Recent zbMATH articles in MSC 53https://www.zbmath.org/atom/cc/532021-04-16T16:22:00+00:00WerkzeugThe action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface.https://www.zbmath.org/1456.200422021-04-16T16:22:00+00:00"Bowditch, Brian H."https://www.zbmath.org/authors/?q=ai:bowditch.brian-h"Sakuma, Makoto"https://www.zbmath.org/authors/?q=ai:sakuma.makotoSummary: Let \(\Sigma\) be a complete finite-area orientable hyperbolic surface with one cusp, and let \(\mathcal{R}\) be the space of complete geodesic rays in \(\Sigma\) emanating from the puncture. Then there is a natural action of the mapping class group of \(\Sigma\) on \(\mathcal{R}\). We show that this action is ``almost everywhere'' wandering.Affine structures on Lie groupoids.https://www.zbmath.org/1456.220012021-04-16T16:22:00+00:00"Lang, Honglei"https://www.zbmath.org/authors/?q=ai:lang.honglei"Liu, Zhangju"https://www.zbmath.org/authors/?q=ai:liu.zhangju"Sheng, Yunhe"https://www.zbmath.org/authors/?q=ai:sheng.yunheAuthors' abstract: We study affine structures on a Lie groupoid, including affine \(k\)-vector fields, \(k\)-forms and \((p, q)\)-tensors. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher-Nijenhuis bracket are graded strict Lie 2-algebras, and affine \((1, 1)\)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
Reviewer: Iakovos Androulidakis (Athína)Pseudoconvexity for the special Lagrangian potential equation.https://www.zbmath.org/1456.350892021-04-16T16:22:00+00:00"Harvey, F. Reese"https://www.zbmath.org/authors/?q=ai:harvey.reese"Lawson, H. Blaine jun."https://www.zbmath.org/authors/?q=ai:lawson.h-blaine-junSummary: The Special Lagrangian Potential Equation for a function \(u\) on a domain \(\Omega \subset \mathbb{R}^n\) is given by \(\mathrm{tr}\{\arctan (D^2 u)\} =\theta\) for a contant \(\theta \in (-n \frac{\pi}{2}, n \frac{\pi}{2})\). For \(C^2\) solutions the graph of \(Du\) in \(\Omega \times \mathbb{R}^n\) is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting. The result carries over to many related equations -- for example, those obtained by taking \(\sum_k \arctan \lambda_k^{\mathfrak{g}} =\theta\) where \(\mathfrak{g}: \mathrm{Sym}^2(\mathbb{R}^n)\rightarrow \mathbb{R}\) is a Gårding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is \(\sum_j \arctan \kappa_j = \theta\) where \(\kappa_1,\dots, \kappa_n\) are the principal curvatures of the graph of \(u\) in \(\Omega \times \mathbb{R}\). We also discuss the inhomogeneous Dirichlet Problem
\[
\mathrm{tr}\{\arctan (D^2_x \,u)\} = \psi (x)
\]
where \(\psi : {\overline{\Omega}} \rightarrow (-n \frac{\pi}{2}, n \frac{\pi}{2})\). This equation has the feature that the pull-back of \(\psi\) to the Lagrangian submanifold \(L\equiv\text{graph} (Du)\) is the phase function \(\theta\) of the tangent spaces of \(L\). On \(L\) it satisfies the equation \(\nabla \psi = -JH\) where \(H\) is the mean curvature vector field of \(L\).Minimal projectivity condition for a smooth mapping and the Gronwall problem.https://www.zbmath.org/1456.530132021-04-16T16:22:00+00:00"Shelekhov, A. M."https://www.zbmath.org/authors/?q=ai:shelekhov.aleksandr-mikhailovich|shelekhov.alexander-mSummary: In this paper, the following assertion is proved: Let \(GW\) and \(\widetilde{GW}\) be Grassmannian three-webs defined respectively in domains \(D\) and \(\tilde{D}\) of a Grassmannian manifold of straight lines of the projective space \(P^{r+1}\); \(\Phi : D \rightarrow \tilde{D}\) be a local diffeomorphism that maps foliations of the web \(GW\) to foliations of the web \(\widetilde{GW} \). Then \(\Phi\) maps bundles of lines to bundles of lines, i.e., induces a point transformation, which is a projective transformation. In the case where \(r = 1\), the proof is much more complicated than in the multidimensional case. In the case where \(r = 1\), the dual theorem is formulated as follows: Let \(LW\) be a rectilinear three-web on a plane, i.e., three families of lines in the general position, and let this web be not regular, i.e., not locally diffeomorphic to a three-web formed by three families of parallel straight lines. Then each local diffeomorphism that maps a three-web \(LW\) to another rectilinear three-web \(\widetilde{LW}\) is a projective transformation. As a consequence, we obtain the positive solution of the Gronwall problem [\textit{T. H. Gronwall}, Journ. de Math. (6) 8, 59--102 (1912; JFM 43.0159.03)]: If \(W\) is a linearizable irregular three-web and \(\theta\) and \(\tilde{\theta}\) are local diffeomorphisms that map the three-web \(W\) to some rectilinear three-webs, then \(\tilde{\theta} = \pi \circ \theta \), where \(\pi\) is a projective transformation.A class of inverse curvature flows in \(\mathbb{R}^{n+1}\). II.https://www.zbmath.org/1456.530762021-04-16T16:22:00+00:00"Hu, Jin-Hua"https://www.zbmath.org/authors/?q=ai:hu.jinhua"Mao, Jing"https://www.zbmath.org/authors/?q=ai:mao.jing"Tu, Qiang"https://www.zbmath.org/authors/?q=ai:tu.qiang"Wu, Di"https://www.zbmath.org/authors/?q=ai:wu.diSummary: : We consider closed, star-shaped, admissible hypersurfaces in \(\mathbb{R}^{n+1}\) expanding along the flow \(\dot{X}=|X|^{\alpha-1}F^{-\beta}\), \(\alpha\leq 1\), \(\beta> 0\), and prove that for the case \(\alpha\leq 1,\beta> 0, \alpha+\beta\leq 2\), this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case \(\alpha\leq 1, \alpha+\beta> 2\), if furthermore the initial closed hypersurface is strictly convex, then the strict convexity is preserved during the evolution process and the flow blows up at finite time.Concise notes on special holonomy with an emphasis on Calabi-Yau and \(G_2\)-manifolds.https://www.zbmath.org/1456.530062021-04-16T16:22:00+00:00"Oliveira, Gonçalo"https://www.zbmath.org/authors/?q=ai:oliveira.goncaloSummary: These are notes for a very short introduction to some selected topics on special Riemannian holonomy with a focus on Calabi-Yau and \(G_2\)-manifolds. No material in these notes is original and more on it can be found in the papers/books of Bryant, Hitchin, Joyce and Salamon referenced during the text.Open and closed mirror symmetry.https://www.zbmath.org/1456.530692021-04-16T16:22:00+00:00"Amorim, Lino"https://www.zbmath.org/authors/?q=ai:amorim.linoSummary: Mirror symmetry predicts a deep correspondence between the symplectic geometry of a space and the algebraic geometry of its ``mirror''. There are different versions of this correspondence, from the equality of some numerical invariants, first predicted by physicists, to categorical versions proposed by Kontsevich. This paper reviews some of these versions and illustrates them on a relatively simple example: a sphere with three orbifold points (on the symplectic side). We explain how to construct the ``mirror'' space, state the mirror predictions and describe an approach to prove them.Arsove-Huber theorem in higher dimensions.https://www.zbmath.org/1456.310052021-04-16T16:22:00+00:00"Ma, Shiguang"https://www.zbmath.org/authors/?q=ai:ma.shiguang"Qing, Jie"https://www.zbmath.org/authors/?q=ai:qing.jieThe aim is to extend the Arsove-Huber theory of surfaces to higher dimensions. A basic tool is the \(n\)-Laplace equation \[\text{div}(|\nabla u|^{n-2}\nabla u)= 0\] in exactly \(n\) dimensions (the so-called borderline case). The Brezis-Merle inequality (a refined version of Trudinger's inequality) is applied in \(n\) dimensions. An ingredient is the Wolff potential.
The theory is applied for hypersurfaces in a hyperbolic space, having nonnegative Ricci curvature. Even some unpublished results are announced.
For the entire collection see [Zbl 1446.58001].
Reviewer: Peter Lindqvist (Trondheim)On LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)4-dimensional manifolds with nonnegative scalar curvature and CMC boundary.https://www.zbmath.org/1456.530332021-04-16T16:22:00+00:00"Wang, Yaohua"https://www.zbmath.org/authors/?q=ai:wang.yaohuaRicci soliton vector fields of spherically symmetric static spacetimes.https://www.zbmath.org/1456.530342021-04-16T16:22:00+00:00"Tahirullah"https://www.zbmath.org/authors/?q=ai:tahirullah."Ali, Ahmad T."https://www.zbmath.org/authors/?q=ai:ali.ahmad-tawfik"Khan, Suhail"https://www.zbmath.org/authors/?q=ai:khan.suhailOn the geometry and entropy of non-Hamiltonian phase space.https://www.zbmath.org/1456.820232021-04-16T16:22:00+00:00"Sergi, Alessandro"https://www.zbmath.org/authors/?q=ai:sergi.alessandro"Giaquinta, Paolo V."https://www.zbmath.org/authors/?q=ai:giaquinta.paolo-vOptimal lower bounds for Donaldson's J-functional.https://www.zbmath.org/1456.530592021-04-16T16:22:00+00:00"Sjöström Dyrefelt, Zakarias"https://www.zbmath.org/authors/?q=ai:sjostrom-dyrefelt.zakariasIn this paper the author provides an explicit formula for the optimal lower bound of Donaldson's J-functional, with the hope of finding explicitly
the optimal constant in the definition of coercivity, which always exists and takes negative values. This constant is positive precisely if the J-equation admits a solution.
Firstly, the explicit formula leads to new existence criteria for constant scalar curvature Kähler (cscK) metrics in terms of Tian's alpha invariant. Secondly, the above formula
enables us to discuss Calabi dream manifolds and an analogous notion for the J-equation,
and show that for surfaces the optimal bound is an explicitly computable rational function
which typically tends to minus infinity as the underlying class approaches the boundary of
the Kähler cone, even when the underlying Kähler classes admit cscK metrics.
Finally, the author shows that if the Lejmi-Székelyhidi conjecture [\textit{M. Lejmi} and \textit{G. Székelyhidi}, Adv. Math. 274, 404--431 (2015; Zbl 1370.53051)] holds, then the optimal bound
coincides with its algebraic counterpart, namely the set of J-semistable classes equals to the closure
of the set of uniformly J-stable classes in the Kähler cone, and there exists an optimal degeneration
for uniform J-stability.
Reviewer: Quanting Zhao (Wuhan)Morse theory for minimal surfaces in manifolds.https://www.zbmath.org/1456.580122021-04-16T16:22:00+00:00"Kim, Hwajeong"https://www.zbmath.org/authors/?q=ai:kim.hwajeongSummary: A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \(\mathcal{E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \(\mathcal{E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \(\mathcal{E}\) is non-degenerated.Shifted derived Poisson manifolds associated with Lie pairs.https://www.zbmath.org/1456.530652021-04-16T16:22:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Stiénon, Mathieu"https://www.zbmath.org/authors/?q=ai:stienon.mathieu"Xu, Ping"https://www.zbmath.org/authors/?q=ai:xu.pingSummary: We study the shifted analogue of the ``Lie-Poisson'' construction for \(L_\infty\) algebroids and we prove that any \(L_\infty\) algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair \((L, A)\), the space \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A))\) admits a degree \((+1)\) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential \(d_A^{\mathrm{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\rightarrow\Omega^{\bullet+1}_A(\Lambda^\bullet (L/A))\) as unary \(L_\infty\) bracket. This degree \((+1)\) derived Poisson algebra structure on \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley-Eilenberg hypercohomology \(\mathbb{H}(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A)),d_A^{\mathrm{Bott}})\) admits a canonical Gerstenhaber algebra structure.A singular radial connection over \(\mathbb B^5\) minimizing the Yang-Mills energy.https://www.zbmath.org/1456.580152021-04-16T16:22:00+00:00"Petrache, Mircea"https://www.zbmath.org/authors/?q=ai:petrache.mirceaSummary: We prove that the pullback of the \(\mathrm{SU}(n)\)-soliton of Chern number \(c_2=1\) over \(\mathbb S^4\) via the radial projection \(\pi :\mathbb B^5{\setminus }\{0\}\to \mathbb S^4\) minimizes the Yang-Mills energy under a topologically fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can have singular sets of codimension 5.Multi-Regge limit of the two-loop five-point amplitudes in \(\mathcal{N} = 4\) super Yang-Mills and \(\mathcal{N} = 8\) supergravity.https://www.zbmath.org/1456.831122021-04-16T16:22:00+00:00"Caron-Huot, Simon"https://www.zbmath.org/authors/?q=ai:caron-huot.simon"Chicherin, Dmitry"https://www.zbmath.org/authors/?q=ai:chicherin.dmitry"Henn, Johannes"https://www.zbmath.org/authors/?q=ai:henn.johannes-m"Zhang, Yang"https://www.zbmath.org/authors/?q=ai:zhang.yang"Zoia, Simone"https://www.zbmath.org/authors/?q=ai:zoia.simoneSummary: In previous work [\textit{E. D'Hoker} et al.,ibid. 2020, No. 8, Paper No. 135, 80 p. (2020; Zbl 1454.83159); \textit{C. R. Mafra} and \textit{O. Schlotterer}, ibid. 2015, No. 10, Paper No. 124, 29 p. (2015; Zbl 1388.83860)], the two-loop five-point amplitudes in \(\mathcal{N} = 4\) super Yang-Mills theory and \(\mathcal{N} = 8\) supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the \(\mathcal{N} = 4\) super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.Positively curved Killing foliations via deformations.https://www.zbmath.org/1456.530232021-04-16T16:22:00+00:00"Caramello, Francisco C. jun.."https://www.zbmath.org/authors/?q=ai:caramello.francisco-c-jun"Töben, Dirk"https://www.zbmath.org/authors/?q=ai:toben.dirkThis paper contain several interesting results on the (transverse) geometry and topology of compact manifolds with a Killing foliation with positive transverse sectional curvature. Recall that a Killing foliation is a complete Riemannian foliation with globally constant Molino sheaf; a class that includes Riemannian foliations on simply connected manifolds and foliations induced by isometric actions. A key step behind the main results is to deform the foliation into one with closed leaves while keeping transverse geometric properties, using a method of \textit{A. Haefliger} and \textit{E. Salem} [Ill. J. Math. 34, No. 4, 706--730 (1990; Zbl 0701.53053)]. One may then apply an orbifold version of the maximal symmetry-rank classification in positive curvature to show that the manifold fibers over finite quotients of spheres or weighted complex projective spaces if the closure of the original foliation has maximal dimension. Several other consequences involving the basic Euler characteristic are also established.
