=ai:zafiris.eliasIn this short note, dedicated to the memory of Professor Anastasios Mallios, the author provides a concise list with the most conceptual and technical innovations that Abstract Differential Geometry (ADG) has contributed to some fundamental physical theories, like general relativity and quantum mechanics.
For the entire collection see [Zbl 1466.46001].
Reviewer: Ioan Bucataru (Iaşi)Anti-de Sitter geometry and Teichmüller theoryhttps://www.zbmath.org/1483.530022022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Seppi, Andrea"https://www.zbmath.org/authors/?q=ai:seppi.andreaSummary: The aim of this chapter is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.
For the entire collection see [Zbl 1470.57002].Monochromatic random waves for general Riemannian manifoldshttps://www.zbmath.org/1483.530032022-05-16T20:40:13.078697Z"Canzani, Yaiza"https://www.zbmath.org/authors/?q=ai:canzani.yaizaSummary: This is a survey article on some of the recent developments on monochromatic random waves defined for general Riemannian manifolds. We discuss the conditions needed for the waves to have a universal scaling limit, we review statistics for the size of their zero set and the number of their critical points, and we discuss the structure of their zero set as described by the diffeomorphism types and the nesting configurations of its components.
For the entire collection see [Zbl 1473.53004].Pedal coordinates and free double linkagehttps://www.zbmath.org/1483.530042022-05-16T20:40:13.078697Z"Blaschke, Petr"https://www.zbmath.org/authors/?q=ai:blaschke.petr"Blaschke, Filip"https://www.zbmath.org/authors/?q=ai:blaschke.filip"Blaschke, Martin"https://www.zbmath.org/authors/?q=ai:blaschke.martinAuthors' abstract: A free double linkage is a mechanical system with three point-masses, two of which are linked to the central node by massless rigid rods. Using the technique of pedal coordinates the authors investigate the orbits of a free double linkage. They provide a geometrical construction for them and also show a surprising connection between this mechanical system and orbits around a Black Hole and solutions of the Dark Kepler problem.
Reviewer: Ergin Bayram (Samsun)Generalized normal ruled surface of a curve in the Euclidean 3-spacehttps://www.zbmath.org/1483.530052022-05-16T20:40:13.078697Z"Kaya, Onur"https://www.zbmath.org/authors/?q=ai:kaya.onur"Önder, Mehmet"https://www.zbmath.org/authors/?q=ai:onder.mehmetSummary: In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space \(E^3\). We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.Stochastic test of a minimal surfacehttps://www.zbmath.org/1483.530062022-05-16T20:40:13.078697Z"Klimentov, D. S."https://www.zbmath.org/authors/?q=ai:klimentov.dmitrii-sergeevichSummary: The paper aims to obtain a stochastic test for minimal surfaces. Such a test is formulated in terms of transition densities of stochastic processes. Two fundamental forms of the surface generate these processes. This work exhausts the problem of stochastic test for regular minimal surfaces.
For the entire collection see [Zbl 1470.47003].An improved bound on the optimal paper Moebius bandhttps://www.zbmath.org/1483.530072022-05-16T20:40:13.078697Z"Schwartz, Richard Evan"https://www.zbmath.org/authors/?q=ai:schwartz.richard-evanAuthor's abstract: We show that a smooth embedded paper Möbius band must have aspect ratio at least
\[
\lambda_1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\sqrt[4]{3}\sqrt{2}-\sqrt[2]{2}\sqrt{3}-3} =1.69497\ldots\tag{\(\ast\)}
\]
This bound comes more than \(3/4\) of the way from the old known bound of \(\pi/2=1.5708\ldots\) to the conjectured bound of \(\sqrt{3}=1.732\ldots\)
Reviewer: Ivan C. Sterling (St. Mary's City)On \(f\)-rectifying curves in the Euclidean 4-spacehttps://www.zbmath.org/1483.530082022-05-16T20:40:13.078697Z"Iqbal, Zafar"https://www.zbmath.org/authors/?q=ai:iqbal.zafar"Sengupta, Joydeep"https://www.zbmath.org/authors/?q=ai:sengupta.joydeepSummary: A rectifying curve in the Euclidean 4-space \(\mathbb{E}^4\) is defined as an arc length parametrized curve \(\gamma\) in \(\mathbb{E}^4\) such that its position vector always lies in its rectifying space (i.e., the orthogonal complement \(N_{\gamma}^{\bot}\) of its principal normal vector field \(N_\gamma)\) in \(\mathbb{E}^4\). In this paper, we introduce the notion of an \(f\)-rectifying curve in \(\mathbb{E}^4\) as a curve \(\gamma\) in \(\mathbb{E}^4\) parametrized by its arc length \(s\) such that its \(f\)-position vector \(\gamma_f\), defined by \(\gamma_f\) (s) = \( \int f(s)d\gamma\) for all \(s\), always lies in its rectifying space in \(\mathbb{E}^4\), where \(f\) is a nowhere vanishing integrable function in parameter \(s\) of the curve \(\gamma \). Also, we characterize and classify such curves in \(\mathbb{E}^4\).Minimal surfaces in hyperbolic 3-manifoldshttps://www.zbmath.org/1483.530092022-05-16T20:40:13.078697Z"Coskunuzer, Baris"https://www.zbmath.org/authors/?q=ai:coskunuzer.barisThe main result of this interesting paper is contained in Theorem 1.2 : Let \(M\) be a complete hyperbolic \(3\)-manifold with finitely generated fundamental group. If \(M\) is not an exceptional manifold, then it contains a smoothly embedded closed minimal surface. The exceptional manifolds are devided into
Type I: if \(M\) is a geometrically infinite product manifold, and
Type II: if \(M\) is a handlebody (Schottky manifold).
Other papers of the author directly connected to this topic are [Trans. Am. Math. Soc. 364, No. 3, 1211--1224 (2012; Zbl 1242.53069); Trans. Am. Math. Soc. 371, No. 2, 1253--1269 (2019; Zbl 1475.53067)].
Reviewer: Dorin Andrica (Riyadh)Heinz-type mean curvature estimates in Lorentz-Minkowski spacehttps://www.zbmath.org/1483.530102022-05-16T20:40:13.078697Z"Honda, Atsufumi"https://www.zbmath.org/authors/?q=ai:honda.atsufumi"Kawakami, Yu"https://www.zbmath.org/authors/?q=ai:kawakami.yu"Koiso, Miyuki"https://www.zbmath.org/authors/?q=ai:koiso.miyuki"Tori, Syunsuke"https://www.zbmath.org/authors/?q=ai:tori.syunsukeSummary: We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.\(d\)-minimal surfaces in three-dimensional singular semi-Euclidean space \(\mathbb{R}^{0,2,1}\)https://www.zbmath.org/1483.530112022-05-16T20:40:13.078697Z"Sato, Yuichiro"https://www.zbmath.org/authors/?q=ai:sato.yuichiroSummary: In this paper, we study surfaces in singular semi-Euclidean space \(\mathbb{R}^{0,2,1}\) endowed with a degenerate metric. We define d-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that \(d\)-minimal surfaces in \(\mathbb{R}^{0,2,1}\) and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space \(\mathbb{R}_1^4\) are in one-to-one correspondence.First explicit constrained Willmore minimizers of non-rectangular conformal classhttps://www.zbmath.org/1483.530122022-05-16T20:40:13.078697Z"Heller, Lynn"https://www.zbmath.org/authors/?q=ai:heller.lynn"Ndiaye, Cheikh Birahim"https://www.zbmath.org/authors/?q=ai:ndiaye.cheikh-birahimIn this interesting paper the authors study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes \((0, b)\) with \(b \sim 1\) the homogeneous tori \(f^ b\) are known to be the unique constrained Willmore minimizers (up to invariance). The authors generalize this result and show that the candidates constructed in [\textit{L. Heller} and \textit{F. Pedit}, in: Willmore energy and Willmore conjecture. Boca Raton, FL: CRC Press. 119--138 (2018; Zbl 1384.53007)] are indeed constrained Willmore minimizers in certain non-rectangular conformal classes \((a, b)\). It is shown that the minimal Willmore energy \(\omega(a, b)\) is real analytic and concave in \(a \in (0, a^b)\) for some \(a^b > 0\) and fixed \(b \sim 1\), \(b \ne 1\).
Reviewer: Andreas Arvanitoyeorgos (Patras)Curves in the Lorentz-Minkowski plane with curvature depending on their positionhttps://www.zbmath.org/1483.530132022-05-16T20:40:13.078697Z"Castro, Ildefonso"https://www.zbmath.org/authors/?q=ai:castro.ildefonso"Castro-Infantes, Ildefonso"https://www.zbmath.org/authors/?q=ai:castro-infantes.ildefonso"Castro-Infantes, Jesús"https://www.zbmath.org/authors/?q=ai:castro-infantes.jesusSummary: Motivated by the classical Euler elastic curves, \textit{D. A. Singer} posed in [Am. Math. Mon. 106, No. 9, 835--841 (1999; Zbl 1037.53500)] the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in \(\mathbb{L}^2\) whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.On the Darboux vector of ruled surfaces in Galilean spacehttps://www.zbmath.org/1483.530142022-05-16T20:40:13.078697Z"Ekici, Cumali"https://www.zbmath.org/authors/?q=ai:ekici.cumali"Dede, Mustafa"https://www.zbmath.org/authors/?q=ai:dede.mustafa"Özüsağlam, Erdal"https://www.zbmath.org/authors/?q=ai:ozusaglam.erdalSummary: In this paper, we investigate the Darboux vector of ruled surfaces in Galilean space. There are three types of ruled surfaces in Galilean space. We obtained the relationship between the Darboux and Frenet vectors of each type of ruled surfaces in Galilean space. In addition, an example is constructed and plotted.Evolutoids of the mixed-type curveshttps://www.zbmath.org/1483.530152022-05-16T20:40:13.078697Z"Zhao, Xin"https://www.zbmath.org/authors/?q=ai:zhao.xin"Pei, Donghe"https://www.zbmath.org/authors/?q=ai:pei.dongheSummary: The evolutoid of a regular curve in the Lorentz-Minkowski plane \(\mathbb{R}_1^2\) is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in \(\mathbb{R}_1^2\). As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in \(\mathbb{R}_1^2\) and give the conception of the \(\sigma \)-transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.On geodesic triangles with right angles in a dually flat spacehttps://www.zbmath.org/1483.530162022-05-16T20:40:13.078697Z"Nielsen, Frank"https://www.zbmath.org/authors/?q=ai:nielsen.frank.1|nielsen.frank-s|nielsen.frankThis chapter concerns geodesic triangles in information geometry. Specifically, the main ingredients involve the dualistic structure and the geodesic triangles the author considered live on the dually flat spaces, i.e., Bregman manifolds. The main output is the construction of geodesic triangles with 1, 2, and 3 interior right angles, respectively. The geodesic triangles for which the dual Pythagorean theorems hold are also constructed in this chapter.
For the entire collection see [Zbl 1473.53007].
Reviewer: Ruobing Zhang (Stony Brook)An elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensorhttps://www.zbmath.org/1483.530172022-05-16T20:40:13.078697Z"Wang, Tianbo"https://www.zbmath.org/authors/?q=ai:wang.tianbo"Yang, Dinglin"https://www.zbmath.org/authors/?q=ai:yang.dinglin"Li, Chen"https://www.zbmath.org/authors/?q=ai:li.chen.1|li.chen"Shi, Diwei"https://www.zbmath.org/authors/?q=ai:shi.diweiSummary: Based on the Cayley-Hamilton theorem and fixed-point method, we provide an elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor in a three-dimensional (3D) inner-product space, which avoids introducing the generating function and Taylor series expansion. The proof is also extended to any finite-dimensional inner-product space.Metric connections with parallel skew-symmetric torsionhttps://www.zbmath.org/1483.530182022-05-16T20:40:13.078697Z"Cleyton, Richard"https://www.zbmath.org/authors/?q=ai:cleyton.richard"Moroianu, Andrei"https://www.zbmath.org/authors/?q=ai:moroianu.andrei"Semmelmann, Uwe"https://www.zbmath.org/authors/?q=ai:semmelmann.uweThis paper is devoted to the study of geometries with parallel skew-symmetric torsion. Such geometry is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Beside the trivial case of Levi-Civita connection, there are many other examples of such geometries.
In [the first author and \textit{A. Swann}, Math. Z. 247, No. 3, 513--528 (2004; Zbl 1069.53041)] the classification of metric connections with parallel torsion and irreducible holonomy representation is presented. The main result of the present paper is the classification in the case of connections with parallel skew-symmetric torsion whose holonomy representation is reducible.
Reviewer: Miroslaw Doupovec (Brno)The basics of information geometryhttps://www.zbmath.org/1483.530192022-05-16T20:40:13.078697Z"Caticha, Ariel"https://www.zbmath.org/authors/?q=ai:caticha.arielSummary: To what extent can we distinguish one probability distribution from another? Are there quantitative measures of distinguishability? The goal of this tutorial is to approach such questions by introducing the notion of the ``distance'' between two probability distributions and exploring some basic ideas of such an ``information geometry''.
For the entire collection see [Zbl 1470.00021].Entropic dynamics: from entropy and information geometry to Hamiltonians and quantum mechanicshttps://www.zbmath.org/1483.530202022-05-16T20:40:13.078697Z"Caticha, Ariel"https://www.zbmath.org/authors/?q=ai:caticha.ariel"Bartolomeo, Daniel"https://www.zbmath.org/authors/?q=ai:bartolomeo.daniel"Reginatto, Marcel"https://www.zbmath.org/authors/?q=ai:reginatto.marcelSummary: Entropic Dynamics is a framework in which quantum theory is derived as an application of entropic methods of inference. There is no underlying action principle. Instead, the dynamics is driven by entropy subject to the appropriate constraints. In this paper we show how a Hamiltonian dynamics arises as a type of non-dissipative entropic dynamics. We also show that the particular form of the ``quantum potential'' that leads to the Schrödinger equation follows naturally from information geometry.
For the entire collection see [Zbl 1470.00021].A special form of SPD covariance matrix for interpretation and visualization of data manipulated with Riemannian geometryhttps://www.zbmath.org/1483.530212022-05-16T20:40:13.078697Z"Congedo, Marco"https://www.zbmath.org/authors/?q=ai:congedo.marco"Barachant, Alexandre"https://www.zbmath.org/authors/?q=ai:barachant.alexandreSummary: Currently the Riemannian geometry of symmetric positive definite (SPD) matrices is gaining momentum as a powerful tool in a wide range of engineering applications such as image, radar and biomedical data signal processing. If the data is not natively represented in the form of SPD matrices, typically we may summarize them in such form by estimating covariance matrices of the data. However once we manipulate such covariance matrices on the Riemannian manifold we lose the representation in the original data space. For instance, we can evaluate the geometric mean of a set of covariance matrices, but not the geometric mean of the data generating the covariance matrices, the space of interest in which the geometric mean can be interpreted. As a consequence, Riemannian information geometry is often perceived by non-experts as a ``black-box'' tool and this perception prevents a wider adoption in the scientific community. Hereby we show that we can overcome this limitation by constructing a special form of SPD matrix embedding both the covariance structure of the data and the data itself. Incidentally, whenever the original data can be represented in the form of a generic data matrix (not even square), this special SPD matrix enables an exhaustive and unique description of the data up to second-order statistics. This is achieved embedding the covariance structure of both the rows and columns of the data matrix, allowing naturally a wide range of possible applications and bringing us over and above just an interpretability issue. We demonstrate the method by manipulating satellite images (pansharpening) and event-related potentials (ERPs) of an electroencephalography brain-computer interface (BCI) study. The first example illustrates the effect of moving along geodesics in the original data space and the second provides a novel estimation of ERP average (geometric mean), showing that, in contrast to the usual arithmetic mean, this estimation is robust to outliers. In conclusion, we are able to show that the Riemannian concepts of distance, geometric mean, moving along a geodesic, etc. can be readily transposed into a generic data space, whatever this data space represents.
For the entire collection see [Zbl 1470.00021].From almost Gaussian to Gaussianhttps://www.zbmath.org/1483.530222022-05-16T20:40:13.078697Z"Costa, Max H. M."https://www.zbmath.org/authors/?q=ai:costa.max-henrique-m"Rioul, Olivier"https://www.zbmath.org/authors/?q=ai:rioul.olivierSummary: We consider lower and upper bounds on the difference of differential entropies of a Gaussian random vector and an approximately Gaussian random vector after they are ``smoothed'' by an arbitrarily distributed random vector of finite power. These bounds are important to establish the optimality of the corner points in the capacity region of Gaussian interference channels. A problematic issue in a previous attempt to establish these bounds was detected in 2004 and the mentioned corner points have since been dubbed ``the missing corner points''. The importance of the given bounds comes from the fact that they induce Fano-type inequalities for the Gaussian interference channel. Usual Fano inequalities are based on a communication requirement. In this case, the new inequalities are derived from a non-disturbance constraint. The upper bound on the difference of differential entropies is established by the data processing inequality (DPI). For the lower bound, we do not have a complete proof, but we present an argument based on continuity and the DPI.
For the entire collection see [Zbl 1470.00021].Geometry of \(F\)-likelihood estimators and \(F\)-max-ent theoremhttps://www.zbmath.org/1483.530232022-05-16T20:40:13.078697Z"Harsha, K. V."https://www.zbmath.org/authors/?q=ai:harsha.k-v"Subrahamanian Moosath, K. S."https://www.zbmath.org/authors/?q=ai:moosath.k-s-subrahamanianSummary: We consider a family of probability distributions called \(F\)-exponential family which has got a dually flat structure obtained by the conformal flattening of the \((F,G)\)-geometry. Geometry of \(F\)-likelihood estimator is discussed and the \(F\)-version of the maximum entropy theorem is proved.
For the entire collection see [Zbl 1470.00021].Entropic quantization of scalar fieldshttps://www.zbmath.org/1483.530242022-05-16T20:40:13.078697Z"Ipek, Selman"https://www.zbmath.org/authors/?q=ai:ipek.selman"Caticha, Ariel"https://www.zbmath.org/authors/?q=ai:caticha.arielSummary: Entropic Dynamics is an information-based framework that seeks to derive the laws of physics as an application of the methods of entropic inference. The dynamics is derived by maximizing an entropy subject to constraints that represent the physically relevant information that the motion is continuous and non-dissipative. Here we focus on the quantum theory of scalar fields. We provide an entropic derivation of Hamiltonian dynamics and using concepts from information geometry derive the standard quantum field theory in the Schrödinger representation.
For the entire collection see [Zbl 1470.00021].Information geometry of Bayesian statisticshttps://www.zbmath.org/1483.530252022-05-16T20:40:13.078697Z"Matsuzoe, Hirosh"https://www.zbmath.org/authors/?q=ai:matsuzoe.hiroshSummary: A survey of geometry of Bayesian statistics is given. From the viewpoint of differential geometry, a prior distribution in Bayesian statistics is regarded as a volume element on a statistical model. In this paper, properties of Bayesian estimators are studied by applying equiaffine structures of statistical manifolds. In addition, geometry of anomalous statistics is also studied. Deformed expectations and deformed independeces are important in anomalous statistics. After summarizing geometry of such deformed structues, a generalization of maximum likelihood method is given. A suitable weight on a parameter space is important in Bayesian statistics, whereas a suitable weight on a sample space is important in anomalous statistics.
For the entire collection see [Zbl 1470.00021].Statistics on Lie groups: a need to go beyond the pseudo-Riemannian frameworkhttps://www.zbmath.org/1483.530262022-05-16T20:40:13.078697Z"Miolane, Nina"https://www.zbmath.org/authors/?q=ai:miolane.nina"Pennec, Xavier"https://www.zbmath.org/authors/?q=ai:pennec.xavierSummary: Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group \(G\) is a manifold that carries an additional group structure. Statistics on \textit{Riemannian} manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall by others. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is \textit{compatible with the group structure}, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group \(G\). The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
For the entire collection see [Zbl 1470.00021].The geometrical structure of quantum theory as a natural generalization of information geometryhttps://www.zbmath.org/1483.530272022-05-16T20:40:13.078697Z"Reginatto, Marcel"https://www.zbmath.org/authors/?q=ai:reginatto.marcelSummary: Quantum mechanics has a rich geometrical structure which allows for a geometrical formulation of the theory. This formalism was introduced by Kibble and later developed by a number of other authors. The usual approach has been to start from the standard description of quantum mechanics and identify the relevant geometrical features that can be used for the reformulation of the theory. Here this procedure is inverted: the geometrical structure of quantum theory is derived from information geometry, a geometrical structure that may be considered more fundamental, and the Hilbert space of the standard formulation of quantum mechanics is constructed using geometrical quantities. This suggests that quantum theory has its roots in information geometry.
For the entire collection see [Zbl 1470.00021].Failures of information geometryhttps://www.zbmath.org/1483.530282022-05-16T20:40:13.078697Z"Skilling, John"https://www.zbmath.org/authors/?q=ai:skilling.johnSummary: Information \(H\) is a unique relationship between probabilities, based on the property of \textit{independence} which is central to scientific methodology. Information Geometry makes the tempting but fallacious assumption that a local metric (conventionally based on information) can be used to endow the space of probability distributions with a preferred global Riemannian metric. No such global metric can conform to \(H\), which is ``from-to'' asymmetric whereas geometrical length is by definition symmetric. Accordingly, \textit{any} Riemannian metric will contradict the required structure of the very distributions which are supposedly being triangulated. It follows that probabilities do not form a metric space. We give counter-examples in which alternative formulations of information, and the use of information geometry, lead to unacceptable results.
For the entire collection see [Zbl 1470.00021].Information geometric density estimationhttps://www.zbmath.org/1483.530292022-05-16T20:40:13.078697Z"Sun, Ke"https://www.zbmath.org/authors/?q=ai:sun.ke"Marchand-Maillet, Stéphane"https://www.zbmath.org/authors/?q=ai:marchand-maillet.stephaneSummary: We investigate kernel density estimation where the kernel function varies from point to point. Density estimation in the input space means to find a set of coordinates on a statistical manifold. This novel perspective helps to combine efforts from information geometry and machine learning to spawn a family of density estimators. We present example models with simulations. We discuss the principle and theory of such density estimation.
For the entire collection see [Zbl 1470.00021].Harmonic maps relative to \(\alpha \)-connections of statistical manifoldshttps://www.zbmath.org/1483.530302022-05-16T20:40:13.078697Z"Uohashi, Keiko"https://www.zbmath.org/authors/?q=ai:uohashi.keikoSummary: In this paper, we study harmonic maps relative to \(\alpha \)-connections, but not necessarily standard harmonic maps. A standard harmonic map is defined by the first variation of the energy functional of a map. A harmonic map relative to an \(\alpha \)-connection is defined by an equation similar to a first variational equation, though it is not induced by the first variation of the standard energy functional. In this paper, we define energy functionals of maps relative to \(\alpha \)-connections of statistical manifolds. Next, we show that, for harmonic maps relative to \(\alpha \)-connections, the Euler-Lagrange equations are induced by first variations of energy functionals relative to \(\alpha \)-connections.