Reviewer: Renato G. Bettiol (New York)Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric.https://www.zbmath.org/1456.530442021-04-16T16:22:00+00:00"Lee, Hyunjin"https://www.zbmath.org/authors/?q=ai:lee.hyunjin"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinSummary: In this paper, first, we investigate the commuting property between the normal Jacobi operator \(\bar{R}_N\) and the structure Jacobi operator \(R_\xi\) for Hopf real hypersurfaces in the complex quadric \(Q^m = \mathrm{SO}_{m+2}/ \mathrm{SO}_m \mathrm{SO}_2\) for \(m \geqslant 3\), which is defined by \(\bar{R}_N R_\xi =R_\xi \bar{R}_N\). Moreover, a new characterization of Hopf real hypersurfaces with \(\mathfrak{A}\)-principal singular normal vector field in the complex quadric \(Q^m\) is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric \(Q^m\) with commuting Jacobi operators.A nilpotency index of conformal manifolds.https://www.zbmath.org/1456.814362021-04-16T16:22:00+00:00"Komargodski, Zohar"https://www.zbmath.org/authors/?q=ai:komargodski.zohar"Razamat, Shlomo S."https://www.zbmath.org/authors/?q=ai:razamat.shlomo-s"Sela, Orr"https://www.zbmath.org/authors/?q=ai:sela.orr"Sharon, Adar"https://www.zbmath.org/authors/?q=ai:sharon.adarSummary: We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index.Reduction of Nambu-Poisson manifolds by regular distributions.https://www.zbmath.org/1456.530672021-04-16T16:22:00+00:00"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaSummary: The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by \textit{R. Ibáñez} et al. [Rep. Math. Phys. 42, No. 1-2, 71--90 (1998; Zbl 0931.37024)]. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.Spinor modules for Hamiltonian loop group spaces.https://www.zbmath.org/1456.580052021-04-16T16:22:00+00:00"Loizides, Yiannis"https://www.zbmath.org/authors/?q=ai:loizides.yiannis"Meinrenken, Eckhard"https://www.zbmath.org/authors/?q=ai:meinrenken.eckhard"Song, Yanli"https://www.zbmath.org/authors/?q=ai:song.yanli.1|song.yanliThis paper studies the spinor modules theory of loop groups.
Let \( G \) be a compact, connected Lie group and let the loop group \( LG \) be the Banach Lie group of \(G\)-valued loops of a fixed Sobolev class \( S > 1/2 \). The authors prove that the tangent bundle of any proper Hamiltonian loop group space \(M\) possesses a canonically defined \(LG-\)equivariant completion \(\overline{T}M\), such
that any weakly symplectic 2-form \(\omega\) of any proper Hamiltonian loop group space extends to a strongly symplectic 2-form on \(\overline{T}M\).
Furthermore, it is proved that the bundle \(\overline{T}M\) possesses a distinguished \(LG-\)invariant polarization and a global \(LG-\)invariant \(\omega-\)compatible complex structure \(J\)
within this polarization class, unique up to homotopy. This leads to the definition
of \( LG-\)equivariant spinor bundle \( \mathrm{S}_{\overline{T}M} \),
which is used to construct the twisted \( \mathrm{Spin}_c \)-structure for the associated quasi-Hamiltonian \(G\)-space \(M\). This is is a way to get a finite-dimensional version of the spinor module \( \mathrm{S}_{\overline{T}M} \).
The authors also discuss \textquoteleft abelianization procedure\textquoteright\, which is another way to get a finite-dimensional version of \( \mathrm{S}_{\overline{T}M} \). The idea is to shift
to a finite-dimensional maximal torus \(T \subseteq LG-\)invariant submanifold of \(M,\) and construct an equivalent
\(\mathrm{Spin}_c \)-structure on that
submanifold. More precisely, if the moment map \(\Phi\) of a proper Hamiltonian
\(LG\)-space is transverse to the Lie algebra \( \mathfrak{t}^* \) (as a space of constant connections valued
in the Lie algebra of the maximal torus \( T \)), then the pre-image \(\Phi^{-1} (\mathfrak{t}^*)\)
is a finite-dimensional pre-symplectic manifold that
inherits a \(T\)-equivalent \(\mathrm{Spin}_c \)-structure.
Reviewer: Kaveh Eftekharinasab (Kyiv)Positive loops of loose Legendrian embeddings and applications.https://www.zbmath.org/1456.530622021-04-16T16:22:00+00:00"Liu, Guogang"https://www.zbmath.org/authors/?q=ai:liu.guogangThe article under review is concerned with positive loops of loose Legendrian embeddings. Let \((M, \xi = \ker \alpha)\) be a contact manifold, and \(L \subset M\) be a Legendrian submanifold. A Legendrian isotopy \(\phi: L \times [0,1] \to M\) is called positive if \(\alpha(\partial_t \phi_t)>0\). Meanwhile, the concept of loose Legendrian submanifolds in higher-dimensions was introduced by \textit{E. Murphy} in [``Loose Legendrian embeddings in high dimensional contact manifolds'', Preprint, \url{arXiv:1201.2245}]. The main result of this article is that for a contact manifold \((M, \xi)\) of dimension greater or equal to 5, any loose Legendrian submanifold \(L\) admits a contractible positive loop of Legendrian embeddings based at \(L\). Note that without the looseness assumption, \textit{F. Laudenbach} [in: New perspectives and challenges in symplectic field theory. Dedicated to Yakov Eliashberg on the occasion of his 60th birthday. Providence, RI: American Mathematical Society (AMS). 299--305 (2009; Zbl 1187.53080)] proves that \(L\) always admits positive loops of Legendrian immersions.
This main result can be regarded as an extension of certain flexibility, since the concept of higher dimensional loose submanifolds is a generalization of the stabilization of Legendrian submanifolds in dimension three. In particular, loose Legendrian submanifolds are flexible in the sense that they satisfy h-principle. As a comparison, a result from \textit{V. Colin} et al. [Int. Math. Res. Not. 2017, No. 20, 6231--6254 (2017; Zbl 1405.53108)] shows that the stabilization of the zero-section \(L\) of \(T^*S^1 \times \mathbb R\), denoted by \(S(L)\), admits a positive loop of Legendrian embeddings based at \(S(L)\).
Next, an application of this main result is a holomorphic curve free proof of the existence of tight contact structures in every dimension. Recall that the concept of an overtwisted or tight contact structure in higher dimension was studied in [\textit{M. S. Borman} et al., Acta Math. 215, No. 2, 281--361 (2015; Zbl 1344.53060)]. Explicitly, this article shows that for any \(n \geq 1\), the contact manifold \((S^{n-1} \times \mathbb R^n, \xi_{\mathrm{std}})\) is tight. Its proof is deeply rooted in the generating function theory.
Last but not least, this main result is clearly related to the orderability concept introduced by \textit{Y. Eliashberg} and \textit{L. Polterovich} [Geom. Funct. Anal. 10, No. 6, 1448--1476 (2000; Zbl 0986.53036)] on the universal cover \(\widetilde{\mathrm{Cont}}_0(M, \xi)\). In this article, a (possibly) different notation is introduced also on \(\widetilde{\mathrm{Cont}}_0(M, \xi)\), called strong orderability. It is defined via a canonical lift from a contact isotopy in \((M, \xi)\) to a Legendrian isotopy in the contact product \((M \times M \times \mathbb R, \pi_1^*\alpha - e^s \pi_2^*\alpha)\), together with a partial order on the level of Legendrian submanifolds [\textit{V. Chernov} and \textit{S. Nemirovski}, Geom. Topol. 14, No. 1, 611--626 (2010; Zbl 1194.53066)]. Moreover, this article shows that overtwisted contact manifolds are not strongly orderable. This result makes an effort towards the interesting study of whether all overtwisted contact manifolds are non-orderable (in Eliashberg-Polterovich's sense).
Reviewer: Jun Zhang (Montréal)Spectral theories and topological strings on del Pezzo geometries.https://www.zbmath.org/1456.831062021-04-16T16:22:00+00:00"Moriyama, Sanefumi"https://www.zbmath.org/authors/?q=ai:moriyama.sanefumiSummary: Motivated by understanding M2-branes, we propose to reformulate partition functions of M2-branes by quantum curves. Especially, we focus on the backgrounds of \textit{P. del Pezzo} [Nap. Rend. 24, 212--216 (1885; JFM 17.0514.01)] geometries, which enjoy Weyl group symmetries of exceptional algebras. We construct quantum curves explicitly and turn to the analysis of classical phase space areas and quantum mirror maps. We find that the group structure helps in clarifying previous subtleties, such as the shift of the chemical potential in the area and the identification of the overall factor of the spectral operator in the mirror map. We list the multiplicities characterizing the quantum mirror maps and find that the decoupling relation known for the BPS indices works for the mirror maps. As a result, with the group structure we can present explicitly the statement for the correspondence between spectral theories and topological strings on del Pezzo geometries.On TCS \(G_2\) manifolds and 4D emergent strings.https://www.zbmath.org/1456.831092021-04-16T16:22:00+00:00"Xu, Fengjun"https://www.zbmath.org/authors/?q=ai:xu.fengjunSummary: In this note, we study the Swampland Distance Conjecture in TCS \(G_2\) manifold compactifications of M-theory. In particular, we are interested in testing a refined version --- the Emergent String Conjecture, in settings with 4d \(N = 1\) supersymmetry. We find that a weakly coupled, tensionless fundamental heterotic string does emerge at the infinite distance limit characterized by shrinking the \(K3\)-fiber in a TCS \(G_2\) manifold. Such a fundamental tensionless string leads to the parametrically leading infinite tower of asymptotically massless states, which is in line with the Emergent String Conjecture. The tensionless string, however, receives quantum corrections. We check that these quantum corrections do modify the volume of the shrinking \(K3\)-fiber via string duality and hence make the string regain a non-vanishing tension at the quantum level, leading to a decompactification. Geometrically, the quantum corrections modify the metric of the classical moduli space and are expected to obstruct the infinite distance limit. We also comment on another possible type of infinite distance limit in TCS \(G_2\) compactifications, which might lead to a weakly coupled fundamental type II string theory.Mimetic Einstein-Cartan-Sciama-Kibble (ECSK) gravity.https://www.zbmath.org/1456.830652021-04-16T16:22:00+00:00"Izaurieta, Fernando"https://www.zbmath.org/authors/?q=ai:izaurieta.fernando"Medina, Perla"https://www.zbmath.org/authors/?q=ai:medina.perla"Merino, Nelson"https://www.zbmath.org/authors/?q=ai:merino.nelson"Salgado, Patricio"https://www.zbmath.org/authors/?q=ai:salgado.patricio"Valdivia, Omar"https://www.zbmath.org/authors/?q=ai:valdivia.omarSummary: In this paper, we formulate the Mimetic theory of gravity in first-order formalism for differential forms, i.e., the mimetic version of Einstein-Cartan-Sciama-Kibble (ECSK) gravity. We consider different possibilities on how torsion is affected by Weyl transformations and discuss how this translates into the interpolation between two different Weyl transformations of the spin connection, parameterized with a zero-form parameter \(\lambda\). We prove that regardless of the type of transformation one chooses, in this setting torsion remains as a non-propagating field. We also discuss the conservation of the mimetic stress-energy tensor and show that the trace of the total stress-energy tensor is not null but depends on both, the value of \(\lambda\) and spacetime torsion.\(\mathcal{N} = 1\) supersymmetric double field theory and the generalized Kerr-Schild ansatz.https://www.zbmath.org/1456.830072021-04-16T16:22:00+00:00"Lescano, Eric"https://www.zbmath.org/authors/?q=ai:lescano.eric"Rodríguez, Jesús A."https://www.zbmath.org/authors/?q=ai:rodriguez.jesus-aSummary: We construct the \(\mathcal{N} = 1\) supersymmetric extension of the generalized Kerr-Schild ansatz in the flux formulation of Double Field Theory. We show that this ansatz is compatible with \(\mathcal{N} = 1\) supersymmetry as long as it is not written in terms of generalized null vectors. Supersymmetric consistency is obtained through a set of conditions that imply linearity of the generalized gravitino perturbation and unrestricted perturbations of the generalized background dilaton and dilatino. As a final step we parametrize the previous theory in terms of the field content of the low energy effective 10-dimensional heterotic supergravity and we find that the perturbation of the 10-dimensional vielbein, Kalb-Ramond field, gauge field, gravitino and gaugino can be written in terms of vectors, as expected.Real hypersurfaces with isometric Reeb flow in Kähler manifolds.https://www.zbmath.org/1456.530422021-04-16T16:22:00+00:00"Berndt, Jürgen"https://www.zbmath.org/authors/?q=ai:berndt.jurgen"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinThe paper under review consists of two main parts. In the first part of the article, the authors develop a general structure theory for real hypersurfaces in Kähler manifolds for which the Reeb flow preserves the induced metric. In the second part of the article, the authors apply this theory to classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type, obtain the following interesting classification result:
\textbf{Theorem.} Let \(M\) be a connected orientable real hypersurface in an irreducible Hermitian symmetric space \(\bar{M}\) of compact type. If the Reeb flow on \(M\) is an isometric flow, then \(M\) is congruent to an open part of a tube of radius \(0 < t < \pi/\sqrt{2}\) around the totally geodesic submanifold \(\Sigma\) in \(\bar{M}\), where
(i) \(\bar{M} = \mathbb{C} P^r = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_1\mathrm{U}_r)\) and \(\Sigma = \mathbb{C} P^k\), \(0 \leq k \leq r-1\);
(ii) \(\bar{M} = G_k(\mathbb{C}^{r+1}) = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_k\mathrm{U}_{r+1-k})\) and \(\Sigma = G_k(\mathbb{C}^r)\), \(2 \leq k \leq \frac{r+1}{2}\);
(iii) \(\bar{M} = G_2^+(\mathbb{R}^{2r}) = \mathrm{SO}_{2r}/\mathrm{SO}_{2r-2}\mathrm{SO}_2\) and \(\Sigma = \mathbb{C} P^{r-1}\), \(3 \leq r\);
(iv) \(\bar{M} = \mathrm{SO}_{2r}/\mathrm{U}_r\) and \(\Sigma = \mathrm{SO}_{2r-2}/\mathrm{U}_{r-1}\), \(5 \leq r\).
Conversely, the Reeb flow on any of these hypersurfaces is an isometric flow.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)Hyperkähler cones and instantons on quaternionic Kähler manifolds.https://www.zbmath.org/1456.530402021-04-16T16:22:00+00:00"Devchand, Chandrashekar"https://www.zbmath.org/authors/?q=ai:devchand.chandrashekar"Pontecorvo, Massimiliano"https://www.zbmath.org/authors/?q=ai:pontecorvo.massimiliano"Spiro, Andrea"https://www.zbmath.org/authors/?q=ai:spiro.andrea-fIn this paper a new method to construct Yang-Mills instantons over quaternionic pseudo-Riemannian Kähler manifolds is introduced by extending a technique motivated by supersymmetry for constructing Yang-Mills instantons over pseudo-Riemannian hyper-Kähler manifods by the same authors.
By a \textit{quaternionic pseudo-Riemannian Kähler manifold} of signature \((p,q)\) one means a real \(4n\)-dimensional pseudo-Riemannian manifold \((M,g)\) whose holonomy group is isomorphic to \(\mathrm{Sp}(1)\mathrm{Sp}(p,q)\) satisfying \(p+q=n\). Such manifolds are automatically Einstein hence solving the Yang-Mills self-duality equations over them is interesting from both a mathematical and a physical viewpoint. The authors' method in the spirit of the classical Atiyah-Ward correspondence rests on a bijection between gauge equivalence classes of Yang-Mills instantons with arbitrary compact structure group \(G\) over \((M,g)\) and certain holomorphic objects over a twistor space-like complex manifold \(H(S(M))\) associated to \((M,g)\). This twistor space is constructed in two steps. First, given \((M,g)\) one takes the so-called \textit{Swann bundle} over \(M\), i.e., a certain \({\mathbb H}^*/{\mathbb Z}_2\)-bundle \(\pi :S(M)\rightarrow M\). It has the structure of a hyper-Kähler cone over \(M\). Secondly, one considers its \textit{harmonic space} \(H(S(M))\) which is a topologically trivial \(\mathrm{SL}(2;{\mathbb C})\)-bundle over \(S(M)\) carrying the unique non-product complex structure which makes \(H(S(M))\) a holomorphic bundle over the classical twistor space \(Z(S(M))=S(M)\times{\mathbb C}P^1\) of the Swann bundle; the fibers of this vector bundle are isomorphic to the Borel subgroup \(B\subset\mathrm{SL}(2;{\mathbb C})\) since \(\mathrm{SL}(2;{\mathbb C})/B\cong{\mathbb C}P^1\). The key observation is that there is a one-to-one correspondence between (local) \(G\)-instantons over \((M,g)\) and certain (local) holomorphic maps, called (supersymmetric) \textit{prepotentials}, from \(H(S(M))\) into the complexified Lie algebra of \(G\). Combining this method with results of Narasimhan and Ramanan the construction settles down to a set of data on certain local maps \(M\supset U\rightarrow \mathrm{Mat}_{k\times m} ({\mathbb C})\) and in this form the construction resembles the classical ADHM construction.