For the entire collection see [Zbl 1470.00021].Reference duality and representation duality in information geometryhttps://www.zbmath.org/1483.530312022-05-16T20:40:13.078697Z"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun|zhang.jun.9|zhang.jun.10|zhang.jun.7|zhang.jun.2|zhang.jun.3|zhang.jun.8|zhang.jun.1|zhang.jun.4|zhang.jun.5|zhang.jun.6Summary: Classical information geometry prescribes, on the parametric family of probability functions \(M_\theta \): (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of \(\alpha \)-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of (non-symmetric) divergence functions \(( \alpha \)-divergence) defined on \(M_\theta \times M_\theta \), which induce the metric and the dual connections. The role of \(\alpha\) parameter, as used in \(\alpha \)-connection and in \(\alpha \)-embedding, is not commonly differentiated. For instance, the case with \(\alpha = \pm 1\) may refer either to dually-flat \((e\)- or \(m\)-) connections or to exponential and mixture families of density functions. Here we illuminate that there are two distinct types of duality in information geometry, one concerning the referential status of a point (probability function, normalized or denormalized) expressed in the divergence function (``reference duality'') and the other concerning the representation of probability functions under an arbitrary monotone scaling (``representation duality''). They correspond to, respectively, using \(\alpha\) as a mixture parameter for constructing divergence functions or as a power exponent parameter for monotone embedding of probability functions. These two dualities are coupled into referential-representational biduality for manifolds of denormalized probability functions with \(\alpha \)-Hessian structure (i.e, transitively flat \(\alpha \)-geometry) and for manifolds induced from homogeneous divergence functions with \(( \alpha,\beta )\)-parameters but one-parameter family of \(( \alpha \cdot \beta )\)-connections.
For the entire collection see [Zbl 1470.00021].Symmetries of Sasakian generalized Sasakian-space-form admitting generalized Tanaka-Webster connectionhttps://www.zbmath.org/1483.530322022-05-16T20:40:13.078697Z"Lalmalsawma, Chawngthu"https://www.zbmath.org/authors/?q=ai:lalmalsawma.chawngthu"Singh, Jay Prakash"https://www.zbmath.org/authors/?q=ai:singh.jay-prakashSummary: The object of this paper is to study certain symmetric properties of Sasakian generalized Sasakian-space-form with respect to generalized Tanaka-Webster connection. We studied semi-symmetry and Ricci semi-symmetry of Sasakian generalized Sasakian-spaceform with respect to generalized Tanaka-Webster connection. Further we obtain results for Ricci pseudosymmetric and Ricci-generalized pseudosymmetric Sasakian generalized Sasakian-space-form.Poisson-Voronoi tessellation on a Riemannian manifoldhttps://www.zbmath.org/1483.530332022-05-16T20:40:13.078697Z"Calka, Pierre"https://www.zbmath.org/authors/?q=ai:calka.pierre"Chapron, Aurélie"https://www.zbmath.org/authors/?q=ai:chapron.aurelie"Enriquez, Nathanaël"https://www.zbmath.org/authors/?q=ai:enriquez.nathanaelThis paper pioneers the study of the Poisson-Voronoi tessellation on Riemannian manifolds from the perspective of stochastic geometry.
Classically, the Poisson-Voronoi tessellation is defined on Euclidean spaces as follows: first, start with a Poisson point process \(\mathcal{P}_\lambda\) with intensity \(\lambda\). Then, associate each point \(x \in \mathbb{R}^d\) to its nearest neighbor in \(\mathcal{P}\). Thus, each point \(p \in \mathcal{P}\) is associated with a cell consisting of all points closest to it
\[ C(p,\mathcal{P}_\lambda) = \{x \in \mathbb{R}^d: d(x,p) \leq d(x,p')\, \forall p' \in \mathcal{P}\}. \]
The collection of all such cell forms a tessellation of \(\mathbb{R}^d\), called the Poisson-Voronoi tessellation. For Euclidean spaces, \(d\) is the Euclidean distance. In this paper, \(\mathbb{R}^d\) is replaced by a Riemannian manifold \(M\), and \(d\) is the Riemannian metric of \(M\).
Typical results in stochastic geometry concern statistics of the tessellation. This paper focuses on statistics of the typical cell. The typical cell \(\mathcal{C}^{M}_{x_0,\lambda}\) is defined as the cell that contains \(x_0\) when the \(x_0\) is added to \(\mathcal{P}_\lambda\). Intuitively, it captures the local geometry of the tessellation around a point \(x_0 \in M\). The main results of this paper include high-intensity asymptotics for the mean number of vertices and density of vertices of the typical cell \(\mathcal{C}^{M}_{x_0,\lambda}\), under some curvature assumptions on \(M\). Results on these quantities have previously been obtained only for Euclidean spaces, and for two non-Euclidean manifolds only, namely the sphere and the hyperbolic space.
Reviewer: Ngoc Mai Tran (Bonn)Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpointhttps://www.zbmath.org/1483.530342022-05-16T20:40:13.078697Z"Honda, Atsufumi"https://www.zbmath.org/authors/?q=ai:honda.atsufumi"Izumiya, Shyuichi"https://www.zbmath.org/authors/?q=ai:izumiya.shyuichi"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Teramoto, Keisuke"https://www.zbmath.org/authors/?q=ai:teramoto.keisukeSummary: A surface in the Lorentz-Minkowski 3-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a lightlike locus, and using it, we define two lightlike ruled surfaces along a lightlike locus which can be regarded as lightlike approximations of the surface along the lightlike locus. We study a relationship of singularities of these lightlike surfaces and differential geometric properties of the lightlike locus. We also consider the intersection curve of two lightlike approximations, which gives a model curve of the lightlike locus.Bochner-Chen ideal submanifoldshttps://www.zbmath.org/1483.530352022-05-16T20:40:13.078697Z"Kılıç, Erol"https://www.zbmath.org/authors/?q=ai:kilic.erol"Gülbahar, Mehmet"https://www.zbmath.org/authors/?q=ai:gulbahar.mehmetSummary: In this paper, we investigate ideal invariant submanifolds, ideal anti-invariant submanifolds and ideal CR-submanifolds of Bochner Kaehler manifolds. Moreover, some characterizations related with the holomorphic sectional curvature and the anti-holomorphic sectional curvatureare obtained for Bochner Kaehler manifolds.Certain characterization of real hypersurfaces of type A in a nonflat complex space formhttps://www.zbmath.org/1483.530362022-05-16T20:40:13.078697Z"Ki, U-Hang"https://www.zbmath.org/authors/?q=ai:ki.u-hangSummary: Let \(M\) be a real hypersurface with almost contact metric structure \((\phi, \xi, \eta, g)\) in a nonflat complex space form \(M_n (c)\). We denote \(S\) and \(R_\xi\) by the Ricci tensor of \(M\) and by the structure Jacobi operator with respect to the vector field \(\xi\) respectively. In this paper, we prove that \(M\) is a Hopf hypersurface of type \(A\) in \(M_n (c)\) if it satisfies \(R_\xi \phi = \phi R_\xi\) and at the same time \(R_\xi (S \phi - \phi S) = 0\).Screen semi-slant lightlike submanifolds of indefinite Kaehler manifoldshttps://www.zbmath.org/1483.530372022-05-16T20:40:13.078697Z"Shukla, Shiv Sharma"https://www.zbmath.org/authors/?q=ai:shukla.shiv-sharma"Yadav, Akhilesh"https://www.zbmath.org/authors/?q=ai:yadav.akhilesh-chandraSummary: In this paper, we introduce the notion of screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions \(D_1\), \(D_2\) and \textit{RadTM} on screen semi-slant lightlike submanifolds of an indefinite Kaehler manifold have been obtained. Further we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds.Static Einstein-Maxwell space-time invariant by translationhttps://www.zbmath.org/1483.530382022-05-16T20:40:13.078697Z"Leandro, Benedito"https://www.zbmath.org/authors/?q=ai:leandro.benedito"de Melo, Ana Paula"https://www.zbmath.org/authors/?q=ai:de-melo.ana-paula"Menezes, Ilton"https://www.zbmath.org/authors/?q=ai:menezes.ilton"Pina, Romildo"https://www.zbmath.org/authors/?q=ai:pina.romildo-da-silva|pina.romildo-sSummary: In this paper we study the static Einstein-Maxwell space when it is conformal to an \(n\)-dimensional pseudo-Euclidean space, which is invariant under the action of an \((n-1)\)-dimensional translation group. We also provide a complete classification of such space.A note on the Gannon-Lee theoremhttps://www.zbmath.org/1483.530392022-05-16T20:40:13.078697Z"Schinnerl, Benedict"https://www.zbmath.org/authors/?q=ai:schinnerl.benedict"Steinbauer, Roland"https://www.zbmath.org/authors/?q=ai:steinbauer.rolandSummary: We prove a Gannon-Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity \(C^1\), the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a \(C^1\)-spacetime is a geodesic and hence of \(C^2\)-regularity.On spherically symmetric Finsler metric with scalar and constant flag curvaturehttps://www.zbmath.org/1483.530402022-05-16T20:40:13.078697Z"Solórzano, Newton"https://www.zbmath.org/authors/?q=ai:solorzano.newton-mayer"Leandro, Benedito"https://www.zbmath.org/authors/?q=ai:leandro.beneditoThe authors focus their works on studying suitable systems of partial differential equations describing spherically symmetric Finsler metrics with scalar and constant flag curvature. Therefore, they obtain examples of Douglas metrics, i.e., a more generalized metrics than Berwald metrics, with scalar curvature. Also, they furnish a classification for a few families of Finsler (resp. Douglas) metrics and with scalar flag curvature (resp. with constant flag curvature).
Reviewer: Mohammed El Aïdi (Bogotá)Inevitability of the Poisson bracket structure of the relativistic constraintshttps://www.zbmath.org/1483.530412022-05-16T20:40:13.078697Z"Głowacki, Jan"https://www.zbmath.org/authors/?q=ai:glowacki.janSummary: The purpose of this paper is to shed some fresh light on the long-standing conceptual question of the \textit{origin} of the well-known Poisson bracket structure of the constraints that govern the canonical dynamics of generally relativistic field theories, i.e. geometrodynamics. This structure has long been known to be the same for a wide class of fields that inhabit the space-time, namely those with non-differential coupling to gravity. It has also been noticed that an identical bracket structure can be derived independently of any dynamical theory, by \textit{purely geometrical} considerations in Lorentzian geometry. Here we attempt to provide the missing link between the dynamics and geometry, which we understand to be the \textit{reason} for this structure to be of the specific kind. We achieve this by a careful analysis of the geometrodynamical approach, which allows us to \textit{derive} the structure in question and understand it as a \textit{consistency requirement} for any such theory. In order to stay close to the classical literature on the subject we stick to the metric formulation of general relativity, but the reasoning should carry over to any other formulation as long as the non-metricity tensor vanishes. The discussion section is devoted to derive some interesting consequences of the presented result in the context of reconstructing the Arnowitt-Deser-Misner (ADM) framework, thus providing a precise sense to the inevitability of the Einstein's theory under minimal assumptions.Model Higgs bundles in exceptional components of the \(\mathrm{Sp}(4,\mathbb{R})\)-character varietyhttps://www.zbmath.org/1483.530422022-05-16T20:40:13.078697Z"Kydonakis, Georgios"https://www.zbmath.org/authors/?q=ai:kydonakis.georgiosLet \(\Sigma\) be a closed connected oriented surface of genus \(g\geq 2.\) The character variety \(\mathcal R(G)\) consists of reductive representations of \(\pi_1(\Sigma)\) into \(G\) modulo conjugation. By the non-abelian Hodge correspondence, the character variety \(\mathcal R(G)\) is isomorphic to the moduli space \(\mathcal M(G)\) of polystable \(G\)-Higgs bundles over a Riemann surface \(X=(\Sigma,J)\). When \(G=\mathrm{Sp}(4,\mathbb R)\), the subspace \(\mathcal M^{\max}(\mathrm{Sp}(4,\mathbb R))\) consists of maximal Higgs bundles, that is, with extremal Toledo invariant. It has been shown that \(\mathcal M^{\max}(\mathrm{Sp}(4,\mathbb R))\) contains \(3\cdot 2^{2g}+2g-4\) components. Among them, there are \(2g-3\) exceptional components solely consisting of Zariski dense representations. By the work of Guichard-Wienhard, representations in such components are continuous deformations of hybrid representations which involves a gluing construction for fundamental group representations over a connected sum of surfaces. The representations in the remaining \(3\cdot 2^{2g}-1\) components are deformations of standard representations, which are compositions of \(\mathrm{SL}(2,\mathbb R)\)-representations and embeddings of \(\mathrm{SL}(2,\mathbb R)\rightarrow \mathrm{Sp}(4,\mathbb R).\) The Higgs bundles corresponding to standard representations are also embeddings of \(\mathrm{SL}(2,\mathbb R)\)-Higgs bundles.
The main goal of the current paper is to establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the \(\mathrm{Sp}(4,\mathbb R)\)-Hitchin equations, in analogy with the construction of hybrid representations.
The gluing techniques used here generalize the work of Swoboda. Start with initial parabolic Higgs bundles over two distinct Riemann surfaces \(X_1, X_2\) and model solutions to Hitchin's equation on annuli around the points in the divisors. First, perturb the initial data into model solutions by appropriate complex gauge transformations and construct a pair \((A_R^{\mathrm{app}}, \Phi_R^{\mathrm{app}})\) over \(X_{\sharp}=X_1\sharp X_2\) that coincides with initial data over \(X_1\) and \(X_2\) away from the divisors. Next, find a complex gauge transformation \(g\) such that \(g^*(A_R^{\mathrm{app}}, \Phi_R^{\mathrm{app}})\) is an exact solution to Hitchin's equation and thus finish the gluing construction. The argument showing the existence of \(g\) is translated into a Banach fixed point theorem argument and involves the study of linearization of a relevant elliptic operator.
Also, the author finds an additive formula of topological invariants under the complex connected sum operation of Higgs bundles, in analogous to the additivity property for the Toledo invariant established by Burger et al. With this formula, one can determine the topological invariant for hybrid Higgs bundles from the initial data. With an appropriate choice of initial data, the gluing constructions provide model Higgs bundles in each component of the \(2g-3\) exceptional components of maximal representations.
Reviewer: Qiongling Li (Tianjin)Transversely Hessian foliations and information geometryhttps://www.zbmath.org/1483.530432022-05-16T20:40:13.078697Z"Boyom, Michel Nguiffo"https://www.zbmath.org/authors/?q=ai:boyom.michel-nguiffo"Wolak, Robert"https://www.zbmath.org/authors/?q=ai:wolak.robert-aSummary: A family of probability distributions parametrized by an open domain \(\Lambda\) in \(\mathbb{R}^n\) defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix is positive definite defining in this way a Riemannian metric on \(\Lambda \). If we replace the ``positive definite'' assumption by ``0-deformable'' condition a foliation with a transvesely Hessian structure appears naturally. We develop the study of transversely Hessian foliations in view of applications in information geometry.
For the entire collection see [Zbl 1470.00021].Liouville type theorem for transversally harmonic mapshttps://www.zbmath.org/1483.530442022-05-16T20:40:13.078697Z"Fu, Xueshan"https://www.zbmath.org/authors/?q=ai:fu.xueshan"Jung, Seoung Dal"https://www.zbmath.org/authors/?q=ai:jung.seoung-dalSummary: Let \((M,\mathcal{F})\) be a complete foliated Riemannian manifold and all leaves be compact. Let \((M',\mathcal{F}')\) be a foliated Riemannian manifold of non-positive transversal sectional curvature. Assume that the transversal Ricci curvature \(\mathrm{Ric}^Q\) of \(M\) satisfies \(\mathrm{Ric}^Q\geq -\lambda_0\) at all point \(x\in M\) and \(\mathrm{Ric}^Q>-\lambda_0\) at some point \(x_0\), where \(\lambda_0\) is the infimum of the spectrum of the basic Laplacian acting on \(L^2\)-basic functions on \(M\). Then every transversally harmonic map \(\phi:M \rightarrow M'\) of finite transversal energy is transversally constant.On Ricci curvature of metric structures on \(\mathfrak{g}\)-manifoldshttps://www.zbmath.org/1483.530452022-05-16T20:40:13.078697Z"Rovenski, Vladimir"https://www.zbmath.org/authors/?q=ai:rovenskii.vladimir-yuzefovich"Wolak, Robert"https://www.zbmath.org/authors/?q=ai:wolak.robert-aFor a Lie algebra \(\mathfrak{g}\), a \(\mathfrak{g}\)-manifold means a smooth manifold \(M\) with a \(\mathfrak{g}\)-action, namely, a Lie algebra homomorphism \(\mathfrak{g} \to \mathfrak{X}(M)\), where \(\mathfrak{X}(M)\) is the Lie algebra of vector fields on \(M\). In this article, the authors study global geometry of \(\mathfrak{g}\)-manifolds \(M\) whose \(\mathfrak{g}\)-action is locally free and preserves a Riemannian metric on \(M\). The orbits of such \(\mathfrak{g}\)-action yields the characteristic foliation, which is a Riemannian foliation with totally geodesic leaves. Sasakian manifolds, or more generally, almost \(K\)-contact manifolds are examples of such \(\mathfrak{g}\)-manifolds with \(\mathfrak{g}\) of dimension one. Almost \(\mathcal{K}\)-manifolds are \(\mathfrak{g}\)-manifolds which can be regarded as a higher-dimensional generalization of almost \(K\)-contact manifolds [\textit{D. E. Blair}, J. Differ. Geom. 4, 155--167 (1970; Zbl 0202.20903)].
In the first part, the authors prove several results on the Ricci curvature of \(\mathfrak{g}\)-manifolds whose \(\mathfrak{g}\)-action is locally free and preserves a Riemannian metric. Among other results, they show that, if a compact Lie group \(G\) acts locally free and isometrically on a compact Riemannian manifold \(M\) and the orbit foliation is modeled on a manifold with a positive Ricci curvature, then \(M\) admits a Riemannian metric of positive Ricci curvature, and hence has a finite fundamental group by Myers's theorem. If \(\mathfrak{g}\) is abelian, by showing and using an analogous result in the negative Ricci curvature case, the authors prove that the characteristic foliation of a compact \(\mathfrak{g}\)-manifold given by an isometric action cannot be modeled on a manifold of negative Ricci curvature. These are generalizations of the results due to [\textit{D. Gromoll} and \textit{G. Walschap}, Metric foliations and curvature. Basel: Birkhäuser (2009; Zbl 1163.53001)] for Riemannian submersions with totally geodesic fibers.
In the second part, they study Ricci solitons and Einstein metrics on almost \(\mathcal{S}\)-manifolds. Recall that a \(\mathcal{K}\)-manifold is defined by a smooth \((2n+p)\)-manifold \(M\) endowed with a Riemannian metric \(g\), unit vector fields \(\xi_1, \dots, \xi_p\), \(1\)-forms \(\eta_1, \dots, \eta_p\) and a \((1,1)\)-tensor \(\varphi\) such that \(\xi_i\) is the dual of \(\eta_i\) with respect to \(g\) for \(i=1,\dots,p\) and \(\varphi^2 = - \operatorname{id}_{TM} + \sum_i \eta^i \otimes \xi_i \). An almost \(\mathcal{S}\)-manifold is defined by an almost \(\mathcal{S}\)-manifold such that \(d\eta^i(X,Y)=g(X,\varphi(Y))\) for every \(X,Y \in \mathfrak{X}(M)\) and every \(i=1, \dots, p\). It is easy to see that \(\xi_1, \dots, \xi_p\) are Killing and span the kernel of \(\varphi\), which is an integrable distribution of rank \(p\). In the case where \(p=1\), for \(K\)-contact manifolds, \textit{C. P. Boyer} and \textit{K. Galicki} [Proc. Am. Math. Soc. 129, No. 8, 2419--2430 (2001; Zbl 0981.53027)] proved that Einstein \(K\)-contact manifolds are Sasaki-Einstein manifolds. The authors prove that, among other results, in the case where \(p>1\), compact almost \(\mathcal{S}\)-manifolds do not admit Einstein metrics. By combining a generalization of a result due to [\textit{J. T. Cho} and \textit{R. Sharma}, Int. J. Geom. Methods Mod. Phys. 7, No. 6, 951--960 (2010; Zbl 1202.53063)], the authors prove that there is no compact almost \(\mathcal{S}\)-manifold that is a Ricci soliton for a vector field \(X\) tangent to \(\operatorname{ker} \varphi\).
Reviewer: Hiraku Nozawa (Bures-sur-Yvette)Para-complex structures on linear coframe bundle with Sasakian metrichttps://www.zbmath.org/1483.530462022-05-16T20:40:13.078697Z"Fattayev, Habil"https://www.zbmath.org/authors/?q=ai:fattayev.habil-dSummary: By using a Riemannian metric on a differentiable manifold, the Sasakian metric is introduced on the linear coframe bundle of the Riemannian manifold. Geometric properties of Levi-Civita connection of Sasakian metric are investigated. Also, para-complex structures on the linear coframe bundle with Sasakian metric are constructed and some interesting properties of those structures are studied.On \(L^2\)-harmonic forms of complete almost Kähler manifoldhttps://www.zbmath.org/1483.530472022-05-16T20:40:13.078697Z"Huang, Teng"https://www.zbmath.org/authors/?q=ai:huang.tengSummary: In this article, we study the \(L^2\)-harmonic forms on the complete \(2n\)-dimensional almost Käher manifold \(X\). We observe that the \(L^2\)-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can extend vanishing theorems of \(d\)(bounded) (resp. \(d\)(sublinear)) Kähler manifold proved by Gromov (resp. Cao-Xavier, Jost-Zuo) to almost Kählerian case, that is, the spaces of all harmonic \((p, q)\)-forms on \(X\) vanishing unless \(p+q=n\). We also give a lower bound on the spectra of the Laplace operator to sharpen the Lefschetz vanishing theorem on \(d\)(bounded) case.On contact CR-submanifolds of \((LCS)_n\)-manifoldshttps://www.zbmath.org/1483.530482022-05-16T20:40:13.078697Z"Hui, Shyamal Kumar"https://www.zbmath.org/authors/?q=ai:hui.shyamal-kumar"Atceken, Mehmet"https://www.zbmath.org/authors/?q=ai:atceken.mehmet"Pal, Tanumoy"https://www.zbmath.org/authors/?q=ai:pal.tanumoy"Mishra, Lakshmi Narayan"https://www.zbmath.org/authors/?q=ai:mishra.lakshmi-narayanSummary: The object of the present paper is to study contact CR-submanifolds of \((LCS)_n\)-manifolds. We obtain the integrability conditions of the distributions of contact CR-submanifolds of \((LCS)_n\)-manifolds. Finally, we give an interesting example of a contact CR-submanifold of \((LCS)_7\)-manifold.Curvature properties of Riemannian manifolds with circulant structureshttps://www.zbmath.org/1483.530492022-05-16T20:40:13.078697Z"Razpopov, Dimitar"https://www.zbmath.org/authors/?q=ai:razpopov.dimitar"Dzhelepov, Georgi"https://www.zbmath.org/authors/?q=ai:dzhelepov.georgi-dSummary: We study a Riemannian manifold \(M\) equipped with a circulant structure \(Q\), which is an isometry with respect to the metric. We consider two types of such manifolds: a 3-dimensional manifold \(M\) where the third power of \(Q\) is the identity, and a 4-dimensional manifold \(M\) where the fourth power of \(Q\) is the identity. In a single tangent space of \(M\) we have a special tetrahedron constructed by vectors of a \(Q\)-basis. The aim of the present paper is to find relations among the sectional curvatures of the 2-planes associated with the four faces of this tetrahedron and its cross sections passing through the medians and the edges of these faces.\(M\)-projective curvature tensor on an \((\mathrm{LCS})_{2n+1}\)-manifoldhttps://www.zbmath.org/1483.530502022-05-16T20:40:13.078697Z"Shanmukha, B."https://www.zbmath.org/authors/?q=ai:shanmukha.b"Venkatesha, V."https://www.zbmath.org/authors/?q=ai:venkatesha.vishnuvardhana-s-v|venkatesha.venkateshaSummary: In this paper, we study \(M\)-projective curvature tensors on an \((\mathrm{LCS})_{2n+1}\)-manifold. Here we study \(M\)-projectively Ricci symmetric and \(M\)-projectively flat admitting spacetime.On the \(F\) structures of the space \(T(Lm(Vn))\)https://www.zbmath.org/1483.530512022-05-16T20:40:13.078697Z"Todua, G."https://www.zbmath.org/authors/?q=ai:todua.gocha|todua.g-sh|todua.g-tSummary: There are constructed lifts of tensor fields \(a^i_j\), \(a_\alpha^i\), \(a^i_{\overline{j}}\), \(a^i_{\overline{\alpha}}\), \(a_j^{\overline{i}}\), \(a_{\alpha}^{\overline{i}}\), \(a_{\overline{j}}^{\overline{i}}\), \(a_{\overline{\alpha}}^{\overline{i}}\), \(a_j^\beta\), \(a^\beta_\alpha\), \(a^\beta_{\overline{j}}\), \(a^\beta_{\overline{\alpha}}\), \(a^{\overline{\beta}}_j\), \(a_{\alpha}^{\overline{\beta}}\), \(a^{\overline{\beta}}_{\overline{j}}\), \(a_{\overline{\alpha}}^{\overline{\beta}}\). There are defined \(F\) structures on the space \(a_j^{\overline{\beta}}\) and there is proved, that real-valued \(F\) structures exist only for \(\lambda = -1\).Affine connections with torsion in (para-)complexified structureshttps://www.zbmath.org/1483.530522022-05-16T20:40:13.078697Z"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.4"Khan, Gabriel"https://www.zbmath.org/authors/?q=ai:khan.gabriel-j-hConsider a Nijenhuis tensor \(N_L\) on a manifold \(M\) with a\((1,1)\)-tensor \(L\):
\[N_L(X,Y)=-L^2[X,Y]+L[X,LY]+L[LX,Y]-[LX,LY]\]
The operator \(L\) is called integrable if \(N_L=0\). By the famous theorem of Newlander-Nirenberg, if \(L=J\) is an almost complex structure, this is the obstruction for it to be a complex structure. The authors define two conditions for a pair \((\nabla,L)\) as below:
(MC1): \(T^{\nabla}(LX,Y)+T^{\nabla}(X,LY)=0\)
(MC2): \(T^{\nabla}(LX,Y)=LT^{\nabla}(X,Y)+(\nabla_XL)Y-(\nabla_YL)X,\) for the torsion tensor \(T^\nabla\) of \(\nabla\).