Reviewer: Gabor Etesi (Budapest)An Obata-type characterisation of Calabi metrics on line bundles.https://www.zbmath.org/1456.530392021-04-16T16:22:00+00:00"Ginoux, Nicolas"https://www.zbmath.org/authors/?q=ai:ginoux.nicolas"Habib, Georges"https://www.zbmath.org/authors/?q=ai:habib.georges"Pilca, Mihaela"https://www.zbmath.org/authors/?q=ai:pilca.mihaela"Semmelmann, Uwe"https://www.zbmath.org/authors/?q=ai:semmelmann.uweA well-known theorem of \textit{M. Obata} in the 1960's [J. Math. Soc. Japan 14, 333--340 (1962; Zbl 0115.39302)] states that the only complete Riemannian manifold \((M^n,g)\) on which a nonconstant real-valued \(C^2\)-function \(u\) exists whose Hessian satisfies \(\nabla^2 u =-u\cdot\mathrm{Id}\) is the round sphere up to homothety on the metric. The theorem has been generalized to Kähler manifolds in several ways. In the present article the authors characterise complete Kähler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is nonnegative, complex
linear, has pointwise two eigenvalues and whose gradient is a Hessian eigenvector.
Reviewer: Andreas Arvanitoyeorgos (Patras)3-Kenmotsu manifolds.https://www.zbmath.org/1456.530242021-04-16T16:22:00+00:00"Attarchi, Hassan"https://www.zbmath.org/authors/?q=ai:attarchi.hassanA \(3\)-Kenmotsu manifold, as defined by the author, is a manifold endowed with three Kenmotsu structures \((\phi_i, \eta, \xi, g)\) (\(i=1,2,3\)), such that \(\phi_1, \phi_2, \phi_3\) satisfy the quaternionic relations.
After proving some properties of these structures, the author gives an example on an open set of \(\mathbb{ R}^5\).
Reviewer: Gianluca Bande (Cagliari)Scattering equations in AdS: scalar correlators in arbitrary dimensions.https://www.zbmath.org/1456.830962021-04-16T16:22:00+00:00"Eberhardt, Lorenz"https://www.zbmath.org/authors/?q=ai:eberhardt.lorenz"Komatsu, Shota"https://www.zbmath.org/authors/?q=ai:komatsu.shota"Mizera, Sebastian"https://www.zbmath.org/authors/?q=ai:mizera.sebastianSummary: We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two current algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.T duality and Wald entropy formula in the Heterotic Superstring effective action at first-order in \(\alpha '\).https://www.zbmath.org/1456.830972021-04-16T16:22:00+00:00"Elgood, Zachary"https://www.zbmath.org/authors/?q=ai:elgood.zachary"Ortín, Tomás"https://www.zbmath.org/authors/?q=ai:ortin.tomasSummary: We consider the compactification on a circle of the Heterotic Superstring effective action to first order in the Regge slope parameter \(\alpha '\) and re-derive the \(\alpha '\)-corrected Buscher rules first found in [\textit{E. Bergshoeff} et al., Classical Quantum Gravity 13, No. 3, 321--343 (1996; Zbl 0849.53074)], proving the T duality invariance of the dimensionally-reduced action to that order in \(\alpha '\). We use Iyer and Wald's prescription to derive an entropy formula that can be applied to black hole solutions which can be obtained by a single non-trivial compactification on a circle and discuss its invariance under the \(\alpha '\)-corrected T duality transformations. This formula has been successfully applied to \(\alpha '\)-corrected 4-dimensional non-extremal Reissner-Nordström black holes in [\textit{P. A. Cano} et al., J. High Energy Phys. 2020, No. 2, Paper No. 31, 31 p. (2020; Zbl 1435.83077)] and we apply it here to a heterotic version of the Strominger-Vafa 5-dimensional extremal black hole.Renormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor.https://www.zbmath.org/1456.813172021-04-16T16:22:00+00:00"Pulmann, Ján"https://www.zbmath.org/authors/?q=ai:pulmann.jan"Ševera, Pavol"https://www.zbmath.org/authors/?q=ai:severa.pavol"Youmans, Donald R."https://www.zbmath.org/authors/?q=ai:youmans.donald-rSummary: We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim \(\sigma\)-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant \(\sigma\)-model or possibly by a more general AKSZ model.Equigeodesics on generalized flag manifolds with \(\text{G}_2\)-type \(\mathfrak{t} \)-roots.https://www.zbmath.org/1456.530412021-04-16T16:22:00+00:00"Statha, Marina"https://www.zbmath.org/authors/?q=ai:statha.marinaLet \(M=G/K\) be a reductive homogeneous space with reductive decomposition \(\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}\), \(\mathfrak{m}\cong T_oM\), \(o=eK\). A curve in \(M\) of the form \(\gamma (t)=(\exp tX)\cdot o\) is called an equigeodesic curve if \(\gamma (t)\) is a geodesc with respect to any \(G\)-invariant metric on \(G/K\). The vector \(X\in\mathfrak{m}\) is called equigeodesic vector. The concept is similar (still different) to a geodesic orbit (g.o.) space \((G/K, g)\), where all geodesics (with respect to the \(G\)-invariant metric \(g\)) in \(G/K\) are homogeneous, i.e., of the form \(\gamma (t)=(\exp tX)\cdot o\).
Equigeodesic curves were first studied by \textit{N. Cohen} et al. [Houston J. Math. 37, No. 1, 113--125 (2011; Zbl 1228.53053)], where they gave a simple algebraic characterization of equigeodesic vectors.
Equigeodesic vectors are called structural (they belong to certain subspaces of \(\mathfrak{m}\), so one needs to identify them) or algebraic (they are obtained as solutions of non linear algebraic systems).
In the present paper the author obtains a characterization of structural equigeodesic vectors for generalized flag manifolds \(G/K=G/C(T)\), where \(T\) is a torus in the compact simple Lie group \(G\), whose system of \(\mathfrak{t}\)-roots is of type \(G_2\), i.e. of the form \(\{\xi_1, \xi_2, \xi_1+\xi_2, 2\xi_1+\xi_2, 3\xi_1+\xi_2, 3\xi_1+2\xi_2\}\).
These generalized flag manifolds are \(F_4/(\mathrm{U}(3)\times \mathrm{U}(1)), E_6/(\mathrm{U}(3)\times \mathrm{U}(3)), E_7/(\mathrm{U}(6)\times \mathrm{U}(1))\) and \(E_8/(E_6\times \mathrm{U}(1)\times \mathrm{U}(1))\).
For these spaces she presents subspaces of \(\mathfrak{m}\) that contain structural equigeodesic vectors. This is achieved by taking into account the description of generalized flag manifolds in Lie terms, hence providing a Lie theoretic description of subspaces of \(\mathfrak{m}\) whose vectors are structural equigeodesc vectors.
Reviewer: Andreas Arvanitoyeorgos (Patras)Positive harmonic functions on groups and covering spaces.https://www.zbmath.org/1456.530532021-04-16T16:22:00+00:00"Polymerakis, Panagiotis"https://www.zbmath.org/authors/?q=ai:polymerakis.panagiotisSummary: We show that if \(p : M \to N\) is a normal Riemannian covering, with \(N\) closed, and \(M\) has exponential volume growth, then there are non-constant, positive harmonic functions on \(M\). This was conjectured by \textit{T. Lyons} and \textit{D. Sullivan} [J. Differ. Geom. 19, 299--323 (1984; Zbl 0554.58022)].Bootstrap bounds on closed Einstein manifolds.https://www.zbmath.org/1456.830852021-04-16T16:22:00+00:00"Bonifacio, James"https://www.zbmath.org/authors/?q=ai:bonifacio.james"Hinterbichler, Kurt"https://www.zbmath.org/authors/?q=ai:hinterbichler.kurtSummary: A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.Conformal vector fields on Finsler square metrics.https://www.zbmath.org/1456.530212021-04-16T16:22:00+00:00"Rajeshwari, M. R."https://www.zbmath.org/authors/?q=ai:rajeshwari.m-r"Narasimhamurthy, S. K."https://www.zbmath.org/authors/?q=ai:narasimhamurthy.senajji-kampalappa|narasimhamurthy.senajji-kamplappa"Roopa, M. K."https://www.zbmath.org/authors/?q=ai:roopa.m-kOn a Finsler manifold \((M,F)\) a vector field \(V=V^{i}\frac{\partial }{\partial x^{i}}\) is a conformal vector field with the conformal factor \(\rho
=\rho (x)\) if and only if
\[
V^{i}\frac{\partial F^{2}}{\partial x^{i}}+y^{i}\frac{\partial V^{j}}{
\partial x^{i}}\frac{\partial F^{2}}{\partial y^{j}}=4\rho F^{2},
\]
see [\textit{Z. Shen} and \textit{Q. Xia}, Sci. China, Math. 55,
No. 9, 1869--1882 (2012; Zbl 1267.53027)].
In their paper, for a Finsler manifold endowed with a square metric (i.e. \(
F=\alpha (1+\beta /\alpha )^{2}\), where \(\alpha =\sqrt{a_{ij}(x)y^{i}y^{j}}\)
is a Riemannian metric and \(\beta =b_{i}(x)y^{i}\) is a 1-form) the authors
describe the PDE system that characterizes the conformal
vector fields of conformal factor \(\rho .\) Afterward, considering a Finsler
square metric of weakly isotropic flag curvature (i.e. \(K_{F}=3\theta
/F+\sigma \), where \(\sigma =\sigma (x)\) is a scalar function and \(\theta
=\theta _{i}(x)y^{i}\) is a 1-form on \(M\)) and imposing the condition \(
b=||\beta ||_{\alpha }=\) constant, by solving the preceding system of PDEs,
they give the explicit local expressions of the corresponding conformal
vector fields.
Reviewer: Mircea Neagu (Braşov)\( T\overline{T} \)-deformation of \(q\)-Yang-Mills theory.https://www.zbmath.org/1456.830692021-04-16T16:22:00+00:00"Santilli, Leonardo"https://www.zbmath.org/authors/?q=ai:santilli.leonardo"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-j"Tierz, Miguel"https://www.zbmath.org/authors/?q=ai:tierz.miguelSummary: We derive the \(T\overline{T} \)-perturbed version of two-dimensional \(q\)-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the \(T\overline{T} \)-deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large \(N\) factorization into chiral and anti-chiral sectors. For the \( \mathrm{U} (N)\) gauge theory on the sphere, we show that the large \(N\) phase transition persists, and that it is of third order and induced by instantons. The effect of the \(T\overline{T} \)-deformation is to decrease the critical value of the 't Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for \( (q,t) \)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large \(N\) limit of Yang-Mills theory, showing that the \(T\overline{T} \)-deformation decreases the contribution of the Boltzmann entropy.Reduced invariants from cuspidal maps.https://www.zbmath.org/1456.140702021-04-16T16:22:00+00:00"Battistella, Luca"https://www.zbmath.org/authors/?q=ai:battistella.luca"Carocci, Francesca"https://www.zbmath.org/authors/?q=ai:carocci.francesca"Manolache, Cristina"https://www.zbmath.org/authors/?q=ai:manolache.cristinaSummary: We consider genus one enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the reduced genus one invariants of the quintic threefold, providing a modular interpretation of the latter.A uniqueness theorem of complete Lagrangian translator in \(\mathbb C^2\).https://www.zbmath.org/1456.530772021-04-16T16:22:00+00:00"Li, Xingxiao"https://www.zbmath.org/authors/?q=ai:li.xingxiao"Liu, Yangyang"https://www.zbmath.org/authors/?q=ai:liu.yangyang"Qiao, Ruina"https://www.zbmath.org/authors/?q=ai:qiao.ruinaSummary: In this paper we study the complete Lagrangian translators in the complex 2-plane \(\mathbb C^2\). As the result, we obtain a uniqueness theorem showing that the plane is the only complete Lagrangian translator in \(\mathbb C^2\) with constant square norm of the second fundamental form. On the basis of this, we can prove a more general classification theorem for Lagrangian \(\xi\)-translators in \(\mathbb C^2\). The same idea is also used to give a similar classification of Lagrangian \(\xi\)-surfaces in \(\mathbb C^2\).Weights, recursion relations and projective triangulations for positive geometry of scalar theories.https://www.zbmath.org/1456.814542021-04-16T16:22:00+00:00"John, Renjan Rajan"https://www.zbmath.org/authors/?q=ai:john.renjan-rajan"Kojima, Ryota"https://www.zbmath.org/authors/?q=ai:kojima.ryota"Mahato, Sujoy"https://www.zbmath.org/authors/?q=ai:mahato.sujoySummary: The story of positive geometry of massless scalar theories was pioneered in [\textit{N. Arkani-Hamed} et al., J. High Energy Phys. 2018, No. 5, Paper No. 96, 78 p. (2018; Zbl 1391.81200)] in the context of bi-adjoint \(\varphi^3\) theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [\textit{R. Kojima}, J. High Energy Phys. 2020, No. 8, Paper No. 54, 34 p. (2020; Zbl 1454.81236)], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations to these theories. We then give a detailed analysis of how the recursion relations in \(\varphi^p\) theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following recent development we also extend our analysis to one-loop integrands in the quartic theory.Warped flatland.https://www.zbmath.org/1456.830582021-04-16T16:22:00+00:00"Detournay, Stéphane"https://www.zbmath.org/authors/?q=ai:detournay.stephane"Merbis, Wout"https://www.zbmath.org/authors/?q=ai:merbis.wout"Ng, Gim Seng"https://www.zbmath.org/authors/?q=ai:ng.gim-seng"Wutte, Raphaela"https://www.zbmath.org/authors/?q=ai:wutte.raphaelaSummary: We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincaré algebra. The latter can be obtained as a certain scaling limit of Warped \( \mathrm{AdS}_3\) space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes.Harmonic mean curvature flow and geometric inequalities.https://www.zbmath.org/1456.530752021-04-16T16:22:00+00:00"Andrews, Ben"https://www.zbmath.org/authors/?q=ai:andrews.ben"Hu, Yingxiang"https://www.zbmath.org/authors/?q=ai:hu.yingxiang"Li, Haizhong"https://www.zbmath.org/authors/?q=ai:li.haizhongSummary: We employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total curvature, which is the integral of Gaussian curvature on the hypersurface. The resulting inequality allows us to use the inverse mean curvature flow to prove Alexandrov-Fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.Categorical mirror symmetry on cohomology for a complex genus 2 curve.https://www.zbmath.org/1456.530702021-04-16T16:22:00+00:00"Cannizzo, Catherine"https://www.zbmath.org/authors/?q=ai:cannizzo.catherineSummary: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs \(X\) and \(Y\) such that the complex geometry on \(X\) mirrors the symplectic geometry on \(Y\). It allows one to deduce symplectic information about \(Y\) from known complex properties of \(X\). \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] conjectured that a complex invariant on \(X\) (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of \(Y\) (the Fukaya category, see [\textit{D. Auroux}, Bolyai Soc. Math. Stud. 26, 85--136 (2014; Zbl 1325.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{D. McDuff} et al., Virtual fundamental cycles in symplectic topology. New York, NY: American Mathematical Society (2019)]). This is known as homological mirror symmetry. In this project, we first use the construction of ``generalized SYZ mirrors'' for hypersurfaces in toric varieties following \textit{M. Abouzaid} et al. [Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)], in order to obtain \(X\) and \(Y\) as manifolds. The complex manifold is the genus 2 curve \(\Sigma_2\) (so of general type \(c_1 < 0\)) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model \((Y, v_0)\) equipped with a holomorphic function \(v_0 : Y \to \mathbb{C}\) which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of \(D^b Coh(\Sigma_2)\) into a cohomological Fukaya-Seidel category of \(Y\) as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{M. Abouzaid} and \textit{P. Seidel}, ``Lefschetz fibration methods in wrapped Floer cohomology'', in preparation].Curvature-torsion quasitensor of Laptev fundamental-group connection.https://www.zbmath.org/1456.530142021-04-16T16:22:00+00:00"Shevchenko, Yu. I."https://www.zbmath.org/authors/?q=ai:shevchenko.yu-iSummary: We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coefficients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are obtained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations.
In two special cases, a space with a fundamental connection is a space with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connection that has not only a curvature tensor, but also a torsion tensor.Convergence to equilibrium of gradient flows defined on planar curves.https://www.zbmath.org/1456.530782021-04-16T16:22:00+00:00"Novaga, Matteo"https://www.zbmath.org/authors/?q=ai:novaga.matteo"Okabe, Shinya"https://www.zbmath.org/authors/?q=ai:okabe.shinyaSummary: We consider the evolution of open planar curves by the steepest descent flow of a geometric functional, with different boundary conditions. We prove that, if any set of stationary solutions with fixed energy is finite, then a solution of the flow converges to a stationary solution as time goes to infinity. We also present a few applications of this result.Deformation classes in generalized Kähler geometry.https://www.zbmath.org/1456.530572021-04-16T16:22:00+00:00"Gibson, Matthew"https://www.zbmath.org/authors/?q=ai:gibson.matthew-r"Streets, Jeffrey"https://www.zbmath.org/authors/?q=ai:streets.jeffrey-dSummary: We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.On the Tachibana numbers of closed manifolds with pinched negative sectional curvature.https://www.zbmath.org/1456.530322021-04-16T16:22:00+00:00"Stepanov, S. E."https://www.zbmath.org/authors/?q=ai:stepanov.sergey-e"Tsyganok, I. I."https://www.zbmath.org/authors/?q=ai:tsyganok.irina-iA conformal Killing form is a natural generalization of conformal Killing vector fields. These considerations were motivated by existence of various applications for these forms. The vector space of conformal Killing \(p\)-forms on an \(n\)-dimensional (\(1\le p\le n-1\)) closed Riemannian manifold \(M\) has a finite dimension \(t_p(M)\) named the Tachibana number. These numbers are conformal scalar invariants of \(M\) and satisfy the duality theorem: \(t_p(M)=t_{n-p}(M)\). In the present article the authors prove two vanishing theorems. In the first, they prove that there are no nonzero Tachibana numbers on an \(n\)-dimensional (\(n\ge 4\)) closed Riemannian manifold with pinched negative sectional curvature such that \(-1-\delta\le\mathrm{sec}\le -1\) for some pinching constant \(\delta<(n-1)^{-1}\). In the second theorem they show that there are no nonzero Tachibana numbers on a three-dimensional closed Riemannian manifold with negative sectional curvature.
Reviewer: Andreas Arvanitoyeorgos (Patras)A dense geodesic ray in the \(\mathrm{Out}(F_r)\)-quotient of reduced outer space.https://www.zbmath.org/1456.200332021-04-16T16:22:00+00:00"Algom-Kfir, Yael"https://www.zbmath.org/authors/?q=ai:algom-kfir.yael"Pfaff, Catherine"https://www.zbmath.org/authors/?q=ai:pfaff.catherineSummary: In [Ann. Math. Stud. 97, 417--438 (1981; Zbl 0476.32027)] \textit{H. Masur} proved the existence of a dense geodesic in the moduli space for a surface. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron-Frobenius eigenvectors of positive integer \(m\times m\) matrices is dense in the positive cone \(\mathbf{R}^m_+\) (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.Global existence of Landau-Lifshitz-Gilbert equation and self-similar blowup of harmonic map heat flow on \(\mathbb{S}^2\).https://www.zbmath.org/1456.350672021-04-16T16:22:00+00:00"Zhong, Penghong"https://www.zbmath.org/authors/?q=ai:zhong.penghong"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshan"Ma, Xuan"https://www.zbmath.org/authors/?q=ai:ma.xuanSummary: Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau-Lifshitz-Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on \(\mathbb{S}^2\). As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.Connections with parallel skew-symmetric torsion on sub-Riemannian manifolds.https://www.zbmath.org/1456.530272021-04-16T16:22:00+00:00"Galaev, S. V."https://www.zbmath.org/authors/?q=ai:galaev.sergei-vasilevichSummary: On a sub-Riemannian manifold \(M\) of contact type, is considered an \(N\)-connection \(\nabla^N\) defined by the pair \((\nabla, N)\), where is an interior metric connection, \(N:D\to D\) is an endomorphism of the distribution \(D\). It is proved that there exists a unique \(N\)-connection \(\nabla^N\) such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connection with parallel torsion are given.Lifting semi-invariant submanifolds to distribution of almost contact metric manifolds.https://www.zbmath.org/1456.530632021-04-16T16:22:00+00:00"Bukusheva, A. V."https://www.zbmath.org/authors/?q=ai:bukusheva.aliya-vSummary: Let \(M\) be an almost contact metric manifold of dimension \(n = 2m + 1\). The distribution \(D\) of the manifold \(M\) admits a natural structure of a smooth manifold of dimension \(n = 4m + 1\). On the manifold \(M\), is defined a linear connection \(\nabla^N\) that preserves the distribution \(D\); this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves.
The assigment of the linear connection \(\nabla^N\) is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution \(D\).
Certain tensor field of type \((1,1)\) on \(D\) defines a so-called prolonged almost contact metric structure.
Each section \(U\in\Gamma(D)\) of the distribution \(D\) defines a morphism \(U:M\to D\) of smooth manifolds. It is proved that if \(\tilde{M}\subset M\) a semi-invariant submanifold of the manifold \(M\) and \(U\in\Gamma(D)\) is a covariantly constant vector field with respect to the \(N\)-connection \(\nabla^N\), then \(U(\tilde{M})\) is a semi-invariant submanifold of the manifold \(D\) with respect to the prolonged almost contact metric structure.Convex and starshaped sets in manifolds without conjugate points.https://www.zbmath.org/1456.520012021-04-16T16:22:00+00:00"Shenawy, Sameh"https://www.zbmath.org/authors/?q=ai:shenawy.samehSummary: Let \(\mathcal W^n\) be the class of \(C^\infty\) complete simply connected \(n\)-dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let \(W\in\mathcal W^n\) and let \(A\) be a subset of \(W\). This article aims at characterization and building convex and starshaped sets in this class from inside. For example, it is proven that, for a compact starshaped set, the convex kernel is the intersection of stars of extreme points only. Also, if a closed unbounded convex set \(A\) does not contain a totally geodesic hypersurface and its boundary has no geodesic ray, then \(A\) is the convex hull of its extreme points. This result is a refinement of the well-known Karein-Millman theorem.The globalization problem of the Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework.https://www.zbmath.org/1456.530732021-04-16T16:22:00+00:00"Esen, Oğul"https://www.zbmath.org/authors/?q=ai:esen.ogul"de León, Manuel"https://www.zbmath.org/authors/?q=ai:de-leon.manuel"Sardón, Cristina"https://www.zbmath.org/authors/?q=ai:sardon.cristina"Zając, Marcin"https://www.zbmath.org/authors/?q=ai:zajac.marcinSummary: In this paper, we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework and we introduce the notion of \textit{locally conformal k-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton-Jacobi equation, which is specifically proposed in a l.c.k-s manifold.Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates.https://www.zbmath.org/1456.740162021-04-16T16:22:00+00:00"Hornung, Peter"https://www.zbmath.org/authors/?q=ai:hornung.peterSummary: We introduce a natural concept of stationarity for the nonlinear bending theory of elastic plates, and we derive the equilibrium equations satisfied by stationary points. A key ingredient is a geometric result about the continuation of infinitesimal bendings on developable surfaces.Entropy rigidity for 3D conservative Anosov flows and dispersing billiards.https://www.zbmath.org/1456.370322021-04-16T16:22:00+00:00"De Simoi, Jacopo"https://www.zbmath.org/authors/?q=ai:de-simoi.jacopo"Leguil, Martin"https://www.zbmath.org/authors/?q=ai:leguil.martin"Vinhage, Kurt"https://www.zbmath.org/authors/?q=ai:vinhage.kurt"Yang, Yun"https://www.zbmath.org/authors/?q=ai:yang.yunSummary: Given an integer \(k \ge 5\), and a \(C^k\) Anosov flow \(\Phi\) on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if \(\Phi\) is \(C^{k - \varepsilon}\)-conjugate to an algebraic flow, for \(\varepsilon > 0\) arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.Mean curvature rigidity of horospheres, hyperspheres, and hyperplanes.https://www.zbmath.org/1456.530482021-04-16T16:22:00+00:00"Souam, Rabah"https://www.zbmath.org/authors/?q=ai:souam.rabahSummary: We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space \(\mathbb{H}^n\), \(n \ge 3\), admit no perturbations with compact support which increase their mean curvature. This is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space \(\mathbb{R}^n\) admits no mean convex perturbations with compact supports.The reverse isoperimetric inequality for convex plane curves through a length-preserving flow.https://www.zbmath.org/1456.520122021-04-16T16:22:00+00:00"Yang, Yunlong"https://www.zbmath.org/authors/?q=ai:yang.yunlong"Wu, Weiping"https://www.zbmath.org/authors/?q=ai:wu.weipingSummary: By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by \textit{S. Pan} et al. [Math. Inequal. Appl. 13, No. 2, 329--338 (2010; Zbl 1192.52011)], which states that if \(\gamma\) is a convex curve with length \(L\) and enclosed area \(A\), then the best constant \(\varepsilon\) in the inequality
\[
L^2 \le 4 \pi A + \varepsilon |\tilde{A}|
\]
is \(\pi\), where \(\tilde{A}\) denotes the oriented area of the locus of its curvature centers.Floer cohomology, multiplicity and the log canonical threshold.https://www.zbmath.org/1456.140422021-04-16T16:22:00+00:00"McLean, Mark"https://www.zbmath.org/authors/?q=ai:mclean.markThe notions of multiplicity and log canonical threshold are the fundamental notions of
complex hypersurface \(H = \{f = 0\}\) defined by polynomials \(f\) on \(\mathbb C^{n+1}\). The former one is rather classical
which is defined by
\[
\mu_P(f) = \dim_{\mathbb C} \mathcal O_{\mathbb C^n,P}/(\delta f/\delta x_1, \ldots, \delta f/\delta x_n)
\]
at \(P \in H\). The latter notion of log canonical threshold is relative new which is given by
\[
\text{lct}_P(f) = \min \{(E_j) + 1/ \text{ord}_f(E_j): j \in S\}
\]
at \(P \in H\), where \((E_j)_{j \in S}\) are the \emph{resolution divisors} of a log resolution at \(0 \in H\) of the pair
\((\mathbb C^{n+1},H)\), whose precise current definition is
given by \textit{V. V. Shokurov} in birational geometry [Russ. Acad. Sci., Izv., Math. 40, No. 1, 95--202 (1992; Zbl 0785.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105--201, Appendix 201--203 (1992)]. Both are algebraic invariants of hypersurface singularities
in complex algebraic geometry. The main result of the paper in review proves that these are indeed symplectic invariants of
the hypersurface in that when \(f, \, g: \mathbb C^{n+1} \to \mathbb C\) are two polynomials with isolated singular points at \(0\) with
embedded contactomorphic links, the multiplicity and the log canonical threshold of \(f\) and \(g\) are equal.
The main technical ingredient used to prove this result is to find formulas for the multiplicity and log canonical threshold
in terms of a sequence of fixed-point Floer cohomology groups in symplectic topology. The author does this by constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose
\(E^1\) page is explicitly described in terms of a log resolution of \(f\). This spectral sequence is a generalization
of a forumla by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]. The author first carries out a rather detailed technical
symplectic massaging, called \(\omega\)-regularization, of a germ of the neighborhood of intersections of
the symplectic crossing divisor \((V_i)_{i\in S}\) which are transversally intersecting codimension 2 symplectic submanifolds.
Then he applies the geometric notions of Liouville domains and open-books to construct a contact open book that is
well-behaved such that the mapping torus of the Milnor monodromy map is isotopic to the mapping torus of a
symplectomorphism arising from the open book. The paper provides much details of basic constructions in symplectic topology
that is expected to be useful for other similar future applications of symplectic machinery to complex algebraic geometry.
Reviewer: Yong-Geun Oh (Pohang)Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces.https://www.zbmath.org/1456.530292021-04-16T16:22:00+00:00"Li, Chao"https://www.zbmath.org/authors/?q=ai:li.chao|li.chao.3|li.chao.5|li.chao.4|li.chao.1|li.chao.2Summary: In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.Covariant vs contravariant methods in differential geometry.https://www.zbmath.org/1456.530042021-04-16T16:22:00+00:00"Min-Oo, Maung"https://www.zbmath.org/authors/?q=ai:min-oo.maungSummary: This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.Helical extension curve of a space curve.https://www.zbmath.org/1456.530092021-04-16T16:22:00+00:00"Dede, Mustafa"https://www.zbmath.org/authors/?q=ai:dede.mustafaSummary: In this paper, we introduce a new class of curves that we call helical extension curve in Euclidean space. Then we investigate the differential geometric properties of the helical extension curves. The main result of the paper is that the helical extension curve of a helix is a plane curve. Moreover, we give several simple characterizations of helical extension curve of special curves such as spherical curve.Chiral algebra, localization, modularity, surface defects, and all that.https://www.zbmath.org/1456.813682021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Fluder, Martin"https://www.zbmath.org/authors/?q=ai:fluder.martinThe authors study Lagrangian \(\mathcal{N} = 2\) superconformal field theories in four dimensions.
By employing supersymmetric localization on a rigid background of the form \(S^3 \times S^1_y\) they explicitly localize a given Lagrangian superconformal field theory and obtain the corresponding two-dimensional vertex operator algebra VOA (chiral algebra) on the torus \(S^1\times S^1_y\subset S^3\times S^1_y\). To derive the VOA the authors define the appropriate rigid supersymmetric \(S^3 \times S^1_y\) background reproducing the superconformal index. They analyze the supersymmetry algebra and classify the possible fugacities and their preserved subalgebras. Although the minimal amount of supersymmetry needed to retain the VOA construction is \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(1|1)_r\) it appears that it is possible to turn on fugacities preserving an \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(2|1)_r\) subalgebra which can be further broken to the minimal one by defects. Specifically, discrete fugacities \(M,N \in \mathbb{Z}\) can be turned on. The authors argue that these deformations do not affect the VOA construction but change the complex structure of the
torus and affect the boundary conditions (spin structure) upon going around one of the cycles, \(S^1_y\)
The authors address the two-dimensional theory corresponding to the localization of the \(\mathcal{N} = 2\) vector multiplets and hypermultiplets. In the latter case they show that the remnant classical piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus \(S^1\times S^1_y\). The authors show that in the presence of flavor holonomies, which appear as mass-like central charges in the supersymmetry algebra, vertex operators charged under the flavor symmetries fail to remain holomorphic while the sector that remains holomorphic is formed by flavor-neutral operators.
The authors study the modular properties of the four-dimensional Schur index. They introduce formal partition functions \(Z^{(\nu_1,\nu_2)}_{(m,n)}\), which are defined as the partition function in the given spin structure \((\nu_1,\nu_2)\), but with the modified contour of the holonomy integral in the localization formula, labeled by two integers \(m\) and \(n\). The authors suggest that the objects \(Z^{(\nu_1,\nu_2)}_{(m,n)}\) furnish an infinite-dimensional projective representation of \(\mathrm{SL}(2,\mathbb{Z})\).
Finally the authors comment on the flat \(\Omega\)-background underlying the chiral algebra.