It is shown that if MC1 holds for both \(\nabla\) and \(\nabla^L=L^{-1}\circ\nabla\circ L\) or MC2 holds for either \(\nabla\) or \(\nabla^{L}\), then \(N_L=0\). Depending on the type of \(L\), we can complexify the tangent bundle \(TM\) and decompose it into two subbundles with positive and negative eigenvalues, \(T_p^{+}M\) and \(T_p^{-}M\). According to this splitting, the Nijenhuis tensor \(N_L\) can be splitted into three pieces \(N^{(1,1)}\), \(N^{(2,0)}\) and \(N^{(0,2)}\). In this paper, the authors investigate the several cases of integrability in terms of the conditions MC1 and MC2 and the three associated pieces of the Nijenhuis tensor.
For the entire collection see [Zbl 1473.53007].
Reviewer: Yun Myung Oh (Berrien Springs)The isoperimetric problem in Carnot-Carathéodory spaceshttps://www.zbmath.org/1483.530532022-05-16T20:40:13.078697Z"Franceschi, Valentina"https://www.zbmath.org/authors/?q=ai:franceschi.valentinaPansu's conjectured isoperimetric profile of the Heisenberg group \(({\mathbb H}^1,g_{\mathrm{cc}})\), where \(g_{\mathrm{cc}}\) denotes the standard Carnot-Caratheodory metric, remains one of the signature open problems in sub-Riemannian geometric analysis. In this paper, the author provides an overview of several related results obtained in two papers with R. Monti and F. Montefalcone.
In [the author and \textit{R. Monti}, Rev. Mat. Iberoam. 32, No. 4, 1227--1258 (2016; Zbl 1368.49051)], the authors solve the corresponding isoperimetric problem in a class of sub-Riemannian spaces of Grushin type, with applications to the classication of \(x\)-symmetric isoperimetric minimizers in \(H\)-type Carnot groups. Their result generalizes prior work by [\textit{R. Monti} and \textit{D. Morbidelli}, J. Geom. Anal. 14, No. 2, 355--368 (2004; Zbl 1076.53035)].
In [\textit{V. Franceschi} et al., Anal. Geom. Metr. Spaces 7, 109--129 (2019; Zbl 1428.53073)], the authors consider the isoperimetric problem for rotationally symmetric surfaces in a family of Riemannian Heisenberg groups. Building on prior work of \textit{P. Tomter} [Proc. Symp. Pure Math. 54, 485--495 (1993; Zbl 0799.53073)], they considered a two-parameter family of left-invariant Riemannian metrics, denoted \(g_{\epsilon,\sigma}\), on the first Heisenberg group. Here the parameter \(\sigma \ne 0\) governs the degree of non-commutativity of the horizontal vector fields, while \(\epsilon>0\) is a penalty parameter restricting non-horizontal motion. Fixing \(\sigma\), respectively \(\epsilon\), the resulting Riemannian 3-manifold structure converges to the Euclidean space \({\mathbb R}^3\), respectively, to the sub-Riemannian Heisenberg group \({\mathbb H}^1\). Moreover, the Riemannian volume and perimeter functionals, suitably normalized, converge to their respective analogues in these two spaces. It follows that isoperimetric solutions in \(({\mathbb R}^3,g_{\epsilon,\sigma})\) converge to their analogues in the two limit spaces. In particular, under the degenerating limiting operation \(({\mathbb R}^3,g_{\epsilon,\sigma}) \stackrel{\epsilon \to 0}{\longrightarrow} ({\mathbb H}^1,g_{\mathrm{cc}})\), the Riemannian isoperimetric solutions converge to sub-Riemannian ones. The authors consider the isoperimetric problem in \(({\mathbb R}^3,g_{\epsilon,\sigma})\) under a topological assumption on the extremals. They identify isoperimetric minimizers which are topological balls, and recover Pansu's conjectured sub-Riemannian extremals in the limit as \(\epsilon \to 0\).
For the entire collection see [Zbl 1411.35007].
Reviewer: Jeremy Tyson (Urbana)A de Rham decomposition type theorem for contact sub-Riemannian manifoldshttps://www.zbmath.org/1483.530542022-05-16T20:40:13.078697Z"Grochowski, Marek"https://www.zbmath.org/authors/?q=ai:grochowski.marekThe author proves a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that \((M, H, g)\) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field \(\xi\) is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on \(H\). Suppose that there exists a point \(q\in M \) such that the holonomy group \(\Psi(q)\) acts reducibly on \(H(q)\) yielding a decomposition \(H(q)=H_1(q)\oplus\cdots\oplus H_m(q) \) into \(\Psi(q)\)-irreducible factors. Using parallel transport the authors obtain the decomposition \(H = H_1 \oplus\cdots\oplus H_m\) of \(H\) into sub-distributions \(H_i\). Unlike the Riemannian case, the distributions \(H_i\) are not integrable, however they induce integrable distributions \(\delta_i\) on \(M/\xi\), which is locally a smooth manifold. As a result, every point in \(M\) has a neighborhood \(U\) such that \(T (U/\xi) = \delta_1\oplus\cdots\oplus\delta_m\), and the latter decomposition of \(T(U/\xi)\) induces the decomposition of \(U/\xi\) into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. The authors also give a version of the theorem for indefinite metrics.
Reviewer: Peibiao Zhao (Nanjing)Corrigendum to: ``Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space''https://www.zbmath.org/1483.530552022-05-16T20:40:13.078697Z"Huang, Lan-Hsuan"https://www.zbmath.org/authors/?q=ai:huang.lan-hsuan"Lee, Dan A."https://www.zbmath.org/authors/?q=ai:lee.dan-a"Sormani, Christina"https://www.zbmath.org/authors/?q=ai:sormani.christinaSummary: There is an error in the proof of Theorem 1.3 of the original article [the authors, ibid. 727, 269--299 (2017; Zbl 1368.53028)]. Despite the problem, it is rigorously proved in joint work of the first two authors and \textit{R. Perales} [``Intrinsic flat convergence of points and applications to stability of the positive mass theorem'', Preprint, \url{arXiv:2010.07885}] that Theorem 1.3 is true, using recent results of \textit{B. Allen} and \textit{R. Perales} [``Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below'', Preprint, \url{arXiv:2006.13030}] that extend the work of \textit{B. Allen}, \textit{R. Perales} and \textit{C. Sormani} [``Volume above distance below'', Preprint, \url{arXiv:2003.01172}].Remarks on manifolds with two-sided curvature boundshttps://www.zbmath.org/1483.530562022-05-16T20:40:13.078697Z"Kapovitch, Vitali"https://www.zbmath.org/authors/?q=ai:kapovitch.vitali"Lytchak, Alexander"https://www.zbmath.org/authors/?q=ai:lytchak.alexanderThe authors discuss results about the distance function and the cut locus of a closed subset of a smooth Riemannian manifold without assuming completeness. They show that these results also hold for manifolds with a two-sided curvature bound in the sense of Alexandrov. For this class of manifolds they investigate the regularity of harmonic functions in distance coordinates and show that \(C^{1,1}\)-submanifolds have positive reach.
Reviewer: Hans-Bert Rademacher (Leipzig)Spectral properties of Killing vector fields of constant length and bounded Killing vector fieldshttps://www.zbmath.org/1483.530572022-05-16T20:40:13.078697Z"Nikonorov, Yu. G."https://www.zbmath.org/authors/?q=ai:nikonorov.yurii-gSummary: This paper is a survey of recent results related to spectral properties of Killing vector fields of constant length and of some their natural generalizations on Riemannian manifolds. One of the main result is the following: If \(\mathfrak{g}\) is a Lie algebra of Killing vector fields on a given Riemannian manifold \((M, g)\), and \(X\in\mathfrak{g}\) has constant length on \((M, g)\), then the linear operator \(\mathrm{ad}(X):\mathfrak{g}\rightarrow\mathfrak{g}\) has a pure imaginary spectrum [the author, J. Geom. Phys. 145, Article ID 103485, 8 p. (2019; Zbl 1427.53043)]. We discuss also more detailed structure results on the corresponding operator \(\mathrm{ad}(X)\). Related results for geodesic orbit Riemannian spaces are considered. Finitely, we discuss some generalizations obtained recently by \textit{M. Xu} and the author [Asian J. Math. 25, No. 2, 229--242 (2021; Zbl 07429186)] for bounded Killing vector fields.
For the entire collection see [Zbl 1470.47003].The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: conservation laws and their applicationshttps://www.zbmath.org/1483.530582022-05-16T20:40:13.078697Z"Avalos, R."https://www.zbmath.org/authors/?q=ai:avalos.rodrigo"Freitas, A."https://www.zbmath.org/authors/?q=ai:freitas.a-b|freitas.a-d|freitas.a-r-r|freitas.andre|freitas.ana-t|freitas.ana-cristina-moreira|freitas.allan-g|freitas.alex-alves|freitas.augusto-s|freitas.adelaide-valente|freitas.ana-p-c|freitas.amauri-a|freitas.ayres|freitas.a-g-c|freitas.antonio-aThe main purposes of the paper under review are two: (1) to present a Pohozaev-Schoen identity on Asymptotically Euclidean (AE) manifolds and (2) to apply this result to obtain rigidity theorems in the case of AE generalized solitons (Ricci-solitons and Codazzi-solitons), to obtain a generalized almost-Schur-type inequality, and to prove some identities related with rigidity of static potentials on AE manifolds.
The paper is in the spirit of the papers of \textit{E. Barbosa} [Proc. Am. Math. Soc. 140, No. 12, 4319--4322 (2012; Zbl 1273.53042)] and \textit{E. Barbosa} et al. [Commun. Anal. Geom. 28, No. 2, 223--242 (2020; Zbl 1441.53035)].
Although the paper is very technical, it is very well written and motivated.
Reviewer: Fernando Etayo Gordejuela (Santander)An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Lower bounds for Cauchy data on curves in a negatively curved surfacehttps://www.zbmath.org/1483.530602022-05-16T20:40:13.078697Z"Galkowski, Jeffrey"https://www.zbmath.org/authors/?q=ai:galkowski.jeffrey"Zelditch, Steve"https://www.zbmath.org/authors/?q=ai:zelditch.steveSummary: We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.A differential perspective on gradient flows on \(\mathbf{\mathsf{CAT}} (\kappa)\)-spaces and applicationshttps://www.zbmath.org/1483.530612022-05-16T20:40:13.078697Z"Gigli, Nicola"https://www.zbmath.org/authors/?q=ai:gigli.nicola"Nobili, Francesco"https://www.zbmath.org/authors/?q=ai:nobili.francescoAuthors' abstract: We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on \(\mathsf{CAT} (\kappa)\)-spaces and prove that they can be characterized by the same differential inclusion \(y_t^{\prime}\in -\partial^- \mathsf{E} (y_t)\) one uses in the smooth setting and more precisely that \(y_t^{\prime}\) selects the element of minimal norm in \(-\partial^- \mathsf{E} (y_t)\). This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar-Schoen energy functional on the space of \(L^2\) and \(\mathsf{CAT}(0)\) valued maps: we define the Laplacian of such \(L^2\) map as the element of minimal norm in \(-\partial^- \mathsf{E}(u)\), provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is \(L^2\)-dense. Basic properties of this Laplacian are then studied.
Reviewer: Mohammed El Aïdi (Bogotá)Decompositions of the space of Riemannian metrics on a compact manifold with boundaryhttps://www.zbmath.org/1483.530622022-05-16T20:40:13.078697Z"Hamanaka, Shota"https://www.zbmath.org/authors/?q=ai:hamanaka.shotaLet \(M\) is a compact connected oriented smooth \(n\)-manifold, \(n \geq 3\), with smooth non-empty boundary \(\partial M\). The main goal of the paper is to prove analogues of the classical slice theorem of \textit{D. G. Ebin} [Proc. Sympos. Pure Math. 15, 11--40 (1970; Zbl 0205.53702)] and decomposition theorem of \textit{N. Koiso} [Osaka J. Math. 16, 423--429 (1979; Zbl 0416.58007)] for the space of all Riemannian metrics on \(M\) endowed with a fixed conformal class on \(\partial M\). We recall that the theorems of Ebin and Koiso apply to closed manifolds. The precise statement of the boundary condition used in the paper is somewhat technical but important for the proofs and results.
As an application, the author obtains rigidity results for relative constant scalar curvature metrics and gives a characterization of relative Einstein metrics.
Reviewer: Mikhail Belolipetsky (Rio de Janeiro)Reparameterization invariant distance on the space of curves in the hyperbolic planehttps://www.zbmath.org/1483.530632022-05-16T20:40:13.078697Z"Le Brigant, Alice"https://www.zbmath.org/authors/?q=ai:le-brigant.alice"Arnaudon, Marc"https://www.zbmath.org/authors/?q=ai:arnaudon.marc"Barbaresco, Frédéric"https://www.zbmath.org/authors/?q=ai:barbaresco.fredericSummary: This paper focuses on the study of time-varying paths in the two-dimensional hyperbolic space, and its aim is to define a reparameterization invariant distance on the space of such paths. We adapt the geodesical distance on the space of parameterized plane curves given by Bauer et al. to the space Imm([0,1],\( \mathbb{H} )\) of parameterized curves in the hyperbolic plane. We present a definition which enables to evaluate the difference between two curves, and show that it satisfies the three properties of a metric. Unlike the distance of Bauer et al., the distance obtained takes into account the positions of the curves, and not only their shapes and parameterizations, by including the distance between their origins.
For the entire collection see [Zbl 1470.00021].On Lie \(\mathrm{SL}(n,\mathbb{R})\)-foliationhttps://www.zbmath.org/1483.530642022-05-16T20:40:13.078697Z"Ndiaye, A."https://www.zbmath.org/authors/?q=ai:ndiaye.aissatou-mossele|ndiaye.ameth|ndiaye.abdoul-azizSummary: In this paper, we show that any compact manifold that carries a \(SL(n,\mathbb{R})\)-foliation is fibered on the circle \(S^1\). Every manifold in this paper is compact and our Lie group G is connected and simply connected.Quasicircle boundaries and exotic almost-isometrieshttps://www.zbmath.org/1483.530652022-05-16T20:40:13.078697Z"Lafont, Jean-François"https://www.zbmath.org/authors/?q=ai:lafont.jean-francois"Schmidt, Benjamin"https://www.zbmath.org/authors/?q=ai:schmidt.benjamin"van Limbeek, Wouter"https://www.zbmath.org/authors/?q=ai:van-limbeek.wouterSummary: We show that the limit set of an isometric and convex cocompact action of a surface group on a proper geodesic hyperbolic metric space, when equipped with a visual metric, is a Falconer-Marsh (weak) quasicircle. As a consequence, the Hausdorff dimension of such a limit set determines its bi-Lipschitz class. We give applications, including the existence of almost-isometries between periodic negatively curved metrics on \(\mathbb{H}^2\) that cannot be realized equivariantly.A zoo of growth functions of mapping class setshttps://www.zbmath.org/1483.530662022-05-16T20:40:13.078697Z"Manin, Fedor"https://www.zbmath.org/authors/?q=ai:manin.fedorGiven two compact piecewise Riemannian spaces \(X\) and \(Y\), one can consider the mapping class set \([X,Y]\), i.e., the set of homotopy classes of maps \(X\rightarrow Y\). It is natural to count the set \([X,Y]\) with respect to the certain geometric scale and investigate its asymptotic growth. More precisely, one can study the function
\[
f(L):=\#\{g\in [X,Y]\,|\,g\text{ can be represented by an \(L\)-Lipschitz map}\}.
\]
It was conjectured by Gromov that, if \(\pi_1(Y)\) is finite (e.g., \(Y\) is simply connected), then \(f(L)\) grows exactly polynomially. The intuition of the conjecture relies on two assumptions: certain integral homotopy classes are evenly distributed, and these integral classes are realizable by \(L\)-Lipschitz maps. The conjecture has been disproved by the author and \textit{S. Weinberger} in a previous paper [Duke Math. J. 169, No. 10, 1943--1969 (2020; Zbl 07226653)], where they constructed an example whose growth function is exactly \(L^8\log L\), which actually violates the first assumption.
In the paper under review, it is shown that the second assumption is also false in general. The author gives a large amount of examples, constructed in a similar matter, by taking \(X\) to be a wedge of two spheres with different dimensions followed by an attaching of a high-dimensional \(n\)-cell, and taking \(Y\) to be \(X\) glued with an \((n+1)\)-cell which kills \(\pi_n(X)\). These constructions in particular shows an interesting result that for any rational number \(r>4\), there exists a pair \(X,Y\) such that \(f(L)\) asymptotically grows at most \(L^r\) and at least \(L^{r-\epsilon}\) for any \(\epsilon>0\), that is,
\[
\lim_{L\rightarrow \infty}\log_L(f(L))=r.
\]
Besides, the author discusses in the last section some cases where the second assumption holds.
Reviewer: Shi Wang (Bloomington)Isoperimetric inequalities in metric measure spaces [after F. Cavalletti \& A. Mondino]https://www.zbmath.org/1483.530672022-05-16T20:40:13.078697Z"Villani, Cédric"https://www.zbmath.org/authors/?q=ai:villani.cedricSummary: The synthetic theory of Ricci curvature in metric measure spaces obtained its first successes a decade ago, and grew rapidly since then; but it was stumbling on questions which were as fundamental as they appeared tricky, such as the Lévy-Gromov inequality or other geometric inequalities where the effective dimension and the optimal constants are crucial. The recent works of Cavalletti and Mondino, adapting the localization techniques of Klartag, allow to bypass these obstacles, and in particular prove the first nonsmooth version of the Lévy-Gromov inequality.
For the entire collection see [Zbl 1416.00029].Critical point equation on 3-dimensional trans-Sasakian manifoldshttps://www.zbmath.org/1483.530682022-05-16T20:40:13.078697Z"Dey, Dibakar"https://www.zbmath.org/authors/?q=ai:dey.dibakarSummary: The object of the present paper is to characterize \(3\)-dimensional trans-Sasakian manifolds satisfying the critical point equation under the condition \(\phi \operatorname{grad}\alpha = \operatorname{grad}\beta \). Also, we present few examples which verifies our results.Canonical identification at infinity for Ricci-flat manifoldshttps://www.zbmath.org/1483.530692022-05-16T20:40:13.078697Z"Park, Jiewon"https://www.zbmath.org/authors/?q=ai:park.jiewonSummary: We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.Quaternionic contact \(4n+3\)-manifolds and their \(4n\)-quotientshttps://www.zbmath.org/1483.530702022-05-16T20:40:13.078697Z"Kamishima, Yoshinobu"https://www.zbmath.org/authors/?q=ai:kamishima.yoshinobuThe author studies quaternionic contact structure (``qc-structure'' for short) on \((4n+3)\)-manifolds to construct quaternionic Hermitian \(4n\)-manifolds as their quotient.
The qc-structure was first introduced by \textit{O. Biquard} [Métriques d'Einstein asymptotiquement symétriques. Paris: Société Mathématique de France (2000; Zbl 0967.53030)], together with a canonical connection (now known as ``Biquard connection''). A qc-structure on a \((4n+3)\)-manifold \(X\) is a distribution \(D\subset TX\) of codimension 3, such that \(D\) admits a \(n\)-dimensional quaternionic structure \(Q\).
In a previous work of the author with \textit{D. Alekseevsky} [Ann. Mat. Pura Appl. (4) 187, No. 3, 487--529 (2008; Zbl 1223.53054)], a special case of qc-structures, called ``quaternionic CR-structure'' was studied, which corresponds to qc-Einstein manifolds with nonzero qc-scalar curvatures. In this paper, the author studies the complementing vanishing qc-scalar curvature case, which has an equivalent description as ``strict qc-structure''.
In the first part of the paper, the author construct a family of simply connected strict qc solvable Lie groups \(\mathcal{M}_{k,l}\), starting with the standard quaternionic Heisenberg nilpotent Lie group \(\mathcal{M}\). These are then characterized as the only contractible unimodular strict qc-groups. Then, both compact and non-compact uniformizable strict qc-manifolds are classified modeling on \(\mathcal{M}\).
In the second part, the author uses the quotients of a conformal deformation of the standard qc-structure on \(\mathcal{M}\) to construct a family of quaternionic Hermitian metrics on a domain of the standard quaternion space, one of which is a Bochner-flat Kähler metric.
Reviewer: Yalong Shi (Nanjing)Twistors, self-duality, and \(\text{spin}^c\) structureshttps://www.zbmath.org/1483.530712022-05-16T20:40:13.078697Z"LeBrun, Claude"https://www.zbmath.org/authors/?q=ai:lebrun.claude-rSummary: The fact that every compact oriented 4-manifold admits \(\text{spin}^c\) structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and \(\text{spin}^c\) structures in any dimension.A formula for the heat kernel coefficients of the Dirac Laplacians on spin manifoldshttps://www.zbmath.org/1483.530722022-05-16T20:40:13.078697Z"Nagase, Masayoshi"https://www.zbmath.org/authors/?q=ai:nagase.masayoshi"Shirakawa, Takumi"https://www.zbmath.org/authors/?q=ai:shirakawa.takumiSummary: Based on Getzler's rescaling transformation, we obtain a formula for the heat kernel coefficients of the Dirac Laplacian on a spin manifold. One can compute them explicitly up to an arbitrarily high order by using only a basic knowledge of calculus added to the formula.Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaceshttps://www.zbmath.org/1483.530732022-05-16T20:40:13.078697Z"Díaz-Ramos, José Carlos"https://www.zbmath.org/authors/?q=ai:diaz-ramos.jose-carlos"Domínguez-Vázquez, Miguel"https://www.zbmath.org/authors/?q=ai:dominguez-vazquez.miguel"Rodríguez-Vázquez, Alberto"https://www.zbmath.org/authors/?q=ai:rodriguez-vazquez.albertoWe recall first the definition of a cohomogeneity-one action on a Riemannian manifold: this is a proper isometric action whose principal orbits are hypersurfaces. This paper is an important contribution to the folowing problem: find all cohomogeneity one actions on a given Riemannian manifold, or, equivalently, classify the homogeneous hypersurfaces up to isometric congruence. The case of Riemannian symmetric spaces of noncompact type and rank one has been considered. The problem has been solved by \textit{É. Cartan} [Ann. Mat. Pura Appl. (4) 17, 177--191 (1938; JFM 64.1361.02); Ann. Mat. Pura Appl. (4) 17, 177--191 (1938; Zbl 0020.06505)] in the case of a real hyperbolic space. It has been solved by \textit{J. Berndt} and \textit{H. Tamaru} [Trans. Am. Math. Soc. 359, No. 7, 3425--3438 (2007; Zbl 1117.53041)] in the case of a complex hyperbolic space and in the case of the Cayley hyperbolic space. But the case of a quaternionic hyperbolic space remainded unsolved until the authors obtain the results which are the subject of the present paper.