Reviewer: Farhang Loran (Isfahan)Morse spectra, homology measures, spaces of cycles and parametric packing problems.https://www.zbmath.org/1456.530352021-04-16T16:22:00+00:00"Gromov, Misha"https://www.zbmath.org/authors/?q=ai:gromov.mikhaelA motivating question for this long and densely-written paper is the following: For an ensemble of moving particles in a space, what happens if the effectively observable number of states is replaced by the number of effective/persistent degrees of freedom? ``We suggest in this paper several mathematical counterparts to the idea of persistent degrees of freedom and formulate specific questions, many of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in a ball. And often we present variable aspects of the same ideas in different sections of the paper.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton.https://www.zbmath.org/1456.530372021-04-16T16:22:00+00:00"Zeng, Lingzhong"https://www.zbmath.org/authors/?q=ai:zeng.lingzhongSummary: : It may very well be difficult to prove an eigenvalue inequality of Payne-Pólya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-Pólya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.Categorical localization for the coherent-constructible correspondence.https://www.zbmath.org/1456.140472021-04-16T16:22:00+00:00"Ike, Yuichi"https://www.zbmath.org/authors/?q=ai:ike.yuichi"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiKontsevich's homological mirror symmetry(HMS) conjecture states that two categories associated to
a mirror pair are equivalent. For a Calabi-Yau(CY) variety, a mirror is also Calabi-Yau and the conjecture is a
quasi-equivalence between the dg category of coherent sheaves over one and the derived Fukaya category of
the other. For non-CY's, mirrors do not need to be varieties. For a Fano toric variety, its mirror is a
Landau-Ginzburg (LG) model, which is a holomorphic function on \((\mathbb C^\times)^n\) which can be read from
the defining fan of the toric variety which is in fact the specialization of Lagrangian potential
function of the toric \(A\)-model that is the generating function of open Gromov-Witten invariants of
a toric fiber [\textit{C.-H. Cho} and \textit{Y.-G. Oh}, Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055); \textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078)].
For a smooth Fano, it has been proven for many special cases that the dg category of coherent sheaves
\(\mathbf{coh}\, X_\Sigma\) over the toric variety \(X_\Sigma\) associated to the fan \(\Sigma\) is quasi-equivalent to
the Fukaya-Seidel category \(\mathfrak{Fuk}(W_\Sigma)\) of the associated Laurent polynomial \(W_\Sigma\).
When a variety is not complete, \(\mathbf{coh}\, X_\Sigma\) is of infinite dimensional nature and its Fukaya-type
category also should have an infinite-dimensional nature. Such a construction is known to be
(partially) wrapped Fukaya categories. In this regard, the main theme of the present paper in review is
to establish a quasi-isomorphism \(\mathbf{coh}(X\setminus D)\cong \mathbf{coh} X/\mathbf{coh}_D X\) in some special cases
in the microlocal world: Here
\(X\setminus D\) is the complement of a divisor \(D\) and \(\mathbf{coh} X/\mathbf{coh}_D X\) is the dg category of
sheaves supported in \(D\) by relating the isomorphism to a similar isomorphism
\[
W_{\mathbf{s}\setminus\mathbf{r}}(M) \cong W_{\mathbf{s}}(M)/\mathfrak B_{\mathbf{r}}
\]
of \textit{Z. Sylvan} [J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)] in the Fukaya-Seidel side: Here \(\mathbf{s}\) is a collection
of symplectic stops and \(\mathbf{r} \subset\mathbf{s}\) is a sub-collection thereof, and
\(\mathfrak B_{\mathbf{r}}\) is the full subcategory spanned by Lagrangians near the sub-stops \(\mathbf{r}\).
The paper extends a version of coherent-constructible correspondence [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011); \textit{K. Bongartz} et al., Adv. Math. 226, No. 2, 1875--1910 (2011; Zbl 1223.16004)] to the dg category of
\emph{quasi-coherent shaves} over \(X_\Sigma\) in dimension 2.
Reviewer: Yong-Geun Oh (Pohang)Curvature properties of \((t-z)\)-type plane wave metric.https://www.zbmath.org/1456.530172021-04-16T16:22:00+00:00"Eyasmin, Sabina"https://www.zbmath.org/authors/?q=ai:eyasmin.sabina"Chakraborty, Dhyanesh"https://www.zbmath.org/authors/?q=ai:chakraborty.dhyaneshSummary: The objective, in this paper, is to obtain the curvature properties of \((t-z)\)-type plane wave metric studied by Bondi et al. (1959). For this a general \(( t - z )\)-type wave metric is considered and the condition for which it obeys Einstein's empty spacetime field equations is obtained. It is found that the rank of the Ricci tensor of \((t-z)\)-type plane wave metric is 1 and is of Codazzi type. Also it is proved that it is not recurrent but Ricci recurrent, conformally recurrent and hyper generalized recurrent. Moreover, it is semisymmetric and satisfies the Ricci generalized pseudosymmetric type condition \(P\cdot P=-\frac{1}{3} Q(Ric,P)\). It is interesting to note that, physically, the energy momentum tensor describes a radiation field with parallel rays and geometrically it is a Codazzi tensor and semisymmetric. As special case, the geometric structures of Taub's plane symmetric spacetime metric are deduced. Comparisons between \((t-z)\)-type plane wave metric and pp-wave metric with respect to their geometric structures are viewed.Some classes of CR submanifolds with an umbilical section of the nearly Kähler \(\mathbb{S}^3\times\mathbb{S}^3\).https://www.zbmath.org/1456.530192021-04-16T16:22:00+00:00"Djurdjević, Nataša"https://www.zbmath.org/authors/?q=ai:djurdjevic.natasaSummary: Recently, the investigation of a CR submanifolds of the nearly Kähler manifold \(\mathbb{S}^3\times\mathbb{S}^3\) was started. In this paper it is proved that CR submanifolds of the nearly Kähler manifold \(\mathbb{S}^3\times\mathbb{S}^3\) with umbilical sections must have dimension three and then we obtain some examples of them with distinguished vector fields. Also, we classify minimal submanifolds that have a vector field \(E_4\) as an umbilical section. The main result is classification of the three-dimensional umbilical CR submanifolds with totally geodesic and almost complex distribution \(\mathscr{D}_1\).Gap theorems for submanifolds in \(\mathbb{H}^n\times\mathbb{R}\).https://www.zbmath.org/1456.530462021-04-16T16:22:00+00:00"Lin, Hezi"https://www.zbmath.org/authors/?q=ai:lin.hezi"Wang, Xuansheng"https://www.zbmath.org/authors/?q=ai:wang.xuanshengSummary: In this paper, we obtain some gap theorems for complete immersed submanifolds in \(\mathbb{H}^n\times\mathbb{R}\). For this purpose, we first prove the lower bound estimate of the first eigenvalue of submanifolds in a product space satisfying some curvature conditions. Based on this estimate, we get some Bernstein type theorems for submanifolds in \(\mathbb{H}^n(-1)\times\mathbb{R}\) under integral curvature pinching conditions.Symmetry classification of viscid flows on space curves.https://www.zbmath.org/1456.530792021-04-16T16:22:00+00:00"Duyunova, Anna"https://www.zbmath.org/authors/?q=ai:duyunova.anna-andreevna"Lychagin, Valentin"https://www.zbmath.org/authors/?q=ai:lychagin.valentin-v"Tychkov, Sergey"https://www.zbmath.org/authors/?q=ai:tychkov.sergey-nSummary: Symmetries and differential invariants of viscid flows with viscosity depending on temperature on a space curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given.Generally covariant \(N\)-particle dynamics.https://www.zbmath.org/1456.530542021-04-16T16:22:00+00:00"Miller, Tomasz"https://www.zbmath.org/authors/?q=ai:miller.tomasz"Eckstein, Michał"https://www.zbmath.org/authors/?q=ai:eckstein.michal"Horodecki, Paweł"https://www.zbmath.org/authors/?q=ai:horodecki.pawel"Horodecki, Ryszard"https://www.zbmath.org/authors/?q=ai:horodecki.ryszardSummary: A simultaneous description of the dynamics of multiple particles requires a configuration space approach with an external time parameter. This is in stark contrast with the relativistic paradigm, where time is but a coordinate chosen by an observer. Here we show, however, that the two attitudes towards modelling \(N\)-particle dynamics can be conciliated within a generally covariant framework. To this end we construct an `\(N\)-particle configuration spacetime' \(\mathcal{M}_{(N)} \), starting from a globally hyperbolic spacetime \(\mathcal{M}\) with a chosen smooth splitting into time and space components. The dynamics of multi-particle systems is modelled at the level of Borel probability measures over \(\mathcal{M}_{(N)}\) with the help of the global time parameter. We prove that with any time-evolution of measures, which respects the \(N\)-particle causal structure of \(\mathcal{M}_{(N)}\), one can associate a single measure on the Polish space of `\(N\)-particle wordlines'. The latter is a splitting-independent object, from which one can extract the evolution of measures for any other global observer on \(\mathcal{M}\). An additional asset of the adopted measure-theoretic framework is the possibility to model the dynamics of indistinguishable entities, such as quantum particles. As an application we show that the multi-photon and multi-fermion Schrödinger equations, although explicitly dependent on the choice of an external time-parameter, are in fact fully compatible with the causal structure of the Minkowski spacetime.Topological structure of spaces of stability conditions and topological Fukaya type categories.https://www.zbmath.org/1456.530722021-04-16T16:22:00+00:00"Qiu, Yu"https://www.zbmath.org/authors/?q=ai:qiu.yu|qiu.yu.1|qiu.yu.2Summary: This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories \textit{J. Woolf} [J. Lond. Math. Soc., II. Ser. 82, No. 3, 663--682 (2010; Zbl 1214.18010)], \textit{A. King} and \textit{Y. Qiu} [Adv. Math. 285, 1106--1154 (2015; Zbl 1405.16021)], \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{N. Broomhead} et al. [J. Lond. Math. Soc., II. Ser. 93, No. 2, 273--300 (2016; Zbl 1376.16006)], \textit{Y. Qiu} and \textit{J. Woolf} [Geom. Topol. 22, No. 6, 3701--3760 (2018; Zbl 1423.18044)]. The key is to understand Happel-Reiten-Smalø tilting as tiling of cells. Second, we review topological realizations of various Fukaya type categories \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{Y. Qiu} and \textit{Y. Zhou} [Compos. Math. 153, No. 9, 1779--1819 (2017; Zbl 1405.16024)], \textit{Y. Qiu} [Math. Ann. 365, No. 1--2, 595--633 (2016; Zbl 1378.16027), Math. Z. 288, No. 1--2, 39--53 (2018; Zbl 1442.16017)], \textit{Y. Qiu} and \textit{Y. Zhou} [Trans. Am. Math. Soc. 372, No. 1, 635--660 (2019; Zbl 1444.16013)], \textit{F. Haiden} et al. [Publ. Math., Inst. Hautes Étud. Sci. 126, 247--318 (2017; Zbl 1390.32010)], namely cluster/Calabi-Yau and derived categories from surfaces. The corresponding spaces of stability conditions are of `tame' nature and can be realized as moduli spaces of quadratic differentials due to Bridgeland-Smith and Haiden-Katzarkov-Kontsevich [\textit{T. Bridgeland} and \textit{I. Smith}, ``Quadratic differentials as stability conditions'', Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)], Haiden et al. [loc. cit.]; \textit{A. Ikeda} [Math. Ann. 367, No. 1--2, 1--49 (2017; Zbl 1361.14015)], \textit{A. King} and \textit{Y. Qiu} [Invent. Math. 220, No. 2, 479--523 (2020; Zbl 1457.13045)].