The problem leads to a linear algebraic problem about the protohomogeneous subspaces \(V\) in the quaternonic space \(\mathbb{H}^n\). The real subspace \(V\) is said to be protohomogeneous if there exists a connected Lie subgroup of \(\mathrm{Sp}(1)\mathrm{Sp}(n)\) that acts transitively on the unit sphere of \(V\). In order to study such a subspace one considers the quaternionic Kähler angle of \(V\) which is a triple \((\varphi _1,\varphi _2,\varphi _3)\) of real numbers. This concept plays an important role in this topic. Let \(\mathcal{I}\) be the subspace of real endomorphisms of \(\mathbb{H}^n\) generated by right multiplication by the elements \(i,j,k\) of \(\mathbb{H}\), and, for \(v\in V\), consider the symmetric bilinear form \(L_v\) defined on \(\mathcal{I}\) by \[L_v(J,J')=\langle P_J v,P_{J'}v\rangle\quad (J,J'\in \mathcal{J}),\] and \(P_J=\pi _V\circ J\), where \(\pi _V\) is the orthogonal projection on \(V\). One says that \(V\) has constant quaternionic Kähler angle \((\varphi _1,\varphi _2,\varphi _3)\) if, for any \(v\in V\), the symmetric bilinear form \(L_v\) has eigenvalues \(\cos ^2 \varphi _1\), \(\cos ^2 \varphi _2\), \(\cos ^2 \varphi _3\). A protohomogeneous subspace \(V\) of \(\mathbb{H}^n\) has constant quaternionic angle. The first main result is the classification, up to congruence by elements in \(\mathrm{Sp}(1)\mathrm{Sp}(n)\), of the protohomogeneous subspaces of \(\mathbb{H}^n\) with their quaternionic Kähler angles. The largest part of the paper is devoted to the proof of this classification. Then, as a consequence, the classification of a cohomogeneity-one actions on a quaternionic hyperbolic space is obtained.
Reviewer: Jacques Faraut (Paris)On the equivariant cohomology of hyperpolar actions on symmetric spaceshttps://www.zbmath.org/1483.530742022-05-16T20:40:13.078697Z"Goertsches, Oliver"https://www.zbmath.org/authors/?q=ai:goertsches.oliver"Hagh Shenas Noshari, Sam"https://www.zbmath.org/authors/?q=ai:haghshenas-noshari.sam"Mare, Augustin-Liviu"https://www.zbmath.org/authors/?q=ai:mare.augustin-liviuThe authors generalize previous results by showing that any hyperpolar action of a compact connected Lie group on a symmetric space of compact type is Cohen-Macaulay. The proof relies essentially on the classification of hyperploar actions obtained by \textit{A. Kollross} [Transform. Groups 22, No. 1, 207--228 (2017; Zbl 1383.53041)].
Reviewer: Oliver Jones (Camarillo)A two-radii theorem for weighted ball means on symmetric spaceshttps://www.zbmath.org/1483.530752022-05-16T20:40:13.078697Z"Volchkov, V. V."https://www.zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://www.zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: Generalizations of functions with zero ball means on Riemannian symmetric spaces \(X = G/K\) are studied. An analog of the local two-radii theorem for symmetric spaces of noncompact type with a complex group \(G\) is obtained.Classification theorems of complete space-like Lagrangian \(\xi\)-surfaces in the pseudo-Euclidean space \(\mathbb{R}^4_2\)https://www.zbmath.org/1483.530762022-05-16T20:40:13.078697Z"Li, Xingxiao"https://www.zbmath.org/authors/?q=ai:li.xingxiao"Qiao, Ruina"https://www.zbmath.org/authors/?q=ai:qiao.ruina"Liu, Yangyang"https://www.zbmath.org/authors/?q=ai:liu.yangyangSummary: \(\xi\)-submanifolds and \(\xi\)-translators are, respectively, the natural generalizations of self-shrinkers and translators of the mean curvature flow and, in the case of codimension one, they are previously known as \(\lambda\)-hypersurfaces and \(\lambda\)-translators, respectively. In this paper, we study the complete Lagrangian space-like \(\xi\)-surfaces and \(\xi\)-translators in \(\mathbb{R}^4_2\), the pseudo-Euclidean 4-spaces of signature 2 endowed with the canonical complex structure. As the result, we first obtain a classification theorem for all complete Lagrangian space-like \(\xi\)-surfaces in \(\mathbb{R}^4_2\) of constant square norm of the second fundamental form. Then the main idea of the proof also allows us to obtain a similar classification theorem for \(\xi\)-translators in \(\mathbb{R}^4_2\) by a Bernstein-type theorem for space-like translators in a general pseudo-Euclidean space \(\mathbb{R}^{m+p}_p\), which is of independent significance.Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space formshttps://www.zbmath.org/1483.530772022-05-16T20:40:13.078697Z"Loi, Andrea"https://www.zbmath.org/authors/?q=ai:loi.andrea"Mossa, Roberto"https://www.zbmath.org/authors/?q=ai:mossa.robertoThe authors consider Kähler manifolds \((M,g)\) that can be immersed in a complex space form, i.e., a Kähler manifold \((S,g_S)\) of constant holomorphic sectional curvature. The main result is that if the Kähler metric \(g\) is a \textit{Kähler-Ricci soliton}, then it is in fact a Kähler-Einstein metric.
Recall that a Kähler-Ricci soliton on a complex manifold \(M\) is a pair \((g,X)\) of a Kähler metric \(g\) and a holomorphic vector field \(X\) such that \[\mathrm{Ric}(g)=\lambda\,g+\mathcal{L}_Xg \] for some real constant \(\lambda\). We can restate the main result of the paper more explicitly as
Theorem 1. Let \((g,X)\) be a Kähler-Ricci soliton on the complex manifold \(M\). If \((M,g)\) can be immersed in a complex space form, then \(g\) is Kähler-Einstein, i.e., \(\mathrm{Ric}(g)=\lambda\,g\).
It is important to notice that no further assumptions, such as compactness or projectivity, are made on \(M\), so that Theorem 1 is a substantial generalization of previous similar results, such as [\textit{L. Bedulli} and \textit{A. Gori}, Proc. Am. Math. Soc. 142, No. 5, 1777--1781 (2014; Zbl 1292.32006); Adv. Geom. 15, No. 2, 167--172 (2015; Zbl 1311.32010)].
There are two main technical tools in the proof of Theorem 1: the first is the \textit{diastasis function}, a local potential for the Kähler metric \(g\) that is particularly useful to investigate Kähler immersions in complex space forms. It was introduced by \textit{E. Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)], we refer to [\textit{A. Loi} and \textit{M. Zedda}, Kähler immersions of Kähler manifolds into complex space forms. Cham: Springer (2018; Zbl 1432.32032)] for the main properties of the diastasis function and some interesting examples of its use. The second main tool is Umehara's algebra, an \(\mathbb{R}\)-algebra \(\Lambda_p\) of germs of functions for a point \(p\in M\) introduced in [\textit{M. Umehara}, J. Math. Soc. Japan 40, No. 3, 519--539 (1988; Zbl 0651.53046)]. The authors prove, in Proposition 3.1, a particular relation between the diastasis function and Umehara's algebra, that is what ultimately allows one to prove Theorem 1.
Reviewer: Carlo Scarpa (Trieste)Around Efimov's differential test for homeomorphismhttps://www.zbmath.org/1483.530782022-05-16T20:40:13.078697Z"Alexandrov, Victor"https://www.zbmath.org/authors/?q=ai:alexandrov.victor-aThere is a famous result due to Efimov, more precisely the following Theorem: No surface can be \(C^2\)-immersed in Euclidean 3-space so as to be complete in the induced Riemannian metric, with Gauss curvature \(K \le \) constant \(< 0\).
The paper under review starts with a mini-survey of results related to the previous theorem.
Among other things, Efimov established that the condition \(K \le\) constant \(< 0\) is not the only obstacle for the immersibility of a complete surface of negative curvature; he showed that a rather slow change of Gauss curvature is another obstacle. In all those numerous articles, he used to a large extent one and the same method based on the study of the spherical image of a surface. At that study, an essential role belongs to statements that, under some conditions, a locally homeomorphic mapping \(f : \mathbb{R}^2 \to \mathbb{R}^2\) is a global homeomorphism and \(f(\mathbb{R}^2)\) is a convex domain in \(\mathbb{R}^2\).
Two other theorems of Efimov are recalled in the present paper and the author gives an overview on the analogues of these theorems, their generalizations and applications. The article is devoted to presentation of results motivated by the theory of surfaces, the theory of global inverse function, the Jacobian Conjecture, and the global asymptotic stability of dynamical systems, respectively.
Reviewer: Adela-Gabriela Mihai (Bucureşti)New developments on the p-Willmore energy of surfaceshttps://www.zbmath.org/1483.530792022-05-16T20:40:13.078697Z"Aulisa, Eugenio"https://www.zbmath.org/authors/?q=ai:aulisa.eugenio"Gruber, Anthony"https://www.zbmath.org/authors/?q=ai:gruber.anthony"Toda, Magdalena"https://www.zbmath.org/authors/?q=ai:toda.magdalena"Tran, Hung"https://www.zbmath.org/authors/?q=ai:tran.hung-tuan|tran-hung.|tran.hung-thanh|tran-cong-hung.|tran.hung-vinhThe Willmore energy is an important conformal invariant of immersed surfaces \(\Sigma\) in the Euclidean 3-space. The article surveys some recent results [the second author et al., Ann. Global Anal. Geom. 56, No. 1, 147--165 (2019; Zbl 1417.58009)] on the \(p\)-Willmore energy \(W^p(\Sigma) = \int_\Sigma H^p dS\) (i.e., different powers of the mean curvature \(H\) of \(\Sigma\) in the integrand) for nonnegative integer \(p\) of surfaces (with boundary) in 3-dimensional space forms. Section 2 gives the first and second variations of \(W^p\), and presented a flux formula, which reveals (in Section 3) a connection between its critical points and the minimal surfaces. Section 4 reformulates the \(p\)-Willmore flow problem and presents (and visualizes through computer implementation) a model for the \(p\)-Willmore flow of graphs.
For the entire collection see [Zbl 1445.53003].
Reviewer: Vladimir Yu. Rovenskij (Nesher)On free boundary minimal hypersurfaces in the Riemannian Schwarzschild spacehttps://www.zbmath.org/1483.530802022-05-16T20:40:13.078697Z"Barbosa, Ezequiel"https://www.zbmath.org/authors/?q=ai:barbosa.ezequiel-r"Espinar, José M."https://www.zbmath.org/authors/?q=ai:espinar.jose-mariaAuthors' abstract: In contrast with the three-dimensional case (cf. Montezuma in Bull Braz Math Soc), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the \(n\)-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\geq 4\). Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\geq 8\), with Morse index equal to 0. In addition, it is shown that, for \(n\geq 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.
Reviewer: Mohammad Nazrul Islam Khan (Buraidah)Gauss map and the topology of constant mean curvature hypersurfaces of \(\mathbb{S}^7\) and \(\mathbb{CP}^3 \)https://www.zbmath.org/1483.530812022-05-16T20:40:13.078697Z"Bittencourt, Fidelis"https://www.zbmath.org/authors/?q=ai:bittencourt.fidelis"Fusieger, Pedro"https://www.zbmath.org/authors/?q=ai:fusieger.pedro"Longa, Eduardo R."https://www.zbmath.org/authors/?q=ai:longa.eduardo-rosinato"Ripoll, Jaime"https://www.zbmath.org/authors/?q=ai:ripoll.jaime-bruckHaving in mind the classifications of the images of complete minimal surfaces in $\mathbb{R}^3$ by the Gauss map as well as the images of surfaces of constant mean curvatures with the constraint that these images are included in hemispheres, the authors of this paper extend these studies to the case of a Gaussian map $\gamma:M\to S^6$ of an oriented hypersurface $M$ of the unit sphere $S^7$. They show that $\gamma$ is harmonic if and only if $M$ has constant mean curvature. They establish interesting results on the topology and the geometry of the surfaces of constant mean curvatures of the sphere $S^7$, under a hypothesis on the image. By a symmetrization process of Hopf, they define a Gauss map for the hypersurfaces of the projective space $\mathbb{CP}^3$ and they obtain similar results for the constant mean curvature hypersurfaces of this space.
Reviewer: Mohammed Benalili (Tilimsān)Minimal surfaces under constrained Willmore transformationhttps://www.zbmath.org/1483.530822022-05-16T20:40:13.078697Z"Casinhas Quintino, Áurea"https://www.zbmath.org/authors/?q=ai:quintino.aurea-casinhasSummary: The class of constrained Willmore (CW) surfaces in space-forms constitutes a Möbius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation
[\textit{F. Burstall} et al., Contemp. Math. 308, 39--61 (2002; Zbl 1031.53026)],
defined by the action of a loop of flat metric connections, and Bäcklund transformations
[\textit{F. E. Burstall} and the author, Commun. Anal. Geom. 22, No. 3, 469--518 (2014; Zbl 1306.53051)],
defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are
[\textit{J. Richter}, Conformal maps of a Riemannian surface into the space of quaternions. Berlin: TU Berlin, FB Mathematik (1997; Zbl 0896.53005)]
examples of CW surfaces, characterized by the existence of some polynomial conserved quantity
[the author, Constrained Willmore surfaces: symmetries of a Möbius invariant integrable system. University of Bath (PhD Thesis) (2008);
Constrained Willmore surfaces. Symmetries of a Möbius invariant integrable system (to appear). Cambridge: Cambridge University Press (2021; Zbl 07298516);
the author and \textit{S. Duarte Santos}, ``Polynomial conserved quantities for constrained Willmore surfaces'', Preprint, \url{arXiv:1507.01253}].
Both CW spectral deformation and CW Bäcklund transformation preserve the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. A classical result by
\textit{G. Thomsen} [Abh. Math. Semin. Univ. Hamb. 3, 31--56 (1923; JFM 49.0530.02)]
characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation. We define, in this way, a CW spectral deformation and CW Bäcklund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.
For the entire collection see [Zbl 1473.53006].\(\Phi\)-harmonic maps and \(\Phi\)-superstrongly unstable manifoldshttps://www.zbmath.org/1483.530832022-05-16T20:40:13.078697Z"Han, Yingbo"https://www.zbmath.org/authors/?q=ai:han.yingbo"Wei, Shihshu Walter"https://www.zbmath.org/authors/?q=ai:wei.shihshu-walterSummary: We motivate and define \(\Phi\)-energy density, \(\Phi\)-energy, \(\Phi\)-harmonic maps and stable \(\Phi\)-harmonic maps. Whereas harmonic maps or \(p\)-harmonic maps can be viewed as critical points of the integral of the first symmetric function \(\sigma_1\) of a pull-back tensor, \(\Phi\)-harmonic maps can be viewed as critical points of the integral of the second symmetric function \(\sigma_2\) of a pull-back tensor. By an extrinsic average variational method in the calculus of variations
(cf.
[\textit{R. Howard} and \textit{S. W. Wei}, Trans. Am. Math. Soc. 294, 319--331 (1986; Zbl 0588.58015);
\textit{S. W. Wei} and \textit{C.-M. Yau}, J. Geom. Anal. 4, No. 2, 247--272 (1994; Zbl 0851.58014);
\textit{S. W. Wei}, Indiana Univ. Math. J. 47, No. 2, 625--670 (1998; Zbl 0930.58010);
\textit{R. Howard} and \textit{S. W. Wei}, Contemp. Math. 646, 127--167 (2015; Zbl 1361.53047)]
),
we derive the average second variation formulas for \(\Phi\)-energy functional, express them in orthogonal notation in terms of the differential matrix, and find \(\Phi\)-superstrongly unstable \((\Phi\)-SSU manifolds. We prove, in particular that every compact \(\Phi\)-SSU manifold must be \(\Phi\)-strongly unstable (\(\Phi\)-SU), i.e., (a) A compact \(\Phi\)-SSU manifold cannot be the target of any nonconstant stable \(\Phi\)-harmonic maps from any manifold, (b) The homotopic class of any map from any manifold into a compact \(\Phi\)-SSU manifold contains elements of arbitrarily small \(\Phi\)-energy, (c) A compact \(\Phi\)-SSU manifold cannot be the domain of any nonconstant stable \(\Phi\)-harmonic map into any manifold, and (d) The homotopic class of any map from a compact \(\Phi\)-SSU manifold into any manifold contains elements of arbitrarily small \(\Phi\)-energy [cf. Theorem 1.1(a),(b),(c), and (d).] We provide many examples of \(\Phi\)-SSU manifolds, which include but not limit to spheres or some unstable Yang-Mills fields
(cf.
[\textit{J.-P. Bourguignon} et al., Proc. Natl. Acad. Sci. USA 76, 1550--1553 (1979; Zbl 0408.53023);
\textit{J.-P. Bourguignon} and \textit{H. B. Lawson jun.}, Commun. Math. Phys. 79, 189--230 (1981; Zbl 0475.53060);
\textit{S. Kobayashi} et al., Math. Z. 193, 165--189 (1986; Zbl 0634.53022);
\textit{S. W. Wei}, Indiana Univ. Math. J. 33, 511--529 (1984; Zbl 0559.53027);
\textit{L. Wu} et al., ``Discovering geometric and topological properties of ellipsoids by curvatures'', Br. J. Math. Comput. Sci. 8, No. 4, 318--329 (2015)]), and examples of \(\Phi\)-harmonic, or \(\Phi\)-unstable map from or into \(\Phi\)-SSU manifold that are not constant. We establish a link of \(\Phi\)-SSU manifold to \(p\)-SSU manifold and topology. The extrinsic average variational method in the calculus of variations, employed is in contrast to an average method in PDE that we applied in
[\textit{B.-Y. Chen} and \textit{S. W. Wei}, J. Geom. Symmetry Phys. 52, 27--46 (2019; Zbl 1427.53046)] to obtain sharp growth estimates for warping functions in multiply warped product manifolds.On the generalized of \(p\)-harmonic and \(f\)-harmonic mapshttps://www.zbmath.org/1483.530842022-05-16T20:40:13.078697Z"Remli, Embarka"https://www.zbmath.org/authors/?q=ai:remli.embarka"Cherif*, Ahmed Mohammed"https://www.zbmath.org/authors/?q=ai:cherif.ahmed-mohammedSummary: In this paper, we extend the definition of \(p\)-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized \(p\)-harmonic maps. We present some new properties for the generalized stress \(p\)-energy tensor. We also prove that every generalized \(p\)-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.A Pu-Bonnesen inequalityhttps://www.zbmath.org/1483.530852022-05-16T20:40:13.078697Z"Katz, Mikhail G."https://www.zbmath.org/authors/?q=ai:katz.mikhail-g"Sabourau, Stéphane"https://www.zbmath.org/authors/?q=ai:sabourau.stephaneThe classical Pu systolic inequality establishes that
\[
\mathrm{area}(g)-\frac{2}{\pi}\mathrm{sys}(g)^2\geq 0
\]
for every Riemannian metric \(g\) on the real projective plane \({\mathbb R}{\mathbb P}^2\). In the paper under review the authors prove a Bonnesen-type inequality that generalizes Pu's inequality, in terms of the circumscribed \(R\) and inscribed \(r\) radii of the Euclidean embedding of the orientable double cover \(S_g\subset{\mathbb R}^3\). More precisely, they show that if \(({\mathbb R}{\mathbb P}^2,g)\) has positive Gaussian curvature, then there exists a monotone continuous function \(\lambda(t)>0\) for \(t>0\) such that
\[
\dfrac{\mathrm{area}(g)}{\mathrm{sys}(g)^2}-\dfrac{2}{\pi}\geq\lambda\left(\dfrac{R-r}{\mathrm{ sys}}\right),
\]
where \(\lambda(t)\) is asymptotically linear as \(t\to\infty\).
The authors present two different proofs of this result, using an extrinsic argument and an intrinsic approach. The first one exploits powerful tools from convexity such as John's ellipsoid theorem or Blaschke's selection theorem.
Reviewer: Maria A. Hernández Cifre (Murcia)On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in general relativityhttps://www.zbmath.org/1483.530862022-05-16T20:40:13.078697Z"Cederbaum, Carla"https://www.zbmath.org/authors/?q=ai:cederbaum.carla"Sakovich, Anna"https://www.zbmath.org/authors/?q=ai:sakovich.annaThe authors define a new total center of mass for an ``isolated system'': the ``Universe'' is a 4-dimensional Lorentzian manifold \((\mathfrak M^{1,3}, \mathfrak{g})\), endowed with an energy-momentum tensor field \(\mathfrak T\), with an ``initial data set'' given by a spacelike hypersuface \((M^3,g)\), with the second fundamental form \(K\), the scalar local energy density \(\mu\) and the (1-form) local momentum density \(J\). When this configuration is ``asymptotically Euclidean'' and with non-vanishing energy, it gives rise to a (unique) foliation by 2-spheres of constant spacetime mean curvature. This foliation is the main tool for constructing the total center of mass. It is shown that this center of mass behaves as a point particle in Special Relativity (i.e. it transforms equivariantly under the asymptotic Poincaré group of \({\mathfrak M^{1,3}}\)). In particular, it evolves in time under the Einstein evolution equations like a point particle in Special Relativity.
Reviewer: Gabriel Teodor Pripoae (Bucureşti)Erratum to: ``Uniform K-stability and asymptotics of energy functionals in Kähler geometry''https://www.zbmath.org/1483.530872022-05-16T20:40:13.078697Z"Boucksom, Sébastien"https://www.zbmath.org/authors/?q=ai:boucksom.sebastien"Hisamoto, Tomoyuki"https://www.zbmath.org/authors/?q=ai:hisamoto.tomoyuki"Jonsson, Mattias"https://www.zbmath.org/authors/?q=ai:jonsson.mattiasSummary: The goal of this note is to indicate a gap in the proof of Theorem 5.6 of the authors' paper [ibid. 21, No. 9, 2905--2944 (2019; Zbl 1478.53115)], and the consequences it has on other results in the same paper. Let
us stress that the main result (Theorem A), which expresses the slopes at infinity of functionals in algebro-geometric terms, is independent of the flawed result, and thus remains valid.Extremal Kähler metrics induced by finite or infinite-dimensional complex space formshttps://www.zbmath.org/1483.530882022-05-16T20:40:13.078697Z"Loi, Andrea"https://www.zbmath.org/authors/?q=ai:loi.andrea"Salis, Filippo"https://www.zbmath.org/authors/?q=ai:salis.filippo"Zuddas, Fabio"https://www.zbmath.org/authors/?q=ai:zuddas.fabioThe authors contribute to the study of extremal metrics on complex manifolds that are induced by finite or infinite-dimensional complex space forms.
In the case where the extremal metric \(g\) on the complex mainfold \(M\) is induced by an immersion in a finite-dimensional ambient space, they state a conjecture describing \(M\) depending on the sign of the constant holomorphic sectional curvature of the ambient space. They then prove this conjecture under the assumption that \(g\) is a radial metric, i.e., it admits a Kähler potential that depends only on the norm of the local coordinates.
The authors then extend this result to the infinite-dimensional setting by imposing additional conditions on the metric \(g\), namely that it has constant scalar curvature and that the radial potential behaves well. They then analyze the radial Kähler-Einstein metrics induced by infinite-dimensional elliptic complex space forms, showing that they have constant non-positive holomorphic sectional curvature under a reasonable stability condition. Notably, they also give a few counterexamples that validate their choice of assumptions.
Reviewer: Miron Stanciu (Bucureşti)A note on the Sagnac effect in general relativity as a Finslerian effecthttps://www.zbmath.org/1483.530892022-05-16T20:40:13.078697Z"Caponio, Erasmo"https://www.zbmath.org/authors/?q=ai:caponio.erasmo"Masiello, Antonio"https://www.zbmath.org/authors/?q=ai:masiello.antonioSummary: The geometry of the Sagnac effect in a stationary region of a spacetime is reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric defined on a spacelike hypersurface associated to a stationary splitting and related to future-pointing null geodesics of the spacetime. We show also that an analogous asymmetry comes into play in the Sagnac effect for timelike geodesics.Compact hypersurfaces in Randers spacehttps://www.zbmath.org/1483.530902022-05-16T20:40:13.078697Z"Li, Jintang"https://www.zbmath.org/authors/?q=ai:li.jintangThe author of the present paper proved that if the second mean curvature \(H_2\) is constant and the norm square \(S\) of the second fundamental form of \(M\) satisfies a certain inequality, then either \(M\) is a Randers space with constant flag curvature \(1 + H_2\) or the equality holds. The results obtained by him are interesting.
Reviewer: V. K. Chaubey (Gorakhpur)Geometric flux formula for the gravitational Wilson loophttps://www.zbmath.org/1483.530912022-05-16T20:40:13.078697Z"Klitgaard, N."https://www.zbmath.org/authors/?q=ai:klitgaard.n"Loll, R."https://www.zbmath.org/authors/?q=ai:loll.renate"Reitz, M."https://www.zbmath.org/authors/?q=ai:reitz.m"Toriumi, R."https://www.zbmath.org/authors/?q=ai:toriumi.reikoThis paper is concerned with finding quantities, constructed out of gravitational holonomies of the Levi-Civita connection, that can be used to obtain information about the curvature of a given manifold. The authors motivate the search by possible applications in nonperturbative quantum gravity, e.g. in the causal dynamical triangulation approach. The main idea is to use Wilson loops as a basic tool and formulate a certain type of generalized Stokes theorem.
The authors first revisit holonomies for infinitesimal loops on Riemannian manifolds and discuss certain nonabelian version of the Stokes theorem for finite (contractible) loops which are boundaries of a given surface \(S\). Schematically, it can be written as
\[
P e^{-\oint_{\partial S}\Gamma}=\mathcal{P} e^{-\int_S \widetilde{R}},
\]
where \(P\) is the path-ordering and \(\mathcal{P}\) is a specific type of surface-ordering considered by the authors. Here, again schematically, \(\Gamma\) denotes the Levi-Civita connection with curvature \(R\), and \(\widetilde{R}\) is the curvature \(R\) parallely-transported to a chosen base point of the surface \(S\). The right hand side is therefore a highly nonlocal object that encodes information about the curvature. Furthermore, the non-uniqueness of the choice of a surface-ordering leads to problems with the interpretation of this quantity. It is therefore natural to ask for a similar relation between holonomies for finite loops and averaging curvature-like quantities, which is free of this issue even for the case with nonabelian holonomy group.
The authors show that there is a class of manifolds, possessing foliation by a family of totally geodesic surfaces, for which this problem can be solved. They study three and four dimensional manifolds with hypersurface-orthogonal Killing vector fields, whose presence is suficient for the existence of totally geodesic surfaces. The authors then find associated geometric fluxes and use them to prove (with the help of the classical Stokes theorem) that the quantity, defined in terms of the integral from the curvature-like quantity over the surface being the boundary of the given loop, is invariant under smooth deformations of this surface that leave the boundary untouched.
Reviewer: Arkadiusz Bochniak (Kraków)Hamiltonian circle actions with almost minimal isolated fixed pointshttps://www.zbmath.org/1483.530922022-05-16T20:40:13.078697Z"Li, Hui"https://www.zbmath.org/authors/?q=ai:li.hui.3|li.hui|li.hui.2|li.hui.4|li.hui.1|li.hui.5Consider a Hamiltonian action of the circle on a connected compact symplectic manifold \((M, \omega)\) of dimension \(2n\). Then this circle action has at least \(n + 1\) fixed points. Having previously discussed the case when the fixed point set consists of exactly \(n + 1\) isolated points in [``Hamiltonian circle actions with minimal isolated fixed points'', Preprint, \url{arXiv:1407.1948}], the author discusses in this work the case when the fixed point set consists of exactly \(n+2\) isolated points. The first observation is that in such a case \(n\) must be even.
The main result of this work describes conditions on the image of the fixed point set by the moment map, which imply that the cohomology ring and the total Chern class of \(M\) are isomorphic to that of \(\widetilde G_2(\mathbb P^{n+2})\) and that the sets of weights of the circle action on \(M\) are those of a standard action on \(\widetilde G_2(\mathbb P^{n+2})\).
Reviewer: Elizabeth Gasparim (Antofagasta)Extensions of quasi-morphisms to the symplectomorphism group of the diskhttps://www.zbmath.org/1483.530932022-05-16T20:40:13.078697Z"Maruyama, Shuhei"https://www.zbmath.org/authors/?q=ai:maruyama.shuheiSummary: In this paper, we construct quasi-morphisms on the group of symplectomorphisms of the closed disk \(D\). These quasi-morphisms are extensions of the Ruelle invariant and Gambaudo-Ghys quasi-morphisms. As a corollary, we show that the second bounded cohomology \(H_b^2(\operatorname{Symp}(D))\) is infinite dimensional.Lagrangian manifolds and efficient short-wave asymptotics in a neighborhood of a caustic cusphttps://www.zbmath.org/1483.530942022-05-16T20:40:13.078697Z"Dobrokhotov, S. Yu."https://www.zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nazaikinskii, V. E."https://www.zbmath.org/authors/?q=ai:nazaikinskii.vladimir-eSummary: We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type \(A_3\) (standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.Uniqueness of real Lagrangians up to cobordismhttps://www.zbmath.org/1483.530952022-05-16T20:40:13.078697Z"Kim, Joontae"https://www.zbmath.org/authors/?q=ai:kim.joontaeAn antisymplectic involution on a symplectic manifold is an involution which acts on the symplectic structure in an antisymmetric fashion. The nonempty fixed point sets of such involutions are necessarily Lagrangian. If a Lagrangian submanifold is the fixed point set of an antisymplectic involution, it is called real. The main result of the present article is that any two real Lagrangians in a closed symplectic manifold are smoothly cobordant; i.e. they represent the same cobordism class in the nonoriented Thom cobordism ring (see e.g. [\textit{C. T. C. Wall}, Ann. Math. (2) 72, 292--311 (1960; Zbl 0097.38801)] for a complete algebraic description). In case the Lagrangian is not real, this claim need not be true in general. The proof of the main result is through the observation that the cobordism class of the symplectic manifold containing the real Lagrangian \(L\) is equal to the class of \(L \times L\). It follows immediately from the main result that if two real Lagrangians \(L_1\) and \(L_2\) in \(M_1\) and \(M_2\) respectively are cobordant, then \(M_1\) and \(M_2\) are cobordant too.
The main result and various previous results in the literature (e.g. those coming from Smith theory) are brought together smartly in the article to conclude interesting observations regarding real Lagrangians. For example, any real Lagrangian in \(\mathbb{C}P^2\) is Hamiltonian isotopic to \(\mathbb{R}P^2\). As for real Lagrangians in \(S^2\times S^2\), \(L\) is either Hamiltonian isotopic to the antidiagonal sphere, or Lagrangian isotopic to the Clifford torus \(S^1\times S^1\subset S^2\times S^2\). Let us note that the general Lagrangian classification problem is much more complicated. Along that direction the author collects basic examples and diverse counter-examples.
The article is written clearly and is easy-to-read.
Reviewer: Ferit Öztürk (İstanbul)A study of new class of almost contact metric manifolds of Kenmotsu typehttps://www.zbmath.org/1483.530962022-05-16T20:40:13.078697Z"Abood, Habeeb M."https://www.zbmath.org/authors/?q=ai:abood.habeeb-mtashar"Abass, Mohammed Y."https://www.zbmath.org/authors/?q=ai:abass.mohammed-ySummary: In this paper, we characterized a new class of almost contact metric manifolds and established the equivalent conditions of the characterization identity in term of Kirichenko's tensors. We demonstrated that the Kenmotsu manifold provides the mentioned class; i.e., the new class can be decomposed into a direct sum of the Kenmotsu and other classes. We proved that the manifold of dimension 3 coincided with the Kenmotsu manifold and provided an example of the new manifold of dimension 5, which is not the Kenmotsu manifold. Moreover, we established the Cartan's structure equations, the components of Riemannian curvature tensor and the Ricci tensor of the class under consideration. Further, the conditions required for this to be an Einstein manifold have been determined.On higher Dirac structureshttps://www.zbmath.org/1483.530972022-05-16T20:40:13.078697Z"Bursztyn, Henrique"https://www.zbmath.org/authors/?q=ai:bursztyn.henrique"Martinez Alba, Nicolas"https://www.zbmath.org/authors/?q=ai:martinez-alba.nicolas"Rubio, Roberto"https://www.zbmath.org/authors/?q=ai:rubio.robertoIn this paper, the authors define and initiate a systematic investigation of a notion of higher-order Dirac structures that extends that first introduced and studied by \textit{M.~Zambon} [J.~Symplectic Geom.~10, No.~4, 563--599, (2012; Zbl 1260.53134)].
As detailed below, they define a higher Dirac structure (of order \(k\)) on a manifold \(M\) as an involutive subbundle of the higher generalized tangent bundle \(TM\oplus \wedge^kT^\ast M\) satisfying a weakly Lagrangian condition.
The latter weakens the usual Lagrangian condition, for \(k>1\), and coincides with it, for \(k=1\).
Considering involutive subbundles that are weakly Lagrangian, the authors obtain a generalization of the notion of higher Dirac structures that encompasses a larger collection of examples.
In particular, their main motivating example is represented by the higher Poisson structures (that are not given by Lagrangian subbundles).
As symplectic structures provide the geometric language for the Hamiltonian description of classical mechanics, similarly their higher order analogues, namely multisymplectic structures, naturally appear in the geometric formulation of classical field theory [\textit{F.~Cantrijn} et al., Rend.~Semin.~Mat., Torino 54, No.~3, 225--236 (1996; Zbl 0911.58016)].
Precisely, a \textit{multisymplectic structure of order \(k\)} (also called a \textit{\(k\)-plectic structure}) on a manifold \(M\) is a closed form \(\omega\in\Omega^{k+1}(M)\) that is non-degenerate in the sense that \(\omega^\flat:TM\to\wedge^k T^\ast M,\ X\mapsto\iota_X\omega,\) is injective.
Generalising symplectic structures, Poisson structures naturally arise in studying mechanical systems with symmetries.
Then one can naturally wonder what are the higher order analogues of Poisson structures that generalise multisymplectic structures in the same way as Poisson structures extend symplectic structures.
Addressing this question, \textit{H.~Bursztyn, A.~Cabrera} and \textit{D.~Iglesias} [Fields Institute Communications 73, 57--73 (2015; Zbl 1335.53107)] have adopted a point of view inspired by the Lie theory with Poisson structures arising as the infinitesimal, or linearized, counterparts of symplectic groupoids, in the same way as Lie algebras/algebroids arise from Lie groups/groupoids.
This approach has led them to identify the higher order Poisson structures with the infinitesimal counterparts of multisymplectic groupoids (namely Lie groupoids equipped with a multisymplectic structure that is multiplicative, i.e.~compatible with the groupoid structure).
Precisely, a \textit{higher Poisson structure of order \(k\)} (also called a \textit{\(k\)-Poisson structure}) on a manifold \(M\) consists of a vector subbundle \(S\subset\wedge^kT^\ast M\) and a vector bundle morphism \(\Lambda:S\to TM\), covering the identity map \(\operatorname{id}_M\), such that:
\begin{enumerate}
\item \(S^\circ:=\{X\in TM\mid\iota_X\alpha=0\ \text{for all}\ \alpha\in S\}=0\),
\item \(\iota_{\Lambda\alpha}\beta=-\iota_{\Lambda\beta}\alpha\), for all \(\alpha,\beta\in S\),
\item \(\Gamma(S)\) is closed under the following bracket
\[
[\alpha,\beta]=\mathcal{L}_{\Lambda\alpha}\beta-\mathcal{L}_{\Lambda\beta}\alpha-\mathrm{d}(\iota_{\Lambda\alpha}\beta),
\]
and \(\Lambda:\Gamma(S)\to\mathfrak{X}(M)\) preserves the brackets.
\end{enumerate}
Concrete examples of this notion of higher Poisson structures can be found in the context of classical field theory, in particular, in connection with taking the quotients of multisymplectic structures by symmetries.
The concept of Dirac structures was developed by \textit{T.~Courant} and \textit{A.~Weinstein} [in: Actions hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986); Travaux en Cours, Vol.~27, 39--49 (1988; Zbl 0698.58020)] in their work on the geometric formulation of Dirac's theory of constrained mechanical systems and, largely independently, by \textit{I.~Ya.~Dorfman} [in: Nonlinear evolution equations: integrability and spectral methods, Proc.~Workshop, Como/Italy 1988; Manchester University Press. Proc.~Nonlinear Sci., 425--431 (1990; Zbl 0717.58026)] in her work on the theory of integrable systems.
Dirac structures encompass, unifying and generalizing, several geometric structures like foliations, Poisson structures and presymplectic structures (namely closed \(2\)-forms) and represent an essential ingredient in the study of generalized complex geometry.
A \textit{Dirac structure} on a manifold \(M\) is an involutive Lagrangian subbundle \(L\) of the generalized tangent bundle \(\mathbb{T}M:=TM\oplus T^\ast M\).
So, Poisson structures (resp.~presymplectic structures) on \(M\) identify with those Dirac structures \(L\subset\mathbb{T}M\) that are transverse to \(TM\) (resp.~\(T^\ast M\)).
Additionally, Dirac structures are equivalent to presymplectic foliations (namely, (singular) foliations whose leaves are smoothly endowed with a presymplectic structure) exactly like Poisson structures are equivalent to symplectic foliations.
Moreover, Dirac structures have underlying Lie algebroids and, if integrable, they identify with the infinitesimal counterparts of presymplectic groupoids.
Expectedly, suitable analogues of most of these properties will be satisfied by the higher order versions of Dirac structures.
The first instances of higher order analogues of Dirac structures appeared, under the name of \textit{multi-Dirac structures}, in the geometric description of the implicit Euler-Lagrange equations for a large class of field theories~[\textit{J.~Vankerschaver} et al., J.~Math.~Phys.~53, No.~7, 072903, 25 p. (2012; Zbl 1277.70023)].
The equivalent (but simpler to handle) notion of \textit{higher Dirac structure} was introduced and studied by \textit{M.~Zambon} [J.~Symplectic Geom.~10, No.~4, 563--599, (2012; Zbl 1260.53134)].
Precisely, a higher Dirac structure of order \(k\) on a manifold \(M\) is defined as an involutive Lagrangian subbundle \(L\subset TM\oplus \wedge^kT^\ast M\).
Examples of higher order Dirac structures include Dirac structures (for \(k=1\)), multisymplectic structures, closed forms together with a foliation, Poisson bivector fields and a restrictive class of multivector fields.
However, as stressed by the authors of the paper under review, higher Poisson structures do not fit within the framework provided this notion of higher Dirac structures, and therefore they propose a slight extension of such notion.
The general framework for studying higher Dirac structures is provided by the \textit{higher generalized tangent bundle} \(\mathbb{T}M^{(k)}:=TM\oplus\wedge^kT^\ast M\), with natural projections \(\operatorname{pr}_1:\mathbb{T}M^{(k)}\to TM\) and \(\operatorname{pr}_2:\mathbb{T}M^{(k)}\to\wedge^kT^\ast M\).
The latter is canonically equipped with
\begin{itemize}
\item the fiberwise non-degenerate symmetric \(\wedge^{k-1}T^\ast M\)-valued pairing \(\langle-,-\rangle\) given by
\[
\langle X+\alpha,Y+\beta\rangle=\iota_X\beta+\iota_Y\alpha,
\]
for all \(X,Y\in \mathfrak{X}(M)\) and \(\alpha,\beta\in\Omega^k(M)\), and
\item the Loday bracket \([\![-,-]\!]:\Gamma(\mathbb{T}M^{(k)})\times\Gamma(\mathbb{T}M^{(k)})\to\Gamma(\mathbb{T}M^{(k)})\) given by
\[
[\![X+\alpha,Y+\beta]\!]=[X,Y]+\mathcal{L}_X\beta-\iota_Y\mathrm{d}\alpha,
\]
for all \(X,Y\in \mathfrak{X}(M)\) and \(\alpha,\beta\in\Omega^k(M)\).
\end{itemize}
According to Zambon's definition, a higher Dirac structure (of order \(k\)) on \(M\) is a vector subbundle \(L\subset\mathbb{T}M^{(k)}\) such that
\begin{itemize}
\item \(L\) is \textit{involutive}, i.e.~the module of its sections \(\Gamma(L)\) is closed under \([\![-,-]\!]\),
\item \(L\) is \textit{Lagrangian}, i.e.~\(L=L^\perp\), where \(L^\perp\) denotes the orthogonal of \(L\) with respect to \(\langle-,-\rangle\).
\end{itemize}
Starting from the observation that, for \(k=1\), the subbundle \(L\) is Lagrangian if and only if
\[
L\subset L^\perp\quad\text{and}\quad L\cap TM=(\operatorname{pr}_2L)^\circ,
\]
the authors refer to any vector subbundle \(L\subset\mathbb{T}M^{(k)}\) satisfying the above condition as \textit{weakly Lagrangian} and define a higher Dirac structure (of order \(k\)) on \(M\) as an involutive weakly Lagrangian vector subbundle \(L\subset\mathbb{T}M^{(k)}\).
Adopting this extension of Zambon's notion of higher Dirac structures, the authors are able to prove that, on the one hand, higher Poisson structures (of order \(k\)) identify with those higher Dirac structures \(L\subset\mathbb{T}M^{(k)}\) such that \(L\cap TM=0\) and, on the other hand, closed \(k+1\) forms (i.e.~\textit{\(k\)-presymplectic structures}) identify with those higher Dirac structures \(L\subset\mathbb{T}M^{(k)}\) projecting isomorphically onto \(TM\).
Moreover, one recovers the structures defined by Zambon as those higher Dirac structures \(L\subset\mathbb{T}M^{(k)}\) that are properly Lagrangian.
The paper initiates the study of this notion of higher Dirac structures presenting two main new results.
Each higher Dirac structure \(L\subset\mathbb{T}M^{(k)}\) has an associated Lie algebroid which determines a (singular) foliation on \(M\), the \textit{characteristic foliation} of \(L\).
The first main result (Theorem 4.2) studies the leaf-wise geometry of a higher Dirac structure \(L\).
It proves that, for each characteristic leaf \(\mathcal{O}\), there is a cochain complex \(\Omega_{\text{sk}}^\bullet(\mathcal{O};F_{\mathcal{O}})\) and a cochain map \(\Omega_{\text{sk}}^\bullet(\mathcal{O};F_{\mathcal{O}})\longrightarrow\Omega^{\bullet+1}(\mathcal{O})\), so that the higher Dirac structure \(L\) is encoded by a distinguished \(1\)-cocycle \(\epsilon_\mathcal{O}\) of \(\Omega_{\text{sk}}^\bullet(\mathcal{O};F_{\mathcal{O}})\) for each characteristic leaf \(\mathcal{O}\).
Finally, extending the integration of Dirac structure by presymplectic groupoids, the second main result (Theorem 5.3) identifies higher Dirac structures of order \(k\) as the infinitesimal counterparts of \(k\)-presymplectic groupoids (cf.~Definition~5.2).
Reviewer: Alfonso Giuseppe Tortorella (Porto)Multiplicative Nambu structures on Lie groupoidshttps://www.zbmath.org/1483.530982022-05-16T20:40:13.078697Z"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaWeinstein introduced the notion of coisotropic submanifold of a Poisson manifold as a natural generalisation of a Lagrangian submanifold of a symplectic manifold. The condition that the bi-vector field that satisfies \([\Pi, \Pi]= 0\) plays no role in defining a coisotropic submanifold. Thus, the notion of coisotropic submanifolds is well-defined for any bivector field and, more generally, any multivector field. The reader should consult the paper under review for more details.
Nambu-Poisson manifolds are one particular higher-order generalisation of Poisson manifolds. Recall that A Nambu-Poisson manifold of order \(n\) is a manifold equipped with an n-vector field such that the associated \(n\)-array bracket on functions satisfies Filippov's Fundamental Identity. Coisotropic submanifolds of a Nambu-Poisson manifold are submanifolds that are coisotropic for the Nambu tensor.
In the paper under review, the author presents some basic properties of coisotropic submanifolds for a given multivector field and generalises the results of \textit{A. Weinstein} [J. Math. Soc. Japan 40, No. 4, 705--727 (1988; Zbl 0642.58025)]. The notion of a Nambu-Lie groupoid, understood as a Lie groupoid equipped with a multiplicative Nambu tensor is introduced and studied.
Reviewer: Andrew Bruce (Swansea)Kahan discretizations of skew-symmetric Lotka-Volterra systems and Poisson mapshttps://www.zbmath.org/1483.530992022-05-16T20:40:13.078697Z"Evripidou, C. A."https://www.zbmath.org/authors/?q=ai:evripidou.charalampos-a"Kassotakis, P."https://www.zbmath.org/authors/?q=ai:kassotakis.pavlos-g"Vanhaecke, P."https://www.zbmath.org/authors/?q=ai:vanhaecke.polThe primary result of this paper is a characterization of connected, skew-symmetric graphs \(\Gamma\) with the Kahan-Poisson property. The authors show that a graph as such is a cloning of a graph with vertices \(1,2,\ldots,n\), with an arc \(i \to j\) precisely when \(i<j\), and with all arks having the same value. This characterization helps to understand better the integrability of Lotka-Volterra systems as well as their deformations.