For the entire collection see [Zbl 1454.00056].Quadratic Killing normal Jacobi operator for real hypersurfaces in complex Grassmannians of rank 2.https://www.zbmath.org/1456.530452021-04-16T16:22:00+00:00"Lee, Hyunjin"https://www.zbmath.org/authors/?q=ai:lee.hyunjin"Woo, Changhwa"https://www.zbmath.org/authors/?q=ai:woo.changhwa"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinSummary: In this paper, we introduce a new notion of \textit{quadratic Killing} normal Jacobi operator (simply, \textit{Killing} normal Jacobi operator) and its geometric meaning for real hypersurfaces in the complex Grassmannians of rank two \(\mathbb{G}_2^{m+2}(c)\), \(c\neq 0\). In addition, we give two classification theorems for Hopf real hypersurfaces with \textit{quadratic Killing normal Jacobi operator} in complex two-plane Grassmannians of compact type and of non-compact type.Curvature properties of Kantowski-Sachs metric.https://www.zbmath.org/1456.530182021-04-16T16:22:00+00:00"Shaikh, Absos Ali"https://www.zbmath.org/authors/?q=ai:shaikh.absos-ali"Chakraborty, Dhyanesh"https://www.zbmath.org/authors/?q=ai:chakraborty.dhyaneshSummary: In this paper we have investigated the curvature restricted geometric properties of the generalized Kantowski-Sachs (briefly, GK-S) spacetime metric, a warped product of 2-dimensional base and 2-dimensional fibre. It is proved that GK-S metric describes a generalized Roter type, 2-quasi Einstein and \(Ein(3)\) manifold. It also has pseudosymmetric Weyl conformal tensor as well as conharmonic tensor and its conformal 2-forms are recurrent. Further, it realizes the curvature condition \(R\cdot R=Q(S,R)+\mathcal{L}(t,\theta)Q(g,C)\) (see, Theorem 4.1). We have also determined the curvature properties of Kantowski-Sachs (briefly, K-S), Bianchi type-III and Bianchi type-I metrics which are the special cases of GK-S spacetime metric. The sufficient condition under which GK-S metric represents a perfect fluid spacetime has also been obtained.The isoperimetric problem of a complete Riemannian manifold with a finite number of \(C^0\)-asymptotically Schwarzschild ends.https://www.zbmath.org/1456.530302021-04-16T16:22:00+00:00"Muñoz Flores, Abraham Enrique"https://www.zbmath.org/authors/?q=ai:munoz-flores.abraham-enrique"Nardulli, Stefano"https://www.zbmath.org/authors/?q=ai:nardulli.stefanoSummary: We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in \(C^2\)-locally asymptotically Euclidean Riemannian manifolds with a finite number of \(C^0\)-asymptotically Schwarzschild ends. This work extends previous results contained in
[\textit{M. Eichmair} and \textit{J. Metzger}, Invent. Math. 194, No. 3, 591--630 (2013; Zbl 1297.49078); J. Differ. Geom. 94, No. 1, 159--186 (2013; Zbl 1269.53071); \textit{S. Brendle} and \textit{M. Eichmair}, J. Differ. Geom. 94, No. 3, 387--407 (2013; Zbl 1282.53053)]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named \(C^0\)-strongly asymptotic Schwarzschild, extending results of [Zbl 1282.53053]. Such results are of interest in the field of mathematical general relativity.Fukaya categories of two-tori revisited.https://www.zbmath.org/1456.530712021-04-16T16:22:00+00:00"Kajiura, Hiroshige"https://www.zbmath.org/authors/?q=ai:kajiura.hiroshigeSummary: We construct an \(A_\infty\)-structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal \(A_\infty\)-products are obtained as derivatives of those of transversal \(A_\infty\)-products.The \(\mathcal{N}_3=3\to\mathcal{N}_3=4\) enhancement of super Chern-Simons theories in \(D=3\), Calabi HyperKähler metrics and M2-branes on the \(\mathcal{C}(\mathrm{N}^{0,1,0})\) conifold.https://www.zbmath.org/1456.530742021-04-16T16:22:00+00:00"Fré, P."https://www.zbmath.org/authors/?q=ai:fre.pietro-giuseppe"Giambrone, A."https://www.zbmath.org/authors/?q=ai:giambrone.adam"Grassi, P. A."https://www.zbmath.org/authors/?q=ai:grassi.pietro-antonio"Vasko, P."https://www.zbmath.org/authors/?q=ai:vasko.petrSummary: Considering matter coupled supersymmetric Chern-Simons theories in three dimensions we extend the Gaiotto-Witten mechanism of supersymmetry enhancement \(\mathcal{N}_3=3\to\mathcal{N}_3=4\) from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto-Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on \(T^\star\mathbb{P}^n\). In this list we find, for \(n=2\) the resolution of the metric cone on \(\mathrm{N}^{0,1,0}\) which is the unique homogeneous Sasaki-Einstein 7-manifold leading to an \(\mathcal{N}_4=3\) compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in \(D=3\), the geometry of M2-brane solutions and also for the dual description of super Chern-Simons theories on curved HyperKähler manifolds in terms of gauged fixed supergroup Chern-Simons theories.Schrödinger heat kernel upper bounds on gradient shrinking Ricci solitons.https://www.zbmath.org/1456.350992021-04-16T16:22:00+00:00"Wu, Jia-Yong"https://www.zbmath.org/authors/?q=ai:wu.jiayongSummary: In this paper we give new Gaussian type upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons with the scalar curvature bounded above. This result is a little broader than our earlier paper at some cases. The proof uses on a Davies type integral estimate and a local mean value inequality on gradient shrinking Ricci solitons.Real hypersurfaces of the homogeneous nearly Kähler \(\mathbb{S}^3\times\mathbb{S}^3\) with \(\mathcal{P}\)-isotropic normal.https://www.zbmath.org/1456.530152021-04-16T16:22:00+00:00"Djorić, Miloš"https://www.zbmath.org/authors/?q=ai:djoric.milos"Djorić, Mirjana"https://www.zbmath.org/authors/?q=ai:djoric.mirjana"Moruz, Marilena"https://www.zbmath.org/authors/?q=ai:moruz.marilenaSummary: We study hypersurfaces of the homogeneous nearly Kähler manifold \(\mathbb{S}^3\times\mathbb{S}^3\) which have \(\mathcal{P}\)-isotropic normal vector field. We describe the immersion of such hypersurfaces in \(\mathbb{S}^3\times\mathbb{S}^3\) and we give one example. We prove that they cannot be either Hopf or minimal hypersurfaces.A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature.https://www.zbmath.org/1456.530312021-04-16T16:22:00+00:00"Park, Jiewon"https://www.zbmath.org/authors/?q=ai:park.jiewon"Tian, Wenchuan"https://www.zbmath.org/authors/?q=ai:tian.wenchuan"Wang, Changliang"https://www.zbmath.org/authors/?q=ai:wang.changliangSummary: \textit{M. Gromov} and \textit{C. Sormani} [Emerging topics: scalar curvature and convergence. Institute for Advanced Study Emerging
Topics Report (2018)] have conjectured the following compactness theorem on scalar curvature to hold. Given a sequence of compact Riemannian manifolds with nonnegative scalar curvature and bounded area of minimal surfaces, a subsequence is conjectured to converge in the intrinsic flat sense to a limit space, which has nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit space has an \(H^1\) warping function which has nonnegative scalar curvature in a weak sense, and has Euclidean tangent cones almost everywhere.A survey on \(g=1\) Gromov-Witten invariants via MSP.https://www.zbmath.org/1456.140712021-04-16T16:22:00+00:00"Chang, Huai-Liang"https://www.zbmath.org/authors/?q=ai:chang.huai-liang"Li, Wei-Ping"https://www.zbmath.org/authors/?q=ai:li.wei-pingFor the entire collection see [Zbl 1454.00056].A class of Randers metrics of scalar flag curvature.https://www.zbmath.org/1456.530202021-04-16T16:22:00+00:00"Cheng, Xinyue"https://www.zbmath.org/authors/?q=ai:cheng.xinyue"Yin, Li"https://www.zbmath.org/authors/?q=ai:yin.li"Li, Tingting"https://www.zbmath.org/authors/?q=ai:li.tingtingGeneralized Hardy type and Caffarelli-Kohn-Nirenberg type inequalities on Finsler manifolds.https://www.zbmath.org/1456.530612021-04-16T16:22:00+00:00"Wei, Shihshu Walter"https://www.zbmath.org/authors/?q=ai:wei.shihshu-walter"Wu, Bing Ye"https://www.zbmath.org/authors/?q=ai:wu.bingyeConformal vector fields on Finsler manifolds.https://www.zbmath.org/1456.530222021-04-16T16:22:00+00:00"Xia, Qiaoling"https://www.zbmath.org/authors/?q=ai:xia.qiaolingIntrinsic differentiability and intrinsic regular surfaces in Carnot groups.https://www.zbmath.org/1456.352062021-04-16T16:22:00+00:00"Di Donato, Daniela"https://www.zbmath.org/authors/?q=ai:di-donato.danielaSummary: A Carnot group \(\mathbb{G}\) is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as \(\mathbb{C}^1\) surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize the results on \textit{L. Ambrosio} et al. [J. Geom. Anal. 16, No. 2, 187--232 (2006; Zbl 1085.49045)] valid in Heisenberg groups to the more general setting of Carnot groups.Flat rotational surfaces with pointwise 1-type Gauss map via generalized quaternions.https://www.zbmath.org/1456.530162021-04-16T16:22:00+00:00"Kahraman Aksoyak, Ferdag"https://www.zbmath.org/authors/?q=ai:aksoyak.ferdag-kahraman"Yayli, Yusuf"https://www.zbmath.org/authors/?q=ai:yayli.yusufIn this article, the authors consider a class of surfaces in \(\mathbb{R}^4\) endowed with the flat metric:
\(g = dx_0^2 + \alpha dx_1^2 + \beta dx_2^2 + \alpha \beta dx_3^2,\)
where \(\alpha, \beta\) are (nonzero) real numbers. The most important special cases are the Euclidean and split-signature cases, corresponding to \((\alpha,\beta)\) being \((1,1)\) and \((1,-1)\) respectively. Using a previously known \((\alpha, \beta)\) deformation of the quaternions, called generalized quaternions (see for instance the book by \textit{H. Pottmann} and \textit{J. Wallner} [Computational line geometry. Berlin: Springer (2001; Zbl 1006.51015)]), the authors define a class \(\mathcal{C}\) of parametrized flat rotational surfaces in \((\mathbb{R}^4,g)\), depending on a parametrized curve in \(\mathbb{R}^2\) (see Equation (9) in the article).
The notion of a pointwise 1-type Gauss map is a condition on the Laplacian of the Gauss map (see Equation (1) in the article). The expression ``pointwise 1-type Gauss map'' originates in the article by \textit{Y. H. Kim} and \textit{D. W. Yoon} [J. Geom. Phys. 34, No. 3--4, 191--205 (2000; Zbl 0962.53034)].
The authors study and classify the subclass \(\mathcal{C}' \subset \mathcal{C}\) consisting of pointwise 1-type Gauss map surfaces, and obtain a classification result, Theorem 1 in the article.
Reviewer: Joseph Malkoun (Hazmieh)Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space.https://www.zbmath.org/1456.530122021-04-16T16:22:00+00:00"Körpinar, T."https://www.zbmath.org/authors/?q=ai:korpinar.talat"Demirkol, R. C."https://www.zbmath.org/authors/?q=ai:demirkol.ridvan-cemSummary: We first describe the conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space \({\mathbb{E}}_1^4\) by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curves in the Minkowski space \({\mathbb{E}}_1^4\) and investigate its relationship with the energy of the same curves in Bishop vector fields in \({\mathbb{E}}_1^4\). In this case, the energy functionals are computed in terms of the Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space \({\mathbb{E}}_1^4\).Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)A note on almost Ricci solitons.https://www.zbmath.org/1456.530382021-04-16T16:22:00+00:00"Deshmukh, Sharief"https://www.zbmath.org/authors/?q=ai:deshmukh.sharief"Al-Sodais, Hana"https://www.zbmath.org/authors/?q=ai:alsodais.hanaSummary: We find several sufficient conditions on a compact almost Ricci soliton under which it is a trivial Ricci soliton. We also find a sufficient condition under which a compact almost Ricci soliton is isometric to a sphere.Analogs of Korn's inequality on Heisenberg groups.https://www.zbmath.org/1456.530282021-04-16T16:22:00+00:00"Isangulova, D. V."https://www.zbmath.org/authors/?q=ai:isangulova.d-vSummary: We give two analogs of Korn's inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn's inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.\(G_2\)-manifolds and the ADM formalism.https://www.zbmath.org/1456.580102021-04-16T16:22:00+00:00"Chihara, Ryohei"https://www.zbmath.org/authors/?q=ai:chihara.ryoheiA \(G_2\)-manifold is a \(7\)-dimensional Riemannian manifold with holonomy group contained in the exceptional Lie group \(G_2\). The author regards the present paper as a continuation of previous work [``\(G_2\)-metrics arising from non-integrable special Lagrangian fibrations'', Preprint, \url{arXiv:1801.05540}], in which the main result gives a characterization of a certain dynamical system as a constraint Hamiltonian dynamical system related to \(G_2\).
The paper is adequately described in the abstract: ``In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a \(3\)-manifold \(M\) and prove the orbits of the constrained Hamiltonian dynamical system correspond to \(G_2\)-manifolds foliated by hypersurfaces diffeomorphic to \(M\times SO(3)\).''
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Heterotic backgrounds via generalised geometry: moment maps and moduli.https://www.zbmath.org/1456.830872021-04-16T16:22:00+00:00"Ashmore, Anthony"https://www.zbmath.org/authors/?q=ai:ashmore.anthony"Strickland-Constable, Charles"https://www.zbmath.org/authors/?q=ai:strickland-constable.charles"Tennyson, David"https://www.zbmath.org/authors/?q=ai:tennyson.david"Waldram, Daniel"https://www.zbmath.org/authors/?q=ai:waldram.danielSummary: We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an \( \mathrm{SU} (3) \times \mathrm{ Spin} (6 + n)\) structure within \( \mathrm{O}(6,6+ n) \times \mathbb{R}^+\) generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds.Quantitative Tamarkin theory.https://www.zbmath.org/1456.530082021-04-16T16:22:00+00:00"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.8In 1980's, Kashiwara and Schapira developed a powerful theory, called the microlocal sheaf theory,
connecting analysis, symplectic geometry, and partial differential equations.
In symplectic geometry, a central topic is the non-displaceability problems.
In his pioneering work [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)], \textit{M. Gromov} proved the non-squeezing theorem,
which can be thought of as a classical result concerning non-displaceability.
It was \textit{D. Tamarkin} who first illustrated how to use the microlocal sheaf theory to solve non-displaceability problems [Springer Proc. Math. Stat. 269, 99--223 (2018; Zbl 1416.35019)].
Since then, aiming at translating more objects in symplectic geometry into the language of sheaves, extensive works have been done.
The purpose of the book under review is to provide an exposition of
the fast development of this topic, which focuses on the relations
between symplectic geometry and Tamarkin category theory, especially the Guillermou-Kashiwara-Schapira sheaf quantization
based on microlocal sheaf theory.
The book is divided into four parts.
The first part introduces the basic objects in symplectic geometry and
the key concept of singular support in microlocal sheaf theory.
The second part centers on the concepts of derived
category, persistence \textbf{k}-module, and singular support which serve as preparations
for the topics in later chapters.
The third part deals with the Tamarkin category theory.
The fourth part discusses various applications of Tamarkin categories in
symplectic geometry.
A more detailed review of the contents is given below.
The book starts with an introductory Chapter 1, that provides a quite readable overview of the whole book.
It contains a brief review of symplectic geometry and a sheaf-theoretic topics related to symplectic geometry, such as the singular support of a sheaf, the Tamarkin category, and the Hofer norm.
Chapter 2 is about the derived categories and the derived functors.
In particular, it includes an important result called the microlocal Morse lemma, a generalization of the classical Morse lemma to a microlocal formulation.
Based on the microlocal Morse lemma, the Tamarkin category is constructed at the beginning of Chapter 3.
This chapter devotes to a detailed study of the Tamarkin category theory.
Many symplectic-related topics are presented, for instance, the sheaf convolution and composition, Lagrangian Tamarkin categories and so on.
The last chapter is about the applications of Tamarkin categories in
symplectic geometry.
Starting with a presentation of the Guillermou-Kashiwara-Schapira sheaf quantization,
the author introduces many sheaf theoretic objects related to the symplectic geometry, especially, the symplectic geometry of the cotangent bundle.
At last, using sheaf invariants developed in the book, the author presents a new proof of Gromov's non-squeezing theorem.
The book contains an appendix, which presents some details on the relation between persistence modules and constructible sheaves, the computation of the sheaf hom, and the dynamical interpretation of the Guillermou-Kashiwara-Schapira sheaf quantization from the perspective of semi-classical analysis.
Reviewer: Xiaojun Chen (Chengdu)Lorentzian geometry: holonomy, spinors, and Cauchy problems.https://www.zbmath.org/1456.530032021-04-16T16:22:00+00:00"Baum, Helga"https://www.zbmath.org/authors/?q=ai:baum.helga"Leistner, Thomas"https://www.zbmath.org/authors/?q=ai:leistner.thomasThis text comprises the first half of the recently published book ``Geometric Flows and the Geometry of Space-time'' and is based on lecture notes of a summer course given by the authors on holonomy issues in Lorentzian geometry.
Although the holonomy group of a semi-Riemannian manifold is defined in the exact same way as the Riemannian case (generated by parallel transports along loops), the indefiniteness of the metric creates several challenges, starting from the fact that this group is no longer a subgroup of the (classical) orthogonal group, but rather of the group of Lorentz transformations. In particular, notions of reducibility and decomposability become much more delicate. The first part of the text reviews basic concepts in Lorentzian geometry and presents the algebraic classification of connected Lorentzian holonomy groups (Section 3). The close relation between special (Lorentzian) holonomy groups and parallel spinors is discussed, and motivates the second part of the text on constructions of globally hyperbolic Lorentzian manifolds with special holonomy, via solving suitable Cauchy problems.
This text is simultaneously an accessible pathway for students entering the field, and also a valuable and convenient resource for specialists interested in a panoramic view of the latest developments in this area of research.
For the entire collection see [Zbl 1412.53002].
Reviewer: Renato G. Bettiol (New York)Conformal invariance of the Newtonian Weyl tensor.https://www.zbmath.org/1456.830622021-04-16T16:22:00+00:00"Dewar, Neil"https://www.zbmath.org/authors/?q=ai:dewar.neil"Read, James"https://www.zbmath.org/authors/?q=ai:read.jamesSummary: It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is an invariant of this structure.Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term.https://www.zbmath.org/1456.350802021-04-16T16:22:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Požár, Norbert"https://www.zbmath.org/authors/?q=ai:pozar.norbertSummary: A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$.https://www.zbmath.org/1456.570242021-04-16T16:22:00+00:00"Alves, Emília"https://www.zbmath.org/authors/?q=ai:alves.emilia"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoA smooth curve \(\gamma:[0,1]\to \mathbb{S}^3\) in \(4\)-dimensional Euclidean space \(\mathbb{R}^4\)
with image on the sphere \(\mathbb{S}^3\), is said to be locally convex, if the set of vectors
\({\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)\) is a positive basis in \(\mathbb{R}^4\)
for all \(t\in[0,1]\). By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve \(\mathcal{F}_{\gamma}: [0,1]\to SO_4\) in the special orthogonal
group \(SO_4\) to a locally convex curve.
For any matrix \(Q\in SO_4\), let \(\mathcal{L}\mathbb{S}^3(Q)\) denote the space of all locally
convex curves \(\gamma:[0,1]\to \mathbb{S}^3\) where \(\mathcal{F}_{\gamma}(0)=I\) (the identity matrix)
and \(\mathcal{F}_{\gamma}(1)=Q\). It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of \(\mathcal{L}\mathbb{S}^3(Q)\). But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres \(\mathbb{S}^n\).