Reviewer: Iakovos Androulidakis (Athína)Generalized deformation of complex structures on nilmanifoldshttps://www.zbmath.org/1483.531002022-05-16T20:40:13.078697Z"Poon, Yat Sun"https://www.zbmath.org/authors/?q=ai:poon.yat-sunSummary: On a 2-step nilmanifold with abelian complex structure, there exists an invariant \((1, 0)\)-form \(\rho\) such that \(d\rho\) is type-\((1,1)\). It acts on the kernel of \(\rho\) by contraction. When this contraction map is non-degenerate, for any given infinitesimal generalized complex deformation \(\Gamma_1\) we construct a solution \((\Gamma,\overrightarrow{\partial})\) for the extended Maurer-Cartan equation. It amounts to identifying the obstruction \(\overrightarrow{\partial}\) for \(\Gamma_1\) to be integrable, and constructing the deformation \(\Gamma\) when the obstruction vanishes. As a consequence of our explicit solutions for \((\Gamma,\overrightarrow{\partial})\), we prove that on any real six-dimensional 2-step nilmanifold with abelian complex structure, when the contraction map is non-degenerate, every infinitesimal generalized complex deformation sufficiently close to zero is integrable. We also show that in all dimensions, if the contraction map is skew-Hermitian, then every infinitesimal generalized complex deformation sufficiently close to zero is integrable. Moreover the differential graded algebra controlling the generalized deformation of the underlying abelian complex structure is quasi-isomorphic to the one after deformation.\(S^1\)-invariant symplectic hypersurfaces in dimension 6 and the Fano conditionhttps://www.zbmath.org/1483.531012022-05-16T20:40:13.078697Z"Lindsay, Nicholas"https://www.zbmath.org/authors/?q=ai:lindsay.nicholas"Panov, Dmitri"https://www.zbmath.org/authors/?q=ai:panov.dmitriIn the paper under review the authors study symplectic Fano \(6\)-manifolds equipped with a Hamiltonian \(S^1\)-action. Let \(M\) be such a manifold, and let \(M_{\min}\) be the set of points where a Hamiltonian that generates the action attains its minimal value. It turns out that \(M_{\min}\) is connected. The main result (Theorem~1.3) states that \(M_{\min}\) is diffeomorphic to either a del Pezzo surface, a \(2\)-sphere or a point. By the result of [\textit{H. Li}, Proc. Am. Math. Soc. 131, No. 11, 3579--3582 (2003; Zbl 1066.53127)] the fundamental group of \(M\) is isomorphic to that of \(M_{\min}\), hence, as consequence, \(M\) must be simply connected. The Todd genus is also determined (Corollary~1.4). The proof is split into several cases, depending on the dimensions of \(M_{\min}\) and of the set \(M_{\max}\) where the generating Hamiltonian takes its maximum. The arguments are intricate, and rely on several interesting tools such as the notion of a \textit{symplectic fibre}, whose existence is proved under the assumption that \(M_{\min}\) is two dimensional (Theorem~1.5), and the existence of surfaces of fixed points with restricted topology (Theorem~1.8).
Reviewer: Umberto Leone Hryniewicz (Rio de Janeiro)Pseudo-rotations and Steenrod squares revisitedhttps://www.zbmath.org/1483.531022022-05-16T20:40:13.078697Z"Shelukhin, Egor"https://www.zbmath.org/authors/?q=ai:shelukhin.egorSummary: In this note we prove that if a closed monotone symplectic manifold admits a Hamiltonian pseudo-rotation, which may be degenerate, then the quantum Steenrod square of the cohomology class Poincaré dual to the point must be deformed. This result gives restrictions on the existence of pseudo-rotations, implying a form of uni-ruledness by pseudo-holomorphic spheres, and generalizes a recent result of the author. The new component in the proof consists in an elementary calculation with capped periodic orbits.Microlocal category for Weinstein manifolds via the h-principlehttps://www.zbmath.org/1483.531032022-05-16T20:40:13.078697Z"Shende, Vivek"https://www.zbmath.org/authors/?q=ai:shende.vivek-vSummary: On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly with high codimension, exact embedding into a cosphere bundle. Thicken to a hypersurface, and consider the Kashiwara-Schapira stack along the thickened skeleton. Pull back along the inclusion of the original skeleton.
Gromov's h-principle for contact embeddings guarantees existence and uniqueness up to isotopy of such an embedding. The invariance of microlocal sheaves along such an isotopy is well known. We expect, but do not prove here, invariance of the global sections of this co/sheaf of categories under Liouville deformation.Rational equivalence and Lagrangian tori on K3 surfaceshttps://www.zbmath.org/1483.531042022-05-16T20:40:13.078697Z"Sheridan, Nick"https://www.zbmath.org/authors/?q=ai:sheridan.nick"Smith, Ivan"https://www.zbmath.org/authors/?q=ai:smith.ivanFor a fixed choice of symplectic K3 surface \(X\), the authors show that there exist Lagrangian tori with vanishing Maslov class in \(X\) whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres.
Let \(T \subset X\) be a Maslov-zero Lagrangian torus. Let \(Y\) be an algebraic K3 surface mirror to \(X\), in the sense that, for a closed point \(y \in Y\) there exists a homological mirror equivalence \[\mathcal F(X) ^{\text{perf}} \simeq D(Y)\] taking \(T\) to the skyscraper sheaf \(\mathcal O_y\). The main result of this paper is that for any countable subset \(H \subset \mathcal F(X) ^{\text{perf}}\) any open neighbourhood of \(T\) in \(X\) contains a Maslov-zero Lagrangian torus \(T'\) with \([T']\notin H\).
Reviewer: Elizabeth Gasparim (Antofagasta)Lagrangian fibers of Gelfand-Cetlin systemshttps://www.zbmath.org/1483.531052022-05-16T20:40:13.078697Z"Cho, Yunhyung"https://www.zbmath.org/authors/?q=ai:cho.yunhyung"Kim, Yoosik"https://www.zbmath.org/authors/?q=ai:kim.yoosik"Oh, Yong-Geun"https://www.zbmath.org/authors/?q=ai:oh.yong-geunSummary: A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of \textit{T. Nishinou} et al. [Adv. Math. 224, No. 2, 648--706 (2010; Zbl 1221.53122)] on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of an \(r\)-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to \(T^r \times N\) for some smooth manifold \(N\) and \(T^r \cong (S^1)^r\). We also prove that such \(N\)'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.The entropy of the angenent torus is approximately 1.85122https://www.zbmath.org/1483.531062022-05-16T20:40:13.078697Z"Berchenko-Kogan, Yakov"https://www.zbmath.org/authors/?q=ai:berchenko-kogan.yakovSummary: To study the singularities that appear in mean curvature flow, one must understand \textit{self-shrinkers}, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In [Prog. Nonlinear Differ. Equ. Appl. 7, 21--38 (1992; Zbl 0762.53028)], \textit{S. B. Angenent} constructed the first nontrivial example of a self-shrinker, a torus. A key quantity in the study of the formation of singularities is the \textit{entropy}, defined by Colding and Minicozzi based on work of Huisken. The values of the entropy of spheres and cylinders have explicit formulas, but there is no known formula for the entropy of the Angenent torus. In this work, we numerically estimate the entropy of the Angenent torus using the discrete Euler-Lagrange equations.Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutionshttps://www.zbmath.org/1483.531072022-05-16T20:40:13.078697Z"Cesaroni, A."https://www.zbmath.org/authors/?q=ai:cesaroni.annalisa"Kröner, H."https://www.zbmath.org/authors/?q=ai:kroner.heiko"Novaga, M."https://www.zbmath.org/authors/?q=ai:novaga.matteoSummary: We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.Mean curvature type flows of graphs with nonzero Neumann boundary data in product manifold \(M^n \times \mathbb{R} \)https://www.zbmath.org/1483.531082022-05-16T20:40:13.078697Z"Chen, Xiao-Li"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Gao, Ya"https://www.zbmath.org/authors/?q=ai:gao.ya"Lu, Wei"https://www.zbmath.org/authors/?q=ai:lu.wei"Mao, Jing"https://www.zbmath.org/authors/?q=ai:mao.jingLet \((M^n,\sigma)\), \(n\ge 2\), be an \(n\)-dimensional complete Riemannian manifold with metric \(\sigma\) and let \(\Omega\subset M^n\) be a bounded domain with \(C^3\) boundary. The authors study the evolution of a graphic hypersurface of the form \((x,u(x,t))\), \(x\in\Omega\), \(t>0\), in the product space \(M^n\times\mathbb{R}\) with a nonparametric mean curvature type flow. They prove a gradient estimate for the flow under suitable conditions. As a consequence long time existence for the flow is obtained.
Reviewer: Shu-Yu Hsu (Chiayi)A curve shortening equation with time-dependent mobility related to grain boundary motionshttps://www.zbmath.org/1483.531092022-05-16T20:40:13.078697Z"Mizuno, Masashi"https://www.zbmath.org/authors/?q=ai:mizuno.masashi"Takasao, Keisuke"https://www.zbmath.org/authors/?q=ai:takasao.keisukeSummary: A curve shortening equation related to the evolution of grain boundaries is presented. This equation is derived from the grain boundary energy by applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. In the proof of these results, one key ingredient is a new weighted monotonicity formula that incorporates a time-dependent mobility.Evolution of the first eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flowhttps://www.zbmath.org/1483.531102022-05-16T20:40:13.078697Z"Qi, Xuesen"https://www.zbmath.org/authors/?q=ai:qi.xuesen"Liu, Ximin"https://www.zbmath.org/authors/?q=ai:liu.ximinSummary: In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the \(p\)-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao's work. Moreover, we give an example to specify applications of conclusions obtained above.Ancient solutions to the Ricci flow in dimension \(3\)https://www.zbmath.org/1483.531112022-05-16T20:40:13.078697Z"Brendle, Simon"https://www.zbmath.org/authors/?q=ai:brendle.simon.1Ancient \(\kappa\)-solutions are the ancient solutions to the Ricci flow, which are complete, non-flat, \(\kappa\)-non-collapsed and have bounded and non-negative curvature. The purpose of this paper is to give a classification of all non-compact ancient \(\kappa\)-solution in dimension \(3\). In particular, the author proves that a \(3\)-dimensional non-compact ancient \(\kappa\)-solutions \((M,g(t))\) is isometric to either a family of shrinking cylinders (or a quotient thereof ) or to the Bryant soliton.
Reviewer: Chandan Kumar Mondal (Durgapur)Second order renormalization group flowhttps://www.zbmath.org/1483.531122022-05-16T20:40:13.078697Z"Guenther, Christine"https://www.zbmath.org/authors/?q=ai:guenther.christine-mThe Ricci flow appears in physics as the first order approximation of the renormalization group flow equations in quantum field theory. In this article the author introduces the fully nonlinear second order renormalization group flow (RG-2) and compares it with the Ricci flow. The author introduces various known results on RG-2 flow including exitence of special solutions which are fixed points of the RG-2 flow. The author also discusses results of the RG-2 flow on locally homogeneous manifolds, the parabolicity of the RG-2 flow and the short time existence of the RG-2 flow. She also discusses extension of Perelman's Ricci flow entropy to the RG-2 flow. A number of open problems for the RG-2 flow are also given.
For the entire collection see [Zbl 1455.53004].
Reviewer: Shu-Yu Hsu (Chiayi)Identifying shrinking solitons by their asymptotic geometrieshttps://www.zbmath.org/1483.531132022-05-16T20:40:13.078697Z"Kotschwar, Brett"https://www.zbmath.org/authors/?q=ai:kotschwar.brett-lSummary: We discuss the classification problem for complete noncompact shrinking solitons and survey some recent uniqueness results, most obtained as part of a joint project with L. Wang, which explore the extent to which these solitons are determined by their geometry on a neighborhood of infinity.
For the entire collection see [Zbl 1455.53004].Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometryhttps://www.zbmath.org/1483.531142022-05-16T20:40:13.078697Z"Kunikawa, Keita"https://www.zbmath.org/authors/?q=ai:kunikawa.keita"Sakurai, Yohei"https://www.zbmath.org/authors/?q=ai:sakurai.yoheiThis is a quite interesting article, in which Liouville type theorems are proved for the harmonic map heat flow from a time-dependent Riemannian manifold, ancient super-solution of the Ricci flow, i.e. satisfying \[ \partial_t g \geq -2 \mathrm{Ricci} \quad (t\in ]-\infty , 0]). \] As to the target, it is just assumed to be complete and simply-connected with an upper bound on its sectional curvature.
There are two distinct sets of conditions required to conclude that a solution of the backward harmonic map flow must be constant (``backward'' because one of the first things the authors do is to reverse the time parameter), expressed in terms of Perelman's reduced geometry.
i) On the metric flow: admissible (so that the reduced distance is achieved), completeness and more technical conditions (though the significance of the bounds they imply are very clear) on the Müller and trace Harnack quantities.
ii) On the map itself, assumptions are made (in both main theorems) on the asymptotic behaviour of the distance function of the image compared with the reduced distance on the domain.
If, in the first theorem, the target manifold's sectional curvature is assumed to be negative, this condition is then relaxed to a uniform upper bound at the cost of an extra assumption on the image being in a sufficiently small geodesic ball.
The article concludes with a nice discussion of the sharpness of an improved static version of the previous theorem and the Schoen-Uhlenbeck smooth radial harmonic maps from \(\mathbb R^m\) to \(\mathbb S_{+}^m\) for \(m\geq 7\) (if \(m\leq 6\) they must be constant).
This paper is the continuation of a work with the same scope for functions [\textit{K. Kunikawa} and \textit{Y. Sakurai}, J. Geom. Anal. 31, No. 12, 11899--11930 (2021; Zbl 07430439)] and aims at a generalisation to metric flows of \textit{S.-t. Yau}'s Liouville theorem on complete manifolds [Commun. Pure Appl. Math. 28, 201--228 (1975; Zbl 0291.31002)]. The method is again to obtain local gradient estimates and nothing is said about the existence of such flows.
Reviewer: Eric Loubeau (Brest)Hermitian curvature flow on compact homogeneous spaceshttps://www.zbmath.org/1483.531152022-05-16T20:40:13.078697Z"Panelli, Francesco"https://www.zbmath.org/authors/?q=ai:panelli.francesco"Podestà, Fabio"https://www.zbmath.org/authors/?q=ai:podesta.fabioSuppose \(M\) is a \(C\)-space, meaning that \(M\) is a closed simply connected homogeneous complex manifold. Then \(M\) is known to have the structure of a bundle over a flag manifold \(N\) with fiber a complex torus \(F\). The authors moreover assume that \(N\) is a product of irreducible Hermitian symmetric spaces of complex dimension at lesast \(2\), and that none of the factors is a complex quadric.
From this structure it follows that any invariant Hermitian metric \(h\) on \(M\) is specified by an arbitrary Hermitian metric \(h^F\) on the fiber together with an invariant Hermitian metric \(h^N\) on \(N\).
In this context, the authors study a specific Hermition Curvature Flow, introduced by \textit{Y. Ustinovskiy} [Am. J. Math. 141, No. 6, 1751--1775 (2019; Zbl 1435.53072)]. Their main results are summarized in Theorem 4.4 and Proposition 4.5:
Given any initial invariant Hermitian metric \((h_0^N, h_0^F)\) on \(M\), Ustinovskiy's flow has the following properties:
1. The solution \(h(t)\) has the form \((h^N(t), h^F(t))\), where \(h^N(t)\) only depends on \(h_0^N\).
2. \(h^N(t)\) exists on a domain of the form \((-\infty, T)\), while \(h(t)^F\) exists on a domain of the form \((-r,T)\). When \(t\rightarrow T\), \(N\) collapses to a product of Hermitian symmetric spaces, while \(h(t)\) converges to a positive semidefinite Hermitian bilinear form \(\hat{h}\). The distribution given by the kernel of \(\hat{h}\) is integral and the leaves are homogeneous. If the leaves are closed, then \(M\) Gromov-Hausdorff converges to a smooth homogeneous manifold with Riemannian metric.
In addition to these results, they show that \(M\) admits a unique invariant Hermitian metric which is static for the flow. If \(h^N(t)\rightarrow 0\) as \(t\rightarrow T\), then \(h(t)\rightarrow 0\) as well, and the normalized flow with constant volume converges to this static metric.
Reviewer: Jason DeVito (Martin)The anomaly flow on nilmanifoldshttps://www.zbmath.org/1483.531162022-05-16T20:40:13.078697Z"Pujia, Mattia"https://www.zbmath.org/authors/?q=ai:pujia.mattia"Ugarte, Luis"https://www.zbmath.org/authors/?q=ai:ugarte.luisIn the present article, the authors study the so-called anomaly flow on 2-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. This is a coupled flow of a pair of Hermitian metrics defined on a 3-dimensional complex manifold equipped with a (3,0) form and on the fibers of a holomorphic vector bundle over the mentioned complex manifold, respectively.
In particular, they characterize the solutions for a simplified version of the anomaly flow under the hypothesis that the base space is a 2-step nilmanifold of real dimension 6 with the first Betti number greater or equal than 4. Moreover, they study its convergence and they prove that two properties are preserved under the flow. Namely, the balanced condition and the diagonal character of the metrics.
Reviewer: Alberto Rodríguez-Vázquez (A Coruña)Improved regularity estimates for Lagrangian flows on \(\mathrm{RCD}(K,N)\) spaceshttps://www.zbmath.org/1483.531172022-05-16T20:40:13.078697Z"Brué, Elia"https://www.zbmath.org/authors/?q=ai:brue.elia"Deng, Qin"https://www.zbmath.org/authors/?q=ai:deng.qin"Semola, Daniele"https://www.zbmath.org/authors/?q=ai:semola.danieleSummary: This paper gives a contribution to the study of regularity of Lagrangian flows on non-smooth spaces with lower Ricci curvature bounds. The main novelties with respect to the existing literature are the better behaviour with respect to time and the local nature of the regularity estimates. These are obtained by sharpening previous results of the first and third authors, in combination with some tools recently developed by the second author (adapting to the synthetic framework ideas introduced in [\textit{T. H. Colding} and \textit{A. Naber}, Ann. Math. (2) 176, No. 2, 1173--1229 (2012; Zbl 1260.53067)]). The estimates are suitable for applications to the fine study of \(\mathrm{RCD}\) spaces and play a central role in the construction of a parallel transport in this setting.Born sigma-models for para-Hermitian manifolds and generalized T-dualityhttps://www.zbmath.org/1483.531182022-05-16T20:40:13.078697Z"Marotta, Vincenzo Emilio"https://www.zbmath.org/authors/?q=ai:marotta.vincenzo-emilio"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-jSurface tension and \(\Gamma\)-convergence of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic mediahttps://www.zbmath.org/1483.531192022-05-16T20:40:13.078697Z"Morfe, Peter S."https://www.zbmath.org/authors/?q=ai:morfe.peter-sThe paper ``Surface Tension and \(\Gamma\)-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media'' is centered on a problem of \(\Gamma\)-convergence of certain functionals which emerge in phenomenological mesoscopic theory of phase transitions. The departure point of the whole analysis is the family of functionals of the form
\[
\mathcal{F}^{\omega}(u) = \int_{\mathbb{R}^d}\left( \frac{1}{2}\varphi^{\omega}(x,Du(x))^2 + W(u(x))\right) dx\,, \tag{1}
\]
where \(\phi^{\omega}(x,\cdot)\) is a stationary ergodic Finsler metric, \(W\) is a double-well potential with wells of equal depth, and \(u\) is a scalar function taking value in the interval \([-1,1]\). Then, the main goal of the paper is to study the role played by randomness in determining the macroscopic surface tension in the Van der Waals-Cahn-Hilliard problem, the randomness being represented by a probability space \(\left( \Omega, \mathscr{B}, \mathbb{P} \right)\) with an action \(\tau\) of the group \(\mathbb{R}^d\).
The result of the paper is expressed in the language of \(\Gamma\)-convergence. Firstly a rescaled energy functional, \(\mathcal{F}^{\omega}_{\epsilon}(u)\) is defined, where \(\epsilon >0\) represents the length scale of the mesoscopic description, and localized to each open subset \(A\subset \mathbb{R}^d\), obtaining the functional
\[
\mathcal{F}_{\epsilon}^{\omega}(u;A) = \int_{A}\left( \frac{\epsilon}{2}\varphi^{\omega}(\epsilon^{-1}x,Du(x))^2 + \epsilon^{-1} W(u(x))\right) dx\,. \tag{2}
\]
Then, if \(\mathscr{E}\) denotes the functional
\[
\mathscr{E}(u;A)=\begin{cases}
\int_{\partial ^*\left\lbrace u=1 \right\rbrace \cap A} \tilde{\varphi}(\nu_{\left\lbrace u=1 \right\rbrace}(\xi)) \mathcal{H}^{d-1}(d\xi), &u\in BV(A;\left\lbrace -1,1 \right\rbrace)\\
\infty, &\text{otherwise}
\end{cases}, \tag{3}
\]
the main result is contained in the following Theorem 1:
\textbf{Theorem 1.} There is a one-homogeneous convex function \(\tilde{\varphi}\,:\,\mathbb{R}^d \,\rightarrow \,(0,\infty)\) depending only on \(\mathbb{P}\), such that, with probability one, \(\mathcal{F}^{\omega}\xrightarrow{\Gamma} \,\mathscr{E}\). More specifically, there is an event \(\hat{\Omega}\in \Sigma\) such that \(\mathbb{P}(\hat{\Omega})=1\) and no matter the choice of Lipschitz, open, bounded \(A \subseteq \mathbb{R}^d\) or \(\omega\in \hat{\Omega}\), the following occurs:
\begin{itemize}
\item[1.] if \((u_{\epsilon})_{\epsilon >0}\subset H^1(A;[-1,1])\) satisfies
\[
\sup \left\lbrace \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\mid \epsilon >0 \right\rbrace <\infty \tag{4}
\]
then \((u_{\epsilon})_{\epsilon >0}\) is relatively compact in \(L^1(A)\) and all of its limit points are in \(BV(A;\left\lbrace -1,1 \right\rbrace)\).
\item[2.] If \(u\in L^1(A;[-1,1])\) and \((u_{\epsilon})_{\epsilon >0}\subseteq H^1(A;[-1,1])\) satisfies \(u_{\epsilon}\,\rightarrow\, u\) in \(L^1(A)\), then
\[
\mathscr{E}(u;A)\leq \liminf_{\epsilon \rightarrow 0^+} \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\,. \tag{5}
\]
\item[3.] If \(u\in L^1(A;[-1,1])\), then there is a family \((u_{\epsilon})_{\epsilon >0}\subseteq H^1(A;[-1,1])\) such that \(u_{\epsilon}\,\rightarrow\, u\) in \(L^1(A)\) and
\[
\limsup_{\epsilon \rightarrow 0^+} \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\leq \mathscr{E}(u;A)\,. \tag{6}
\]
\end{itemize}
In the above functional \(\mathscr{E}\), \(\partial ^*E\) is the reduced boundary of the Caccioppoli set \(E\), whereas \(\nu_E\) is its normal vector and \(\mathcal{H}^d\) denotes the d-dimensional Hausdorff measure. In the main theorem \(\Sigma\subset \mathscr{B}\) is the \(\sigma\)-algebra of subsets of \(\Omega\) invariant under the action \(\tau\). Therefore, the above theorem states that even in presence of randomness there exists a macroscopic surface tension, the functional \(\mathscr{E}\), obtained as the limit in the sense of \(\Gamma\)-convergence of a mesoscopic energy functional.