In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case \(Q=-I\). The results are technical and cannot be given in detail.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Triple path to the exponential metric.https://www.zbmath.org/1456.830102021-04-16T16:22:00+00:00"Makukov, Maxim"https://www.zbmath.org/authors/?q=ai:makukov.maxim-a"Mychelkin, Eduard"https://www.zbmath.org/authors/?q=ai:mychelkin.eduard-gSummary: The exponential Papapetrou metric induced by scalar field conforms to observational data not worse than the vacuum Schwarzschild solution. Here, we analyze the origin of this metric as a peculiar space-time within a wide class of scalar and antiscalar solutions of the Einstein equations parameterized by scalar charge. Generalizing the three families of static solutions obtained by \textit{I. Z. Fisher} [``Scalar mesostatic field with regard for gravitational effects'', Zh. Èksper. Teor. Fiz. 18, 636--640 (1948), \url{https://cds.cern.ch/record/406391}], \textit{A. I. Janis} et al. [``Reality of the Schwarzschild singularity'', Phys. Rev. Lett. 20, No. 16, 878--880 (1968; \url{doi:10.1103/PhysRevLett.20.878})], and \textit{B. C. Xanthopoulos} and \textit{T. Zannias} [``Einstein gravity coupled to a massless scalar field in arbitrary spacetime dimensions'', Phys. Rev. D (3) 40, No. 8, 2564--2567 (1989; \url{doi:10.1103/PhysRevD.40.2564})], we prove that all three reduce to the same exponential metric provided that scalar charge is equal to central mass, thereby suggesting the universal character of such background scalar field.Minimal submanifolds in a metric measure space.https://www.zbmath.org/1456.530502021-04-16T16:22:00+00:00"Cheng, Xu"https://www.zbmath.org/authors/?q=ai:cheng.xu"Zhou, Detang"https://www.zbmath.org/authors/?q=ai:zhou.detangSummary: In this paper, we survey some of our and related work on minimal submanifolds in a smooth metric measure space, or called, weighted minimal submanifolds in a Riemannian manifold, focusing on the volume estimate of immersed minimal submanifolds.Two-sided conformally recurrent self-dual spaces.https://www.zbmath.org/1456.530602021-04-16T16:22:00+00:00"Chudecki, Adam"https://www.zbmath.org/authors/?q=ai:chudecki.adamSummary: Two-sided conformally recurrent 4-dimensional self-dual spaces are considered. It is shown that such spaces are equipped with nonexpanding congruences of null strings. The general structure of weak nonexpanding hyperheavenly spaces is given. Finally, the general metrics of Petrov-Penrose type \([\text{D}]\otimes[-]\) spaces are presented.Basic inequalities for submanifolds of quaternionic space forms with a quarter-symmetric connection.https://www.zbmath.org/1456.530472021-04-16T16:22:00+00:00"Lone, Mehraj Ahmad"https://www.zbmath.org/authors/?q=ai:lone.mehraj-ahmadSummary: In this paper, we obtain some basic inequalities of curvature invariants for submanifolds of quaternionic space forms with a quarter-symmetric connection.The KW equations and the Nahm pole boundary condition with knots.https://www.zbmath.org/1456.813112021-04-16T16:22:00+00:00"Mazzeo, Rafe"https://www.zbmath.org/authors/?q=ai:mazzeo.rafe-r"Witten, Edward"https://www.zbmath.org/authors/?q=ai:witten.edwardIn this detailed technical paper the authors extend further their previous analysis of the Kapustin-Witten (KW) equations with Nahm pole boundary condition now adapted to general 4-manifolds-with-boundary such that the boundary-3-manifold contains a knot or more generally a link.
The motivation is a conjecture of the second author that the coefficients of the Laurent expansion of the Jones polynomial of a link \(L\subset {\mathbb R}^3\) arise by counting solutions of the KW equations on the half-space \({\mathbb R}^4_+\) obeying a generalized Nahm pole boundary condition on \(\partial {\mathbb R}^4_+={\mathbb R}^3\supset L\) i.e. the Nahm pole boundary condition generalized to be compatible with the extra information of containing a link on the boundary. Roughly this means to prescribe further singularities in the Higgs field part of the KW pair along each link component while the connection part is continuous up to the boundary as before. The conjecture is important because it is well-known that computing the Jones polynomial of a link is an exponentially difficult problem in terms of e.g. the crossing number of any plane diagram of the link.
Reviewer: Gabor Etesi (Budapest)Real hypersurfaces in the complex quadric with generalized Killing shape operator.https://www.zbmath.org/1456.530432021-04-16T16:22:00+00:00"Lee, Hyunjin"https://www.zbmath.org/authors/?q=ai:lee.hyunjin"Hwang, Doo Hyun"https://www.zbmath.org/authors/?q=ai:hwang.doo-hyun"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinSummary: In this paper, we introduce a notion of \textit{generalized Killing shape} operator (or called the quadratic Killing shape operator) and its geometric meaning on real hypersurfaces in the complex quadric. In addition, we give a non-existence theorem for a Hopf real hypersurface with \textit{generalized Killing shape} operator in the complex quadric.Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians.https://www.zbmath.org/1456.530492021-04-16T16:22:00+00:00"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinSummary: In this paper, first we introduce a new notion of \textit{generalized Killing Ricci tensor} for a real hypersurface \(M\) in complex two-plane Grassmannians \(G_2(\mathbb{C}^{m+2})\). Next we give a complete classification for Hopf real hypersurfaces in complex two-plane Grassmannians \(G_2(\mathbb{C}^{m+2})\) with \textit{generalized Killing Ricci tensor}.Geometric matrix midranges.https://www.zbmath.org/1456.150312021-04-16T16:22:00+00:00"Mostajeran, Cyrus"https://www.zbmath.org/authors/?q=ai:mostajeran.cyrus"Grussler, Christian"https://www.zbmath.org/authors/?q=ai:grussler.christian"Sepulchre, Rodolphe"https://www.zbmath.org/authors/?q=ai:sepulchre.rodolphe-jNonexistence of proper \(p\)-biharmonic maps and Liouville type theorems. I: Case of \(p\ge 2\).https://www.zbmath.org/1456.530522021-04-16T16:22:00+00:00"Han, Yingbo"https://www.zbmath.org/authors/?q=ai:han.yingbo"Luo, Yong"https://www.zbmath.org/authors/?q=ai:luo.yongFor \(p >1\), the \(p\)-energy of a map \(u: (M^n, g) \to (N^m, h)\) between Riemannian manifolds is defined by \(E_p(u) = \int_M |\tau(u)|^p dv_g\), where \(\tau(u)\) is the tension field of \(u\). Critical points of the \(p\)-energy functional \(E_p\) are called \(p\)-biharmonic maps.
In this paper, the authors extend a nonexistence result in [\textit{Y. Han} and \textit{W. Zhang}, J. Korean Math. Soc. 52, No. 5, 1097--1108 (2015; Zbl 1325.58006)], and as a corollary, they show a Liouville-type theorem for \(p\)-harmonic maps.
More precisely, let \(u: (M, g) \to (N, h)\) be a \(p\)-biharmonic map, \(p \ge 2\) from
a complete Riemannian manifold \((M, g)\) into a Riemannian manifold \((N, h)\) of nonpositive sectional curvature satisfying
\(\int_M |\tau(u)|^s dv_g <\infty\) for \(p-1 < s\).
If either \(\int_M |du|^q dv_g < \infty\) for \(1 \le q\le \infty\), or \(\mathrm{Vol}(M, g) = \infty\), then \(u\) must be harmonic. In case of \(q = 2\) and
\(s = p+a\) with \(a \ge 0\), the result is due to Han and Zhang [loc. cit.].
As a corollary, the authors prove that if \(u: (M, g) \to (N, h)\) is a \(p\)-biharmonic map, \(p \ge 2\) from
a complete Riemannian manifold \((M, g)\) with nonnegative Ricci curvature into a Riemannian manifold \((N, h)\) of nonpositive sectional curvature
such that \(\int_M \left(|\tau(u)|^s + |du|^q \right)dv_g <\infty\) for \(p-1 < s, 1 < q\), then \(u\) must be a constant map.
Reviewer: Gabjin Yun (Yongin)On one-dimensionality of metric measure spaces.https://www.zbmath.org/1456.530362021-04-16T16:22:00+00:00"Schultz, Timo"https://www.zbmath.org/authors/?q=ai:schultz.timoSummary: In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict \(CD(K,N)\) -space or an essentially non-branching \(MCP(K,N)\)-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching \(MCP(K,N)\)-spaces.Introduction to differential and Riemannian geometry.https://www.zbmath.org/1456.530022021-04-16T16:22:00+00:00"Sommer, Stefan"https://www.zbmath.org/authors/?q=ai:sommer.stefan"Fletcher, Tom"https://www.zbmath.org/authors/?q=ai:fletcher.tom"Pennec, Xavier"https://www.zbmath.org/authors/?q=ai:pennec.xavierThis article is the first chapter of the book by \textit{X. Pennec} (ed.) et al. [Riemannian geometric statistics in medical image analysis. Amsterdam: Elsevier/Academic Press (2020; Zbl 1428.92004)] with 16 chapters.
From the cover of the book:
``Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data.
Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods.
As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces.''
From the introduction of this article: ``The following sections describe these foundational concepts and how they lead to notions commonly associated with geometry: curves, length, distances, geodesics, curvature, parallel transport, and volume form. In addition to the differential and Riemannian structure, we describe one extra layer of structure, Lie groups that are manifolds equipped with smooth group structure. Lie groups and their quotients are examples of homogeneous spaces. The group structure provides relations between distant points on the group and thereby additional ways of constructing Riemannian metrics and deriving geodesic equations.''
The article is structured in 8 sections:
1. Introduction -- 2. Manifolds (2.1 Embedded submanifolds, 2.2 Charts and local euclideaness, 2.3 Abstract manifolds and atlases, 2.4 Tangent vectors and tangent space, 2.5 Differentials and pushforward) -- 3. Riemannian manifolds (3.1 Riemannian metric, 3.2 Curve length and Riemannian distance, 3.3 Geodesics, 3.4 Levi-Cività connection, 3.5 Completeness, 3.6 Exponential and logarithm maps, 3.7 Cut locus) -- 4. Elements of analysis in Riemannian manifolds (4.1 Gradient and musical isomorphisms, 4.2 Hessian and Taylor expansion, 4.3 Riemannian measure or volume form, 4.4 Curvature) -- 5. Lie groups and homogeneous manifolds (5.1 One-parameter subgroups, 5.2 Actions, 5.3 Homogeneous spaces, 5.4 Invariant metrics and geodesics) -- 6. Elements of computing on Riemannian manifolds -- 7. Examples (7.1 The sphere, 7.2 2D Kendall shape space, 7.3 Rotations) -- 8. Additional references -- References (26 references).
For the entire collection see [Zbl 1428.92004].
Reviewer: Ludwig Paditz (Dresden)Continuous singularities in Hamiltonian relative equilibria with abelian momentum isotropy.https://www.zbmath.org/1456.530682021-04-16T16:22:00+00:00"Rodríguez-Olmos, Miguel"https://www.zbmath.org/authors/?q=ai:rodriguez-olmos.miguelSummary: We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.The boundary model for the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\).https://www.zbmath.org/1456.220042021-04-16T16:22:00+00:00"Pieters, Hester"https://www.zbmath.org/authors/?q=ai:pieters.hesterSummary: We prove that the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\) can be measurably realized on the boundary of hyperbolic space. This implies in particular that for \(\mathrm{Isom}^+ (\mathbb H^n)\) the comparison map from continuous bounded cohomology to continuous cohomology is injective in degree 3. We furthermore prove a stability result for the continuous bounded cohomology of \(\mathrm{Isom}(\mathbb H^n)\) and\(\mathrm{Isom}(\mathbb H^n_\mathbb C)\).Letter to the editors.https://www.zbmath.org/1456.300902021-04-16T16:22:00+00:00"Gorbatsevich, V. V."https://www.zbmath.org/authors/?q=ai:gorbatsevich.vladimir-v|gorbatsevich.vladimir-vitalevichCorrection to the author's article [Izv. Math. 83, No. 1, 20--48 (2019; Zbl 1412.30139); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 83, No. 1, 25--85 (2019)].The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps.https://www.zbmath.org/1456.530662021-04-16T16:22:00+00:00"Cruz, Inês"https://www.zbmath.org/authors/?q=ai:cruz.ines"Mena-Matos, Helena"https://www.zbmath.org/authors/?q=ai:mena-matos.helena"Sousa-Dias, Esmeralda"https://www.zbmath.org/authors/?q=ai:sousa-dias.esmeraldaSummary: We consider a family of birational maps \(\varphi_k\) in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family \(\varphi_k\) using Poisson geometry tools, namely the properties of the restrictions of the maps \(\varphi_k\) and their fourth iterate \(\varphi^{(4)}_k\) to the symplectic leaves of an appropriate Poisson manifold \((\mathbb{R}^4_+, P)\). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product \(SL(2, \mathbb{Z})\ltimes\mathbb{R}^2\). The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for \(\varphi_k\) characterized by the parameter values \(k = 1\), \(k = 2\) and \(k\geq 3\).The method of averaging for Poisson connections on foliations and its applications.https://www.zbmath.org/1456.530642021-04-16T16:22:00+00:00"Avendaño-Camacho, Misael"https://www.zbmath.org/authors/?q=ai:avendano-camacho.misael"Hasse-Armengol, Isaac"https://www.zbmath.org/authors/?q=ai:hasse-armengol.isaac"Velasco-Barreras, Eduardo"https://www.zbmath.org/authors/?q=ai:velasco-barreras.eduardo"Vorobiev, Yury"https://www.zbmath.org/authors/?q=ai:vorobiev.yuriiSummary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.On viscosity and equivalent notions of solutions for anisotropic geometric equations.https://www.zbmath.org/1456.350792021-04-16T16:22:00+00:00"De Zan, Cecilia"https://www.zbmath.org/authors/?q=ai:de-zan.cecilia"Soravia, Pierpaolo"https://www.zbmath.org/authors/?q=ai:soravia.pierpaoloSummary: We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.Almost Kenmotsu \((k,\mu)'\)-manifolds with Yamabe solitons.https://www.zbmath.org/1456.530262021-04-16T16:22:00+00:00"Wang, Yaning"https://www.zbmath.org/authors/?q=ai:wang.yaningSummary: Let \((M^{2n+1},\phi ,\xi ,\eta ,g)\) be a non-Kenmotsu almost Kenmotsu \((k,\mu)'\)-manifold. If the metric \(g\) represents a Yamabe soliton, then either \(M^{2n+1}\) is locally isometric to the product space \(\mathbb{H}^{n+1}(-4)\times \mathbb{R}^n\) or \(\eta\) is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.Existence of similar point configurations in thin subsets of \(\mathbb{R}^d\).https://www.zbmath.org/1456.520202021-04-16T16:22:00+00:00"Greenleaf, Allan"https://www.zbmath.org/authors/?q=ai:greenleaf.allan"Iosevich, Alex"https://www.zbmath.org/authors/?q=ai:iosevich.alex"Mkrtchyan, Sevak"https://www.zbmath.org/authors/?q=ai:mkrtchyan.sevakSummary: We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let \(d \ge 2\) and \(E \subset \mathbb{R}^d\) be a compact set. For \(k \ge 1\), define
\[
\Delta_k(E) = \left\{\left(|x^1 - x^2|, \ldots, |x^i - x^j|, \ldots, |x^k - x^{k+1}|\right) : \left\{x^i\right\}_{i = 1}^{k + 1} \subset E \right\} \subset \mathbb{R}^{k(k+1)/2},
\]
the \((k+1)\)-point configuration set of \(E\). For \(k \le d\), this is (up to permutations) the set of congruences of \((k+1)\)-point configurations in \(E\); for \(k > d\), it is the edge-length set of \((k + 1)\)-graphs whose vertices are in \(E\). Previous works by a number of authors have found values \(s_{k, d} < d\) so that if the Hausdorff dimension of \(E\) satisfies \(\dim_{\mathcal H}(E) > s_{k, d} \), then \(\Delta_k(E)\) has positive Lebesgue measure. In this paper we study more refined properties of \(\Delta_k(E)\), namely the existence of similar or multi-similar configurations. For \(r \in \mathbb{R}, r > 0\), let
\[
\Delta_k^r(E) := \{\mathbf{t}\in \Delta_k (E) : r \mathbf{t} \in \Delta_k (E)\} \subset \Delta_k (E).