The above theorem is proven in the last section of the work. In order to get there, the author proves a series of intermediate results which are collected in Sections 3 and 4. The goal of these two sections is to prove that there is a continuous function \(\tilde{\varphi}\) which is, then, used in the surface tension functional. This function is obtained via a procedure which the author calls thermodynamic limit. Firstly, chosen a function \(q\,:\,\mathbb{R}\,\rightarrow\,[-1,1]\) which is used to impose boundary conditions, a random process, called finite-volume surface tension and denoted \(\tilde{\varphi}^{\omega}(e,G,h)\), is obtained as the minimum
\[
\tilde{\varphi}^{\omega}(e,G,h) = \min \left\lbrace \mathcal{F}^{\omega}(u;A)\mid u\in H^1(A;[-1,1]), \; u-q_e \in H^1_0(A) \right\rbrace\,, \tag{7}
\]
where \(A=G\oplus_{e}(-h,h)\) is the following set
\[
G\oplus_{e}(-h,h) = \left\lbrace O_e(y)+te\,\mid\,y\in G,\; t\in (-h,h) \right\rbrace \tag{8}
\]
and \(O_e\,:\,\mathbb{R}^{d-1}\,\rightarrow\,\mathbb{R}^d\) is a linear isometry onto the hyperplane orthogonal to \(e\). Here \(e\) is a unit vector in the unit sphere \(S^{d-1}\subset \mathbb{R}^d\). Then, theorem 2 states:
\textbf{Theorem 2.} For each \(e\in S^{d-1}\), there is an event \(\hat{\Omega}\in \Sigma_e\) satisfying \(\mathbb{P}(\hat{\Omega}_e)=1\) such that if \(\omega\in \hat{\Omega}_e\), then
\[
\begin{split} \tilde{\varphi}(e) &= \lim_{R\rightarrow \infty}R^{1-d}\tilde{\varphi}^{\omega}_{\infty}(e,Q(0,R))\\
&= \lim_{h\rightarrow \infty} \limsup_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),h)\\
&= \lim_{h\rightarrow \infty} \liminf_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),h)\\
&= \lim_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),kR) \,. \end{split}
\tag{9}
\]
In the above theorem, \(\Sigma_e\) is the \(\sigma\)-algebra of subsets invariant under the action \(\tau_x\) with \(x\) orthogonal to \(e\), \(Q(0,R)\subset \mathbb{R}^{d-1}\) is the cube centered at the origin with side length \(R/2\), and \(\tilde{\varphi}^{\omega}_{\infty}\) is the minimum
\[
\tilde{\varphi}^{\omega}_{\infty}(e,G) = \min \left\lbrace \mathcal{F}^{\omega}(u;G\oplus_e\mathbb{R})\mid -1 \leq u \leq 1, \; u-q =0 \; \text{on} \: \partial G \oplus_e \mathbb{R} \right\rbrace\,.
\]
Therefore, the above theorem states the existence of the thermodynamic limit for cubes centered at the origin. Then, the last part of section 3 extends the previous thermodynamic limit to any cube in \(\mathbb{R}^d\) as stated in
\textbf{Proposition 10.} There is an event \(\hat{\Omega}\in \Sigma\) satisfying \(\mathbb{P}(\hat{\Omega})=1\) such that if \(\omega \in \hat{\Omega}\), \(e\in S^{d-1}\), \(x_0\in \mathbb{R}^d\), and \(\rho>0\), then
\[
\tilde{\varphi}(e)\rho^{d-1} = \lim_{R\rightarrow \infty} R^{1-d}\tilde{\Phi}^{\omega}(e,Rx_0,RQ^{e}(x_0,\rho))\,, \tag{10}
\]
where
\[
\tilde{\Phi}^{\omega}(e,x,A)= \min \left\lbrace \mathcal{F}^{\omega}(u;A)\mid u\in H^1(A;[-1,1]), \; u-T_x q_e \in H^1_0(A) \right\rbrace\,. \tag{11}
\]
In the above theorem \(Q^{e}(x_0,\rho)\) denotes the cube in \(\mathbb{R}^d\)
\[
Q^{e}(x_0,\rho)= x+Q(0,R)\oplus_e \left( -\frac{\rho}{2}, \frac{\rho}{2} \right)\,, \tag{12}
\]
and \(T_xq(y) = q(x+y)\). Once the surface tension \(\tilde{\varphi}\) is found the results of Theorem 1 are obtained using some techniques proven in [\textit{N. Ansini} et al., Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 2, 265--296 (2003; Zbl 1031.49021)] of the paper.
The paper is very well organized and the Introduction is a fundamental guide to understand the development of the whole work. Indeed, it summarizes the contents of the main theorems and gives a short sketch of their proofs. Moreover, it provides useful references where finding some pieces of missing information. However, it could be useful starting from Section 2 in order to make contact with the notation of the work. The final appendices provide further definitions making the paper self-consistent. The technical core of the paper is divided into several sections and subsections, which facilitate the reading of the contents, together with a short initial comment anticipating the main results of any section.
Reviewer: Fabio Di Cosmo (Madrid)Deformation analysis of nonuniform lipid membrane subjected to local inflammationshttps://www.zbmath.org/1483.531202022-05-16T20:40:13.078697Z"Yao, Wenhao"https://www.zbmath.org/authors/?q=ai:yao.wenhao"Kim, Chun Il"https://www.zbmath.org/authors/?q=ai:kim.chun-ilSummary: We present complete analytical solutions describing the deformations of both rectangular and circular lipid membranes subjected to local inflammations and coordinate-dependent (nonuniform) property distributions. The membrane energy potential of the Helfrich type is refined to accommodate the coordinate-dependent responses of the membranes. Within the description of the superposed incremental deformations and Monge parametrization, a linearized version of the shape equation describing coordinate-dependent membrane morphology is obtained. The local inflammation of a lipid membrane is accommodated by the prescribed uniform internal pressure and/or lateral pressure. This furnishes a partial differential equation of Poisson type from which a complete analytical solution is obtained by employing the method of variation of parameters. The solution obtained predicts the smooth and coordinate-dependent morphological transitions over the domain of interest and is reduced to those from the classical uniform membrane shape equation when the equivalent energy potential is applied. In particular, the obtained model closely assimilated the pressure-induced inflammations of lipid membranes where only quantitatively equivalent analyses were reported via the impositions of equivalent edge moments. Lastly, we note that the principle of superposition remains valid even in the presence of coordinate-dependent membrane properties.Application of Kähler manifold to signal processing and Bayesian inferencehttps://www.zbmath.org/1483.531212022-05-16T20:40:13.078697Z"Choi, Jaehyung"https://www.zbmath.org/authors/?q=ai:choi.jaehyung"Mullhaupt, Andrew P."https://www.zbmath.org/authors/?q=ai:mullhaupt.andrew-pSummary: We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kähler manifold case. We find conditions for the information geometry of linear systems to be Kähler, and the relation of the Kähler potential to information geometric quantities such as \(\alpha \)-divergence, information distance and the dual \(\alpha \)-connection structure. The Kähler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the \(\alpha \)-generalization of the geometric objects. The Laplace-Beltrami operator is also simplified in the Kähler geometry. One of the goals in information geometry is the construction of Bayesian priors outperforming the Jeffreys prior, which we use to demonstrate the utility of the Kähler structure.
For the entire collection see [Zbl 1470.00021].Nested sampling with demonshttps://www.zbmath.org/1483.531222022-05-16T20:40:13.078697Z"Habeck, Michael"https://www.zbmath.org/authors/?q=ai:habeck.michaelSummary: This article looks at Skilling's nested sampling from a physical perspective and interprets it as a microcanonical demon algorithm. Using key quantities of statistical physics we investigate the performance of nested sampling on complex systems such as Ising, Potts and protein models. We show that releasing multiple demons helps to smooth the truncated prior and eases sampling from it because the demons keep the particle off the constraint boundary. For continuous systems it is straightforward to extend this approach and formulate a phase space version of nested sampling that benefits from correlated explorations guided by Hamiltonian dynamics.
For the entire collection see [Zbl 1470.00021].Flexibility in contact geometry in high dimension [after Borman, Eliashberg and Murphy]https://www.zbmath.org/1483.570322022-05-16T20:40:13.078697Z"Massot, Patrick"https://www.zbmath.org/authors/?q=ai:massot.patrickSummary: Contact structures are hyperplane fields appearing naturally on the boundary of symplectic or holomorphic manifolds, and whose appeal stems from a subtle mixture of rigidity and flexibility. On the rigid side, Gromov's holomorphic curves prove that, in all dimensions, homotopical invariants are not enough to describe deformation classes of contact structures. On the flexible side, which is the topic of this exposition, Borman, Eliashberg and Murphy [\textit{M. S. Borman} et al., Acta Math. 215, No. 2, 281--361 (2015; Zbl 1344.53060)] proved the existence, in all dimensions, of a class of contact structures whose geometry is entirely ruled by algebraic topology. In particular, they give a homotopical characterisation of manifolds carrying contact structures.
For the entire collection see [Zbl 1416.00029].Geometry of surfaces in \(\mathbb{R}^5\) through projections and normal sectionshttps://www.zbmath.org/1483.570352022-05-16T20:40:13.078697Z"Deolindo-Silva, J. L."https://www.zbmath.org/authors/?q=ai:silva.jorge-luiz-deolindo"Sinha, R. Oset"https://www.zbmath.org/authors/?q=ai:sinha.raul-osetThe main focus of the article is the study of geometry of surfaces in \(\mathbb{R}^5\). The first approach used by the authors is to relate the geometry of a surface in \(\mathbb{R}^5\) to that of corresponding (both regular and singular) surfaces in \(\mathbb{R}^4\) obtained by orthogonal projections. In particular, relations between the asymptotic directions of the original surface and those of the projected surface are obtained. It is interesting to note here that the asymptotic directions for surfaces in \(\mathbb{R}^5\), unlike those in \(\mathbb{R}^4\), do not depend only on the second order geometry of the surface.
The authors also establish relations between the umbilical curvatures for surfaces in \(\mathbb{R}^5\) [\textit{S. I. R. Costa} et al., Differ. Geom. Appl. 27, No. 3, 442--454 (2009; Zbl 1176.53015)] and their projected surfaces in \(\mathbb{R}^4\). The authors study the contact between the projected surfaces with spheres in \(\mathbb{R}^4\) and show that there exists a unique umbilical focal hypersphere at a point of the surface if and only if there exists a unique umbilic focal hypersphere at the corresponding point on the projected surface.
Surfaces in \(\mathbb{R}^5\) can also be obtained as normal sections of 3-manifolds in \(\mathbb{R}^6\), so the authors then go on to consider the geometry of surfaces in \(\mathbb{R}^5\) by relating the asymptotic directions at a point in the 3-manifold with asymptotic directions at the corresponding point in the normal section. By introducing an appropriate umbilic curvature for 3-manifolds, they then study the contact with spheres using this invariant and relate it to the contact between spheres and the surface in \(\mathbb{R}^5\) obtained as a normal section.
Reviewer: Graham Reeve (Liverpool)Polytope Novikov homologyhttps://www.zbmath.org/1483.570382022-05-16T20:40:13.078697Z"Pellegrini, Alessio"https://www.zbmath.org/authors/?q=ai:pellegrini.alessioFor a closed smooth oriented and connected finite dimensional manifold \(M\), Sergey P. Novikov associated a homology with each a cohomology class \(a\in H^1_\mathrm{dR}(M)\), the so-called Novikov homology \(HN_\ast(a)\), cf. [\textit{S. P. Novikov}, Sov. Math., Dokl. 24, 222--226 (1981; Zbl 0505.58011); translation from Dokl. Akad. Nauk SSSR 260, 31--35 (1981), Russ. Math. Surv. 37, No. 5, 1--56 (1982; Zbl 0571.58011); translation from Usp. Mat. Nauk 37, No. 5(227), 3--49 (1982)]. Let \(\Phi_a:\pi_1(M)\to\mathbb{R}\) be the period homomorphism, and let \(\pi:\widetilde{M}_a\to M\) be the minimal regular covering with the group of deck transformations \(\Gamma_a\cong\pi_1(M)/\mathrm{Ker}(\Phi_a)\). Then for any representative \(\alpha\in a\) there exists an \(\tilde{f}_\alpha\in C^\infty(\widetilde{M}_a)\) such that \(\pi^\ast\alpha=d\tilde{f}_\alpha\). For a Riemannian metric \(g\) on \(M\) the pair \((\alpha, g)\) is said to be Morse-Smale if \((\tilde{f}_\alpha, \pi^\ast g)\) satisfies the Morse-Smale condition on \(\widetilde{M}_a\). For each \(i\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}\) let \(\mathrm{Crit}_i(\tilde{f}_\alpha)\) denote the critical points of \(\tilde{f}_\alpha\) with Morse index \(i\). The \(i\)th Novikov chain group \(\mathrm{CN}_i(\alpha)\) consists of all formal sums
\[
\xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle
\]
such that \(\{\tilde{x}\mid \xi_{\tilde{x}}\in\mathbb{Z}\setminus\{0\}\,\&\, \tilde{f}_\alpha(\tilde{x})>c\}\) is finite for each \(c\in\mathbb{R}\). The boundary operator \(\partial : \mathrm{CN}_i(\alpha) \to \mathrm{CN}_{i-1}(\alpha)\) is defined by
\[
\partial \xi:=\sum_{\tilde{x}, \, \tilde{y}} \, \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha}) \, \tilde{y},
\]
where \(\#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha})\) counts trajectories of negative gradient of \(\tilde{f}_\alpha\) with respect to \(\tilde{g}:=\pi^\ast g\) with signs from \(\tilde{x}\) to \(\tilde{y}\).
The Novikov ring \(\Lambda_\alpha\) consists of all formal sums
\[
\lambda=\sum_{A\in\Gamma_a}\lambda_A A\in \bigoplus_{A\in\Gamma_a}\mathbb{Z}\langle A\rangle
\]
such that \(\{A\in\Gamma_a\mid \lambda_A\in\mathbb{Z}\setminus\{0\}\,\&\, \Phi_a(A)<c\}\) is finite for each \(c\in\mathbb{R}\). The product is given by the convolution
\[
(\lambda\ast\mu)_A=\sum_{B\in\Gamma_a}\lambda_B\mu_{B^{-1}A}.
\]
According to the obvious action of \(\Lambda_a\) on \(\mathrm{CN}_\ast(\alpha)\), the latter is a finitely generated \(\Lambda_a\)-module. Moreover the boundary operator \(\partial\) is \(\Lambda_a\)-linear, and for each \(i \in \mathbb{N}_0\) the Novikov homology
\[
\mathrm{HN}_i(\alpha,g):=\frac{\ker \partial_i}{\mathrm{im} \, \partial_{i+1}}
\]
carries a \(\Lambda_a\)-module structure. Different choices of cohomologous Morse forms representing \(\alpha\) induce isomorphic Novikov homologies. The isomorphism class is said to be the Novikov homology of pairs \((\alpha, g)\), and denoted by \(\mathrm{HN}_\ast(a)\).
In the paper under review the author generalizes the above Novikov homology and defines polytope Novikov homology. Corresponding to a polytope \(\mathcal{A}=\langle a_0, \dots, a_k \rangle \subset H^1_{\mathrm{dR}}(M)\) with vertices \(a_0,\dots,a_k\), there exists a regular cover \(\pi : \widetilde{M}_{\mathcal{A}} \to M\) with the group of deck transformations
\[ \Gamma_\mathcal{A}\cong {\pi_1(M)}{\bigg /}\bigcap_{l=0}^k \mathrm{Ker}(\Phi_{a_l}), \]
Then for every \(a \in \mathcal{A}\) and for any representative \(\alpha\in a\) there exists a \(\tilde{f}_{\alpha} \in C^{\infty}(\widetilde{M}_{\mathcal{A}})\) such that \(\pi^*\alpha=d\tilde{f}_{\alpha}\). Fix a smooth section \(\theta : \mathcal{A} \longrightarrow \Omega^1(M)\), that is, \(\theta_a\) is a representative of \(a\). For each \(i\in\mathbb{N}_0\) the \(i\)th polytope Novikov chain complex group \(\mathrm{CN}_i(\theta_a,\mathcal{A})\) consists of all formal sums
\[ \xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_{\theta_a})}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle \]
such that
\begin{gather*}
\xi=\sum_{\tilde{x} \in \mathrm{Crit}_i\left(\tilde{f}_{\theta_a}\right)} \xi_{\tilde{x}} \, \tilde{x} \in \mathrm{CN}_i(\theta_a, \mathcal{A}) \iff \forall b \in \mathcal{A}, \forall c \in \mathbb{R} : \\ \#\lbrace \tilde{x} \mid \xi_{\tilde{x}} \neq 0, \; \tilde{f}_\beta(\tilde{x})>c \rbrace < +\infty,
\end{gather*}
where \(\beta \in b\) is any representative. The groups \(\mathrm{CN}_\bullet(\theta_a,\mathcal{A})\) may be equipped with boundary operators \(\partial_{\theta_a} : \mathrm{CN}_\ast(\theta_a,\mathcal{A}) \to \mathrm{CN}_{\ast-1}(\theta_a,\mathcal{A})\) given by
\[ \partial_{\theta_a} \xi:= \sum_{\tilde{x}, \tilde{y}} \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}\left(\tilde{x},\tilde{y};\tilde{f}_{\theta_a}\right) \, \tilde{y}. \]
Let \(\widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\) denote the upward completion of the group ring \(\mathbb{Z}[\Gamma_{\mathcal{A}}]\) with respect to the period homomorphism \(\Phi_b : \Gamma_{\mathcal{A}} \to \mathbb{R}\). Define the polytope Novikov ring \(\Lambda_\mathcal{A}=\bigcap_{b \in \mathcal{A}} \widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\). The above boundary operator \(\partial_{\theta_a}\) is \(\Lambda_{\mathcal{A}}\)-linear. The homology of the chain complex \(\left(\mathrm{CN}_\ast(\vartheta_a,g_{\vartheta_a},\mathcal{A}),\partial \right)\), denoted by \(\mathrm{HN}_\ast(\vartheta_a,\mathcal{A})\), is called the polytope Novikov homology. It is proved that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope.
An important application is to present a novel approach to the (twisted) Novikov Morse Homology Theorem: For any cohomology class \(a \in H^1_{\mathrm{dR}}(M)\) there exists an isomorphism \(\mathrm{HN}_\ast (a) \cong \mathrm{H}_\ast(M,\Lambda_a)\) of Novikov-modules.
The second application is to prove a new polytope Novikov Principle, which generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case [\textit{A. Pajitnov}, Eur. J. Math. 6, No. 4, 1303--1341 (2020; Zbl 1470.57050)].
Reviewer: Guang-Cun Lu (Beijing)Fubini-study metrics and Levi-Civita connections on quantum projective spaceshttps://www.zbmath.org/1483.580012022-05-16T20:40:13.078697Z"Matassa, Marco"https://www.zbmath.org/authors/?q=ai:matassa.marcoIn the study of noncommutative spaces, it is desirable to quantize important notions of commutative spaces. In classical Riemannian geometry, the Riemannian metric and the Levi-Civita connection are fundamental notions. The idea to quantize the Riemannian metric and the Levi-Civita connection to noncommutative algebra is used as algebraic approach, which is explained for instance in [\textit{E.J. Beggs} and \textit{S. Majid}, Quantum Riemannian geometry. Cham: Springer (2020; Zbl 1436.53001)]. In this paper under review, the author introduces analogues of the Fubini-Study metrics and the corresponding Levi-Civita connections on the quantum projective spaces.
Let \(\mathcal{B}\) be the algebra of a generic quantum projective space (see Section 4 for the definition) and \(\Omega\) the degree-one part of a differential calculus over \(\mathcal{B}\). Then a quantum metric can be defined as an element \(g \in \Omega \otimes_{\mathcal{B}} \Omega\) satisfying an appropriate condition. The first main result in this paper is a quantization of the Fubini-Study metric.
Theorem (see Theorem 6.11). Any quantum projective space \(\mathcal{B}\) admits a quantum metric \(g \in \Omega \otimes_{\mathcal{B}} \Omega\). Moreover, in the classical limit it reduces to the Fubini-Study metric.
Let \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) be a connection on \(\Omega\) (the notion of the connection on \(\Omega\) and its tortion can be defined in the standard algebraic sense). To formulate an analogue of the compatibility of \(\nabla\) with \(g\), there are two possibilities: 1) (weak sense) use ``cotortion'', 2) (strong sense) require \(\nabla\) to be a bimodule connection. In this paper, the author defines a quantization of the Levi-Civita connection which is compatible with the quantum Fubini-Study metric \(g\) in the weak and strong sense.
Theorem (see Theorem 7.7). Any quantum projective space \(\mathcal{B}\) admits a connection \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) which is torsion free and cotorsion free. Moreover, in the classical limit it reduces to the Levi-Civita connection for the Fubini-Study metric on the cotangent bundle.
Theorem (see Theorem 8.4). The connection \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) is a bimodule connection and is compatible with the quantum metric, in the sense that \(\nabla g = 0\).
Reviewer: Tatsuki Seto (Tokyo)Compact manifolds with fixed boundary and large Steklov eigenvalueshttps://www.zbmath.org/1483.580072022-05-16T20:40:13.078697Z"Colbois, Bruno"https://www.zbmath.org/authors/?q=ai:colbois.bruno"El Soufi, Ahmad"https://www.zbmath.org/authors/?q=ai:el-soufi.ahmad"Girouard, Alexandre"https://www.zbmath.org/authors/?q=ai:girouard.alexandreSummary: Let \((M,g)\) be a compact Riemannian manifold with boundary. Let \(b>0\) be the number of connected components of its boundary. For manifolds of dimension \(\geq 3\), we prove that for \(j=b+1\) it is possible to obtain an arbitrarily large Steklov eigenvalue \(\sigma_j(M,e^\delta g)\) using a conformal perturbation \(\delta \in C^\infty (M)\) which is supported in a thin neighbourhood of the boundary, with \(\delta =0\) on the boundary. For \(j\leq b\), it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of \(M\). In fact, when working in a fixed conformal class and for \(\delta =0\) on the boundary, it is known that the volume of \((M,e^\delta g)\) has to tend to infinity in order for some \(\sigma _j\) to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.Low correlation noise stability of symmetric setshttps://www.zbmath.org/1483.600332022-05-16T20:40:13.078697Z"Heilman, Steven"https://www.zbmath.org/authors/?q=ai:heilman.steven-mGaussian noise stability is a well-studied topic with connections to geometry of minimal surfaces [\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, in: Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press. 73--143 (2011; Zbl 1261.53006)], hypercontractivity and invariance principles [\textit{E. Mossel} et al., Ann. Math. (2) 171, No. 1, 295--341 (2010; Zbl 1201.60031)], isoperimetric inequalities [\textit{D. M. Kane}, Comput. Complexity 23, No. 2, 151--175 (2014; Zbl 1314.68138)], sharp unique games hardness results in theoretical computer science [\textit{S. Khot} et al., SIAM J. Comput. 37, No. 1, 319--357 (2007; Zbl 1135.68019)], social choice theory, learning theory [\textit{A. R. Klivans} et al. ``Learning geometric concepts via Gaussian surface area'', in: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS'08. Los Alamitos, CA: IEEE Computer Society. 541--550 (2008; \url{doi:10.1109/FOCS.2008.64})] and communication complexity [\textit{A. Chakrabarti} and \textit{O. Regev}, in: Proceedings of the 43rd annual ACM symposium on theory of computing, STOC '11. San Jose, CA, USA, June 6--8, 2011. New York, NY: Association for Computing Machinery (ACM). 51--60 (2011; Zbl 1288.90005)]. The author studies the Gaussian noise stability of subsets \(A\) of Euclidean space satisfying \(A =-A.\) It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Statistical analysis via the curvature of data spacehttps://www.zbmath.org/1483.620102022-05-16T20:40:13.078697Z"Kobayashi, Kei"https://www.zbmath.org/authors/?q=ai:kobayashi.kei"Mitsuru, Orita"https://www.zbmath.org/authors/?q=ai:mitsuru.orita"Wynn, Henry P."https://www.zbmath.org/authors/?q=ai:wynn.henry-pSummary: It has been known that the curvature of data spaces plays a role in data analysis. For example, the Frechet mean (intrinsic mean) always exists uniquely for a probability measure on a non-positively curved metric space. In this paper, we use the curvature of data spaces in a novel manner. A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The population version defines distance by the amount of probability mass accumulated on travelling between two points and geodesic metric arises from the shortest path version. Such metrics are then transformed in a number of ways to produce families of geodesic metric spaces. Empirical versions of the geodesics allow computation of intrinsic means and associated measures of dispersion. A version of the empirical geodesic is introduced based on some metric graphs computed from the sample points. For certain parameter ranges the spaces become CAT(0) spaces and the intrinsic means are unique. In the graph case a minimal spanning tree obtained as a limiting case is CAT(0). In other cases the aggregate squared distance from a test point provides local minima which yield information about clusters. This is particularly relevant for metrics based on so-called metric cones which allow extensions to CAT\(( \kappa )\) spaces. We show how our methods work by using some actual data. This paper is a summary of a longer version. See it for proof of theorems and details.
For the entire collection see [Zbl 1470.00021].Constant-speed ramps for a central force fieldhttps://www.zbmath.org/1483.700152022-05-16T20:40:13.078697Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez-camino.rafael"Perdomo, Óscar"https://www.zbmath.org/authors/?q=ai:perdomo.oscar-marioSummary: We investigate the problem of determining the planar curves that describe ramps where a particle of mass \(m\) moves with constant-speed when is subject to the action of the friction force and a force whose magnitude \(F(r)\) depends only on the distance \(r\) from the origin. In this paper we describe all the constant-speed ramps for the case \(F(r)=-m/r\). We show the circles and the logarithmic spirals play an important role. Not only they are solutions but every other solution approaches either a circle or a logarithmic spiral.Thermodynamics as a theory of measurementhttps://www.zbmath.org/1483.800022022-05-16T20:40:13.078697Z"Lychagin, Valentin"https://www.zbmath.org/authors/?q=ai:lychagin.valentin-vSummary: We are going to a further development of relations between formal measurement theory and thermodynamics. Higher-order geometrical structures on Lagrangian manifolds are introduced and their relations with an accuracy of measurement and confidence domains as well as phase transitions in thermodynamics are discussed. The approach is illustrated for the case of real gases and in detail for methane.Spinorial Snyder and Yang models from superalgebras and noncommutative quantum superspaceshttps://www.zbmath.org/1483.810912022-05-16T20:40:13.078697Z"Lukierski, Jerzy"https://www.zbmath.org/authors/?q=ai:lukierski.jerzy"Woronowicz, Mariusz"https://www.zbmath.org/authors/?q=ai:woronowicz.mariuszSummary: The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.A numbers-based approach to a free particle's proper spacetimehttps://www.zbmath.org/1483.810922022-05-16T20:40:13.078697Z"Ferber, R."https://www.zbmath.org/authors/?q=ai:ferber.reginaldSummary: This paper contains a proposal for a free, nonzero-rest-mass particle's proper spacetime, determined exclusively by the particle's rest mass \(m_0\) and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers \(1, 2, \ldots\) that count de Broglie time units \((h/c^2)(m_0^{-1}\) (see \textit{R. Ferber} in [Found. Phys. Lett. 9, No. 6, 575--586 (1996,; \url{doi:10.1007/BF02190032})]. The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by the Compton unit \((h/c)(m_0^{-1})\). The spatial and temporal coordinates that form the particle's proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term \(m_0c^2\), of an additional energy term of the same order of magnitude, which is related to large intervals of the \(m_0\)-particle's proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed.From 2d droplets to 2d Yang-Millshttps://www.zbmath.org/1483.811002022-05-16T20:40:13.078697Z"Chattopadhyay, Arghya"https://www.zbmath.org/authors/?q=ai:chattopadhyay.arghya"Dutta, Suvankar"https://www.zbmath.org/authors/?q=ai:dutta.suvankar"Mukherjee, Debangshu"https://www.zbmath.org/authors/?q=ai:mukherjee.debangshu"Neetu"https://www.zbmath.org/authors/?q=ai:neetu.babbarSummary: We establish a connection between time evolution of free Fermi droplets and partition function of \textit{generalised q}-deformed Yang-Mills theories on Riemann surfaces. Classical phases of \((0 + 1)\) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces \(\mathcal{H}_+\) and \(\mathcal{H}_-\) associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large \(N\) limit, equal to the quadratic Casimir of \(u(N)\).
We establish an exact mapping between states in \(\mathcal{H}_\pm\) and geometries of droplets. In particular, coherent states in \(\mathcal{H}_\pm\) correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in \(\mathcal{H}_\pm\) is equal to the chiral and anti-chiral partition function of \(2d\) Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space \(\mathcal{H}_+ \otimes \mathcal{H}_-\) admits a \textit{composite} basis, we show that correlation between two classical droplet geometries is equal to the full \(U(N)\) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in \(\mathcal{H}_\pm\) and higher point correlators in \(2d\) Yang-Mills on Riemann surface. There are special states in \(\mathcal{H}_\pm\) whose transition amplitudes are equal to the partition function of 2\textit{d q}-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the \(q\)-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.Two-dimensional quantum Yang-Mills theory and the Makeenko-Migdal equationshttps://www.zbmath.org/1483.811052022-05-16T20:40:13.078697Z"Lévy, Thierry"https://www.zbmath.org/authors/?q=ai:levy.thierrySummary: These notes, echoing a conference given at the Strasbourg-Zurich seminar in October 2017, are written to serve as an introduction to 2-dimensional quantum Yang-Mills theory and to the results obtained in the last five to ten years about its so-called large \(N\) limit.
For the entire collection see [Zbl 1473.53004].D-brane description from nontrivial M2-braneshttps://www.zbmath.org/1483.811182022-05-16T20:40:13.078697Z"Garcia del Moral, M. P."https://www.zbmath.org/authors/?q=ai:garcia-del-moral.maria-pilar"Las Heras, C."https://www.zbmath.org/authors/?q=ai:las-heras.cSummary: We obtain the bosonic D-brane description of toroidally compactified non-trivial M2-branes with the unique property of having a purely discrete supersymmetric regularized spectrum with finite multiplicity. As a byproduct, we generalize the previous Hamiltonian formulation to describe a M2-brane on a completely general constant quantized background \(C_3\) denoted by us as CM2-brane. We show that under this condition, the theory is equivalent to a more restricted one, denoted as an M2-brane with \(C_\pm\) fluxes, which has been shown to have good quantum behavior. As a result, the spectral properties of both sectors must be the same. We then obtain its bosonic D-brane description and discover new symmetries. We find that it contains a new symplectic gauge field in addition to the expected U(1) Dirac-Born-Infeld gauge symmetry and a nontrivial \(U(1)\) associated with the presence of 2-form fluxes. Its bundle description takes on a new structure in the form of a twisted torus bundle. By turning off some of the fields, the D-brane description of other toroidally nontrivial M2-brane sectors can be obtained from this one. The possibility of reinterpreting these sectors in terms of Dp-brane bound states is discussed. These D-brane descriptions constitute String theory sectors with a quantum consistent uplift to M-theory.Holographic chaos, pole-skipping, and regularityhttps://www.zbmath.org/1483.811242022-05-16T20:40:13.078697Z"Natsuume, Makoto"https://www.zbmath.org/authors/?q=ai:natsuume.makoto"Okamura, Takashi"https://www.zbmath.org/authors/?q=ai:okamura.takashiSummary: We investigate the ``pole-skipping'' phenomenon in holographic chaos. According to pole-skipping, the energy-density Green's function is not unique at a special point in the complex momentum plane. This arises because the bulk field equation has two regular near-horizon solutions at the special point. We study the regularity of the two solutions more carefully using curvature invariants. In the upper-half \(\omega\)-plane, one solution, which is normally interpreted as the outgoing mode, is in general singular at the future horizon and produces a curvature singularity. However, at the special point, both solutions are indeed regular. Moreover, the incoming mode cannot be uniquely defined at the special point due to these solutions.Physics of the inverted harmonic oscillator: From the lowest Landau level to event horizonshttps://www.zbmath.org/1483.811392022-05-16T20:40:13.078697Z"Subramanyan, Varsha"https://www.zbmath.org/authors/?q=ai:subramanyan.varsha"Hegde, Suraj S."https://www.zbmath.org/authors/?q=ai:hegde.suraj-s"Vishveshwara, Smitha"https://www.zbmath.org/authors/?q=ai:vishveshwara.smitha"Bradlyn, Barry"https://www.zbmath.org/authors/?q=ai:bradlyn.barrySummary: In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking-Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.A model of spontaneous collapse with energy conservationhttps://www.zbmath.org/1483.811782022-05-16T20:40:13.078697Z"Snoke, D. W."https://www.zbmath.org/authors/?q=ai:snoke.david-wSummary: A model of spontaneous collapse of fermionic degrees of freedom in a quantum field is presented which has the advantages that it explicitly maintains energy conservation and gives results in agreement with an existing numerical method for calculating quantum state evolution, namely the quantum trajectories model.Spacetimes with continuous linear isotropies. I: Spatial rotationshttps://www.zbmath.org/1483.830062022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to \textit{S. W. Goode} and \textit{J. Wainwright} [ibid. 18, 315--331 (1986; Zbl 0584.53029)]. Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis's original condition for that case is necessary as well as sufficient.Microscopic origin of Einstein's field equations and the \textit{raison d'être} for a positive cosmological constanthttps://www.zbmath.org/1483.830082022-05-16T20:40:13.078697Z"Padmanabhan, T."https://www.zbmath.org/authors/?q=ai:padmanabhan.thanu|padmanabhan.t-v"Chakraborty, Sumanta"https://www.zbmath.org/authors/?q=ai:chakraborty.sumantaSummary: In the paradigm of effective field theory, one hierarchically obtains the effective action \(\mathcal{A}_{\mathrm{eff}} [q, \cdots]\) for some low(er) energy degrees of freedom \(q\), by integrating out the high(er) energy degrees of freedom \(\xi\), in a path integral, based on an action \(\mathcal{A} [q, \xi, \cdots]\). We show how one can integrate out a vector field \(v^a\) in an action \(\mathcal{A} [\Gamma, v, \cdots]\) and obtain an effective action \(\mathcal{A}_{\mathrm{eff}}[\Gamma, \cdots]\) which, on variation with respect to the connection \(\Gamma\), leads to the Einstein's field equations and a metric compatible with the connection. The derivation \textit{predicts} a non-zero, positive, cosmological constant, which arises as an integration constant. The Euclidean action \(\mathcal{A} [\Gamma, v, \cdots]\), has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field \(v^a\) can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.A wavefunction description for a localized quantum particle in curved spacetimeshttps://www.zbmath.org/1483.830092022-05-16T20:40:13.078697Z"Perche, T. Rick"https://www.zbmath.org/authors/?q=ai:perche.tales-rick"Neuser, Jonas"https://www.zbmath.org/authors/?q=ai:neuser.jonasComment on: ``Do electromagnetic waves always propagate along null geodesics?''https://www.zbmath.org/1483.830122022-05-16T20:40:13.078697Z"Linnemann, Niels"https://www.zbmath.org/authors/?q=ai:linnemann.niels-s"Read, James"https://www.zbmath.org/authors/?q=ai:read.jamesSpacetimes with continuous linear isotropies. III: Null rotationshttps://www.zbmath.org/1483.830202022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group \(G_r\) with \(r\ge 3\) acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing ``pure radiation'' (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a ``tachyon fluid'').
For Parts I and II, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 6, Paper No. 61, 12 p. (2021; Zbl 1483.83019)].5D \(\mathcal{N} = 1\) super QFT: symplectic quivershttps://www.zbmath.org/1483.830262022-05-16T20:40:13.078697Z"Saidi, E. H."https://www.zbmath.org/authors/?q=ai:saidi.el-hassan"Drissi, L. B."https://www.zbmath.org/authors/?q=ai:drissi.lalla-btissamSummary: We develop a method to build new 5D \(\mathcal{N} = 1\) gauge models based on Sasaki-Einstein manifolds \(Y^{p, q}\). These models extend the standard 5D ones having a unitary \(\mathrm{SU}(p)_q\) gauge symmetry based on \(Y^{p, q} \). Particular focus is put on the building of a gauge family with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds \(\hat{X}(Y^{2r, 0})\) constructed from conical \(Y^{2r, 0}\). By using outer-automorphism symmetries of 5D \(\mathcal{N} = 1\) BPS quivers with unitary \(\mathrm{SU}(2r)\) gauge invariance, we also construct BPS quivers with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) gauge symmetry. Other related aspects are discussed.Exploration of a singular fluid spacetimehttps://www.zbmath.org/1483.830282022-05-16T20:40:13.078697Z"Remmen, Grant N."https://www.zbmath.org/authors/?q=ai:remmen.grant-nSummary: We investigate the properties of a special class of singular solutions for a self-gravitating perfect fluid in general relativity: the singular isothermal sphere. For arbitrary constant equation-of-state parameter \(w=p/\rho\), there exist static, spherically-symmetric solutions with density profile \(\propto 1/r^2\), with the constant of proportionality fixed to be a special function of \(w\). Like black holes, singular isothermal spheres possess a fixed mass-to-radius ratio independent of size, but no horizon cloaking the curvature singularity at \(r=0\). For \(w=1\), these solutions can be constructed from a homogeneous dilaton background, where the metric spontaneously breaks spatial homogeneity. We study the perturbative structure of these solutions, finding the radial modes and tidal Love numbers, and also find interesting properties in the geodesic structure of this geometry. Finally, connections are discussed between these geometries and dark matter profiles, the double copy, and holographic entropy, as well as how the swampland distance conjecture can obscure the naked singularity.On a conformal Schwarzschild-de Sitter spacetimehttps://www.zbmath.org/1483.830402022-05-16T20:40:13.078697Z"Culetu, Hristu"https://www.zbmath.org/authors/?q=ai:culetu.hristuSummary: On the basis of the C-metric, we investigate the conformal Schwarzschild - deSitter spacetime and compute the source stress tensor and study its properties, including the energy conditions. Then we analyze its extremal version \((b^2=27m^2\), where \(b\) is the deS radius and \(m\) is the source mass), when the metric is nonstatic. The weak-field
version is investigated in several frames, and the metric becomes flat with the special choice \(b=1/a\), \(a\) being the constant acceleration of the Schwarzschild-like mass or black hole. This form is Rindler's geometry in disguise and is also conformal to a de Sitter metric where the acceleration plays the role of the Hubble constant. In its time dependent version, one finds that the proper acceleration of a static observer is constant everywhere, in contrast with the standard Rindler case. The timelike geodesics along the z-direction are calculated and proves to be hyperbolae.Stability analysis of geodesics and quasinormal modes of a dual stringy black hole via Lyapunov exponentshttps://www.zbmath.org/1483.830452022-05-16T20:40:13.078697Z"Giri, Shobhit"https://www.zbmath.org/authors/?q=ai:giri.shobhit"Nandan, Hemwati"https://www.zbmath.org/authors/?q=ai:nandan.hemwatiSummary: We investigate the stability of both timelike as well as null circular geodesics in the vicinity of a dual (3+1) dimensional stringy black hole (BH) spacetime by using an excellent tool so-called Lyapunov exponent. The proper time \((\tau)\) Lyapunov exponent \((\lambda_p)\) and coordinate time \((t)\) Lyapunov exponent \((\lambda_c)\) are explicitly derived to analyze the stability of equatorial circular geodesics for the stringy BH spacetime with \textit{electric charge} parameter \((\alpha )\) and \textit{magnetic charge} parameter \((Q)\). By computing
these exponents for both the cases of BH spacetime, it is observed that the coordinate time Lyapunov exponent of magnetically charged stringy BH for both timelike and null geodesics are independent of magnetic charge parameter \((Q)\). The variation of the ratio of Lyapunov exponents with radius of timelike circular orbits \((r_0/M)\) for both the cases of stringy BH are presented. The behavior of instability exponent for null circular geodesics with respect to charge parameters \((\alpha\) and \(Q)\) are also observed for both the cases of BH. Further, by establishing a relation between quasinormal modes (QNMs) and parameters related to null circular geodesics (like angular frequency and Lyapunov exponent), we deduced the QNMs (or QNM frequencies) for a massless scalar field perturbation around \textit{both} the cases of stringy BH spacetime in the eikonal limit. The variation of scalar field potential with charge parameters and angular momentum of perturbation \((l)\) are visually presented and discussed accordingly.Straightforward Hamiltonian analysis of \textit{BF} gravity in \(n\) dimensionshttps://www.zbmath.org/1483.830772022-05-16T20:40:13.078697Z"Montesinos, Merced"https://www.zbmath.org/authors/?q=ai:montesinos.merced"Escobedo, Ricardo"https://www.zbmath.org/authors/?q=ai:escobedo.ricardo"Celada, Mariano"https://www.zbmath.org/authors/?q=ai:celada.marianoSummary: We perform, in a manifestly \(\mathrm{SO}(n-1,1) [\mathrm{SO}(n)]\) covariant fashion, the Hamiltonian analysis of general relativity in \(n\) dimensions written as a constrained \textit{BF} theory. We solve the constraint on the \(B\) field in a way naturally adapted to the foliation of the spacetime that avoids explicitly the introduction of the vielbein. This leads to a form of the action involving a presymplectic structure, which is reduced by doing a suitable parametrization of the connection and then, after integrating out some auxiliary fields, the Hamiltonian form involving only first-class constraints is obtained.Nonstaticity with type II, III, or IV matter field in \(f(R_{\mu\nu\rho\sigma},g^{\mu\nu})\) gravityhttps://www.zbmath.org/1483.830882022-05-16T20:40:13.078697Z"Maeda, Hideki"https://www.zbmath.org/authors/?q=ai:maeda.hidekiSummary: In all \(n(\ge 3)\)-dimensional gravitation theories whose Lagrangians are functions of the Riemann tensor and metric, we show that static solutions are absent unless the total energy-momentum tensor for matter fields is of type I in the Hawking-Ellis classification. In other words, there is no hypersurface-orthogonal timelike Killing vector in a spacetime region with an energy-momentum tensor of type II, III, or IV. This asserts that, if back-reaction is taken into account to give a self-consistent solution, ultra-dense regions with a semiclassical type-IV matter field cannot be static even with higher-curvature correction terms. As a consequence, a static Planck-mass relic is possible as a final state of an evaporating black hole only if the semiclassical total energy-momentum tensor is of type I.Static cosmological solutions in quadratic gravityhttps://www.zbmath.org/1483.830892022-05-16T20:40:13.078697Z"Müller, Daniel"https://www.zbmath.org/authors/?q=ai:muller.daniel"Toporensky, Alexey"https://www.zbmath.org/authors/?q=ai:toporensky.alexey-vSummary: We consider conditions for existence and stability of a static cosmological solution in quadratic gravity. It appears that such a solution for a Universe filled by only one type of perfect fluid is possible in a wide range of the equation of state parameter \(w\) and for both positively and negatively spatially curved Universe. We show that the static solution for the negative curvature is always unstable if we require positive energy density of the matter content. On the other hand, a static solution with positive spatial curvature can be stable under certain restrictions. Stability of this solution with respect to isotropic perturbation requires that the coupling constant with the \(R^2\) therm in the Lagrangian of the theory is positive, and the equations of state parameter \(w\) is located in a rather narrow interval. Nevertheless, the stability condition does not require violation of the strong energy condition. Taking into account anisotropic perturbations leads to further restrictions on the values of coupling constants and the parameter \(w\).Pre-inflationary bounce effects on primordial gravitational waves of \(f(R)\) gravityhttps://www.zbmath.org/1483.830902022-05-16T20:40:13.078697Z"Odintsov, S. D."https://www.zbmath.org/authors/?q=ai:odintsov.sergei-d"Oikonomou, V. K."https://www.zbmath.org/authors/?q=ai:oikonomou.vasilis-kSummary: In this work we shall study a possible pre-inflationary scenario for our Universe and how this might be realized by \(f(R)\) gravity. Specifically, we shall introduce a scenario in which the Universe in the pre-inflationary era contracts until it reaches a minimum magnitude, and subsequently expands, slowly entering a slow-roll quasi-de Sitter inflationary era. This pre-inflationary bounce avoids the cosmic singularity, and for the eras before and after the quasi-de Sitter inflationary stage, approximately satisfies the string theory motivated scale factor duality \(a(t) = a^{-1}(-t)\).
We investigate which approximate forms of \(f(R)\) can realize such a non-singular pre-inflationary scenario, the quasi-de Sitter patch of which is described by an \(R^2\) gravity, thus the exit from inflation is guaranteed. Furthermore, since in string theory pre-Big Bang scenarios lead to an overall amplification of the gravitational wave energy spectrum, we examine in detail this perspective for the \(f(R)\) gravity generating this pre-inflationary non-singular bounce. As we show, in the \(f(R)\) gravity case, the energy spectrum of the primordial gravitational waves background is also amplified, however the drawback is that the amplification is too small to be detected by future high frequency interferometers. Thus we conclude that, as in the case of single scalar field theories, \(f(R)\) gravity cannot produce detectable stochastic gravitational waves and a synergistic theory of scalars and higher order curvature terms might be needed.Particle creation in some LRS Bianchi I modelshttps://www.zbmath.org/1483.830912022-05-16T20:40:13.078697Z"Pimentel, Luis O."https://www.zbmath.org/authors/?q=ai:pimentel.luis-o"Pineda, Flavio"https://www.zbmath.org/authors/?q=ai:pineda.flavioSummary: In this work we consider particle creation by the expansion of the universe, using two Bianchi type I anisotropic models. The particles studied are of spin 0 and 1/2. The cosmological models have rotational symmetry, which allows us to solve exactly the equations of motion. The number density of the created particles is calculated with the method of Bogolubov transformations.Inflation with Gauss-Bonnet and Chern-Simons higher-curvature-corrections in the view of GW170817https://www.zbmath.org/1483.830972022-05-16T20:40:13.078697Z"Venikoudis, S. A."https://www.zbmath.org/authors/?q=ai:venikoudis.s-a"Fronimos, F. P."https://www.zbmath.org/authors/?q=ai:fronimos.f-pSummary: Inflationary era of our Universe can be characterized as semi-classical because it can be described in the context of four-dimensional Einstein's gravity involving quantum corrections. These string motivated corrections originate from quantum theories of gravity such as superstring theories and include higher gravitational terms as, Gauss-Bonnet and Chern-Simons terms. In this paper we investigated inflationary phenomenology coming from a scalar field, with quadratic curvature terms in the view of GW170817. Firstly, we derived the equations of motion, directly from the gravitational action. As a result, formed a system of
differential equations with respect to Hubble's parameter and the inflaton field which was very complicated and cannot be solved analytically, even in the minimal coupling case. Based on the observations from GW170817 event, which have shown that the speed of the primordial gravitational wave is equal to the speed of light, \(c_{\mathcal{T}}^2=1\) in natural units, our equations of motion where simplified after applying the constraint \(c_{\mathcal{T}}^2=1\), the slow-roll approximations and neglecting the string corrections. We described the dynamics of inflationary phenomenology and proved that theories with Gauss-Bonnet term can be compatible with recent observations. Also, the Chern-Simons term leads to asymmetric generation and evolution of the two circular polarization states of gravitational wave. Finally, viable inflationary models are presented, consistent with the observational constraints. The possibility of a blue tilted tensor spectral index is briefly investigated.Quantum phase space description of a cosmological minimal massive bigravity modelhttps://www.zbmath.org/1483.830982022-05-16T20:40:13.078697Z"Vera-Hernández, Julio César"https://www.zbmath.org/authors/?q=ai:vera-hernandez.julio-cesarSummary: Bimetric gravity theories describes gravitational interactions in the presence of an extra spin-2 field. The Hassan-Rosen (HR) nonlinear massive minimal bigravity theory is a ghost-free bimetric theory formulated with respect a flat, dynamical reference metric. In this work the deformation quantization formalism is applied to a HR cosmological model in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in the minisuperspace. The Friedmann-Lemaître-Robertson-Walker model with flat metrics is worked out and the computation of the Wigner functions for the Hartle-Hawking, Vilenkin and Linde wavefunctions are done numerically and, in the Hartle-Hawking case, also analytically. From the stability analysis in the quantum minisuper phase space it is found an interpretation of the mass of graviton as an emergent cosmological constant and as a measure of the deviation of classical behavior of the Wigner functions. Also, from the Hartle-Hawking case, an interesting relation between the curvature and the mass of graviton in a cusp catastrophe surface is discussed.