\]
We show that if \(\dim_{\mathcal H}(E) > s_{k, d}\), for a natural measure \(\nu_k\) on \(\Delta_k(E)\), one has \(\nu_k \left(\delta^r_k(E)\right)\) all \(r \in \mathbb{R}_+\). Thus, in \(E\) there exist many pairs of \((k + 1)\)-point configurations which are similar by the scaling factor \(r\). We extend this to show the existence of multi-similar configurations of any multiplicity. These results can be viewed as variants and extensions, for compact thin sets, of theorems of Furstenberg, Katznelson and Weiss [7], Bourgain [2] and Ziegler [11] for sets of positive density in \(\mathbb{R}^d\).Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations.https://www.zbmath.org/1456.352272021-04-16T16:22:00+00:00"Yang, Hui"https://www.zbmath.org/authors/?q=ai:yang.hui.1"Zou, Wenming"https://www.zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation
\[
(-\Delta)^\sigma u = |x|^\alpha u^p \qquad \text{in } B_1 \backslash \{0\}
\]
with an isolated singularity at the origin, where \(\sigma \in (0, 1)\) and the punctured unit ball \(B_1 \backslash \{0\} \subset \mathbb{R}^n\) with \(n \geq 2\). When \(-2 \sigma < \alpha < 2 \sigma\) and \(\frac{n + \alpha}{n - 2 \sigma} < p < \frac{n + 2 \sigma}{n - 2 \sigma}\), we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by \textit{B. Gidas} and \textit{J. Spruck} [Commun. Pure Appl. Math. 34, 525--598 (1981; Zbl 0465.35003)], but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on \(\mathbb{S}_+^n\). We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.Left-invariant conformal vector fields on non-solvable Lie groups.https://www.zbmath.org/1456.530562021-04-16T16:22:00+00:00"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.1|zhang.hui.6|zhang.hui.7|zhang.hui.2|zhang.hui.9|zhang.hui.11|zhang.hui.4|zhang.hui.5|zhang.hui.8|zhang.hui.3|zhang.hui.10|zhang.hui"Chen, Zhiqi"https://www.zbmath.org/authors/?q=ai:chen.zhiqi"Tan, Ju"https://www.zbmath.org/authors/?q=ai:tan.juSummary: Let \((G,\langle\cdot,\cdot\rangle)\) be a pseudo-Riemannian Lie group of type \((p,q)\) with the Lie algebra \(\mathfrak{g}\). In this paper, we prove that \(G\) is solvable if \((G,\langle\cdot,\cdot\rangle)\) admits a non-Killing left-invariant conformal vector field and \(\dim [\mathfrak{g},\mathfrak{g}]=\dim\mathfrak{g}-\min (p,q)+1\) for \(\min (p,q)\geq 2\). Then we construct a non-solvable pseudo-Riemannian Lie group \(G\) of type \((p,q)\) which admits a non-Killing left-invariant conformal vector field and \(\dim[\mathfrak{g},\mathfrak{g}]=\dim \mathfrak{g}-\min (p,q)+d\) for any \(p,q\geq 3\) and \(2\leq d\leq \min (p,q)-1\). It gives a negative answer to a forthcoming conjecture by H. Zhang and Z. Chen.Simple matrix models for random Bergman metrics.https://www.zbmath.org/1456.824992021-04-16T16:22:00+00:00"Ferrari, Frank"https://www.zbmath.org/authors/?q=ai:ferrari.frank"Klevtsov, Semyon"https://www.zbmath.org/authors/?q=ai:klevtsov.semyon"Zelditch, Steve"https://www.zbmath.org/authors/?q=ai:zelditch.steveCurves on a smooth surface with position vectors Lie in the tangent plane.https://www.zbmath.org/1456.530112021-04-16T16:22:00+00:00"Shaikh, Absos Ali"https://www.zbmath.org/authors/?q=ai:shaikh.absos-ali"Ghosh, Pinaki Ranjan"https://www.zbmath.org/authors/?q=ai:ghosh.pinaki-ranjanSummary: The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length of the position vector, tangential component of the position vector and geodesic curvature of a curve on a surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces.Linear Batalin-Vilkovisky quantization as a functor of \(\infty \)-categories.https://www.zbmath.org/1456.180182021-04-16T16:22:00+00:00"Gwilliam, Owen"https://www.zbmath.org/authors/?q=ai:gwilliam.owen"Haugseng, Rune"https://www.zbmath.org/authors/?q=ai:haugseng.runeThe authors consider a categorical construction of linear Batalin-Vilkovisky quantization in a derived setting.
The basic example that is the starting point for this article is the Weyl quantization, sending a symplectic vector space \(\mathbb R^{2n}\) to the Weyl algebra on \(2n\) generators. One can factor this construction as taking a vector space with a skew-symmetric form first to its Heisenberg Lie algebra and then to its universal envelopping algebra. The specalization at \(\hbar = 0\) of this universal envelopping algebra is a Poisson algebra and the specializiation at \(\hbar = 1\) is its quantizaiton.
The authors consider a special case of the shifted derived versions of this problem: Their starting point are chain complexes equipped with a 1-shifted symmetric pairing. Following the article we will call them quadratic modules for short.
They then construct \(\infty\)-categorical versions of both the Heisenberg Lie algebra (which is actually a shifted \(L_\infty\)-algebra) of a quadratic module, and the universal enveloping \(BD\)-algebra of a shifted Lie algebra. Both of these appear to be of independent interest.
The universal enveloping \(BD\)-algebra is a so-called Beilinson-Drinfeld algebra, a \(k[\hbar]\)-algebra over a certain operad that specialises to a shifted Poisson algebra at \(\hbar = 0\) and to an \(E_0\)-algebra at \(\hbar = 1\). (An \(E_0\)-algebra is just a pointed chain complex, but this is the correct edge case of the notion of \(E_n\)-algebras. The classical, unshifted case involves an unshifted Poisson algebra and an \(E_1\)-algebra (i.e.\ an associative algebra) as specializiations.)
Thus the authors are able to construct linear BV quantization as a symmetric monoidal \(\infty\)-functor from quadratic algebras to \(BD\)-algebras.
The proofs involve a mixture of categorical techniques (model, simplicial and \(\infty\)).
One upside of the \(\infty\)-categorical approach is that by using Lurie's descent theorem the author can consider linear BV quantization for sheaves of quadratic modules on derived stacks. Thus they are able to show that the graded vector bundle \(V \oplus V^\vee[1]\) with its obvious quadratic form quantizes to a line bundle. This is an explicit example of the BV formalism ``behaving like a determinant'', an idea the authors credit to K. Costello. The paper also provides an example that the behaviour for more general 1-shifted symplectic modules is more complicated and the quantization need only be invertible in the formal neighbourhood of a point.
The paper under review contains some interesting discussions in the introduction: Section 1.3 considers higher BV quantizations (which should arise from more general \((1-n)\)-shifted skew-symmetric forms) and a possible application to quantization of AKSZ field theories. Section 1.4 discusses the physical perspective on linear BV quantizations, providing useful context and motivation.
Reviewer: Julian Holstein (Hamburg)The constraint equations in the presence of a scalar field: the case of the conformal method with volumetric drift.https://www.zbmath.org/1456.530552021-04-16T16:22:00+00:00"Vâlcu, Caterina"https://www.zbmath.org/authors/?q=ai:valcu.caterinaThe author applies the conformal method to determine the classical system of constraint equations and certain conditions are imposed on the presence of matter field. A priori estimates for solutions of the Lichnerowicz equation are also discussed. The conformal systems established by Maxwell in the presence of a scalar field and appropriate parameters are established.
Further, in closed Riemannian manifolds of dimension 3, 4 and 5, metrics with and without conformal Killing fields are considered and some important results are established.
Reviewer: Mohammad Nazrul Islam Khan (Buraidah)Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary.https://www.zbmath.org/1456.510072021-04-16T16:22:00+00:00"Burtscher, Annegret"https://www.zbmath.org/authors/?q=ai:burtscher.annegret-y"Ketterer, Christian"https://www.zbmath.org/authors/?q=ai:ketterer.christian"McCann, Robert J."https://www.zbmath.org/authors/?q=ai:mccann.robert-j"Woolgar, Eric"https://www.zbmath.org/authors/?q=ai:woolgar.eric\textit{A. Kasue} [J. Math. Soc. Japan 35, 117--131 (1983; Zbl 0494.53039)] established a sharp estimate for the inscribed radius,
or inradius denoted \(\mathrm{InRad}\), of a smooth \(n\)-dimensional Riemannian manifold \(M\) with nonnegative Ricci curvature and smooth boundary \(\partial M\)\ whose mean curvature is bounded from below by \(n-1\). Exactly speaking, he concluded that
\[
\mathrm{InRad}_{M}\leq 1.
\]
The result was rediscovered by [\textit{M. M. C. Li}, J. Geom. Anal. 24, No. 3, 1490--1496 (2014; Zbl 1303.53053)], being extended to weighted Riemannian manifolds with Bakry-Émery curvature bounds in [\textit{H. Li} and \textit{Y. Wei}, J. Geom. Anal. 25, No. 1, 421--435 (2015; Zbl 1320.53075); Int. Math. Res. Not. 2015, No. 11, 3651--3668 (2015; Zbl 1317.53065); \textit{Y. Sakurai}, Tohoku Math. J. (2) 71, No. 1, 69--109 (2019; Zbl 1422.53029)]. These results are to be seen either as a manifold-with-boundary analogue of Bonnet and Myers' diameter bound or as a Riemannian analogue of the Hawking singularity theorem [\textit{S. W. Hawking}, Proc. R. Soc. Lond., Ser. A 294, 511--521 (1966; Zbl 0139.45803)], whose generalization to a nonsmooth setting is of paramount interest [\textit{M. Graf}, Commun. Math. Phys. 378, No. 2, 1417--1450 (2020; Zbl 1445.53052); \textit{M. Kunzinger} et al., Classical Quantum Gravity 32, No. 7, Article ID 075012, 19 p. (2015; Zbl 1328.83123); \textit{Y. Lu} et al., ``Geometry of weighted Lorentz-Finsler manifolds. I: Singularity theorems'', Preprint, \url{arXiv:1908.03832}].
This paper generalizes Kasue's [loc. cit.] and Li's [loc. cit.] estimate to subsets \(\Omega\)\ of a possibly nonsmooth space \(X\) abiding by a curvature dimension condition \(\mathrm{CD}(K,N)\) with \(K\in\mathbb{R}\) and \(N>1\), provided the topological boundary \(\partial\Omega\) has a lower bound on its inner mean curvature in the sense of [\textit{C. Ketterer}, Proc. Am. Math. Soc. 148, No. 9, 4041--4056 (2020; Zbl 1444.53028)]. The authors' result not only covers Kasue's [loc. cit.] theorem but also holds for a large class of domains in Alexandrov spaces or in Finsler manifolds. Kasue [loc. cit.] as well as Li [loc. cit.] was able to establish a rigidity result analogous to \textit{S.-Y. Cheng}'s theorem [Math. Z. 143, 289--297 (1975; Zbl 0329.53035)] in the Bonnet-Myers context [\textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; JFM 67.0673.01); \textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; Zbl 0025.22704)], namely that, among smooth manifolds, their inscribed radious bound is obtained exactly by the Euclidean unit ball. In the nonsmooth case, there are also truncated cones attaining maximal inradius. The authors establish, under an additional hypothesis known as RCD, that these are the only nonsmooth oprimizers provided \(\Omega\)\ is compact and its interior is connected.
Independently and almost simultaneously, \textit{F. Cavalletti} and \textit{A. Mondino} [Commun. Contemp. Math. 19, No. 6, Article ID 1750007, 27 p. (2017; Zbl 1376.53064); Invent. Math. 208, No. 3, 803--849 (2017; Zbl 1375.53053); Anal. PDE 13, 2091--2147 (2020); ``Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications'', Preprint, \url{arXiv:2004.08934}] have proposed a synthetic new framework for Lorentzian
geometry in which an analogue of the Hawking result is established.
Reviewer: Hirokazu Nishimura (Tsukuba)Contact and Frobenius solvable Lie algebras with abelian nilradical.https://www.zbmath.org/1456.170092021-04-16T16:22:00+00:00"Alvarez, M. A."https://www.zbmath.org/authors/?q=ai:alvarez.maria-alejandra"Rodríguez-Vallarte, M. C."https://www.zbmath.org/authors/?q=ai:rodriguez-vallarte.maria-c"Salgado, G."https://www.zbmath.org/authors/?q=ai:salgado.gilThe authors obtain that complex Frobenius Lie algebras are decomposable, while in the real case, there are exactly two that are indecomposable. Families of both Frobenius and contact solvable Lie algebras are characterized under certain conditions. When these Lie algebras have abelian nilradical, conditions are determined for which they are double extensions of Lie algebras of codimension 2. It is shown that these algebras have a natural Z\(_2\) grading.
Reviewer: Ernest L. Stitzinger (Raleigh)Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space.https://www.zbmath.org/1456.530802021-04-16T16:22:00+00:00"Gürbüz, Nevin"https://www.zbmath.org/authors/?q=ai:gurbuz.nevin"Yoon, Dae Won"https://www.zbmath.org/authors/?q=ai:yoon.dae-wonSummary: In this work, we study Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Later, we derive two formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation.Conjectures and open questions on the structure and regularity of spaces with lower Ricci curvature bounds.https://www.zbmath.org/1456.530052021-04-16T16:22:00+00:00"Naber, Aaron"https://www.zbmath.org/authors/?q=ai:naber.aaronSummary: In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.Symmetries of the simply-laced quantum connections and quantisation of quiver varieties.https://www.zbmath.org/1456.812722021-04-16T16:22:00+00:00"Rembado, Gabriele"https://www.zbmath.org/authors/?q=ai:rembado.gabrieleSummary: We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.Nonlinear flag manifolds as coadjoint orbits.https://www.zbmath.org/1456.370602021-04-16T16:22:00+00:00"Haller, Stefan"https://www.zbmath.org/authors/?q=ai:haller.stefan"Vizman, Cornelia"https://www.zbmath.org/authors/?q=ai:vizman.corneliaIn [Math. Ann. 329, No. 4, 771--785 (2004; Zbl 1071.58005)], the present authors introduced the notion of a nonlinear Grassmannian and studied the Fréchet manifold \(\mathrm{Gras}_n(M)\) of all \(n\)-dimensional oriented compact submanifolds of a smooth closed connected \(m\)-dimensional manifold \(M\).
They showed that every closed \((n+2)\)-form \(\alpha\) on \(M\) defines a closed 2-form \(\widetilde{\alpha}\) on \(\mathrm{Gras}_n(M)\), and if \(\alpha\) is integrable, then \(\widetilde{\alpha}\) is the curvature form of a principal connection on a principal \(S^1\)-bundle over \(\mathrm{Gras}_n(M)\). In the case \(\alpha\) is a closed, integrable volume form, then every connected component \(\mathcal{M}\) of \(\mathrm{Gras}_{m-2}(M)\), equipped with the symplectic form \(\widetilde{\alpha}\), is a prequantizable coadjoint orbit of some central extension of the Hamiltonian group \(\text{Ham}(M,\alpha)\) by \(S^1\).
In this paper, the authors generalize the notion of a nonlinear Grassmannian to the notion of a nonlinear flag manifold.
If \(M\) is a smooth manifold, \(S_1,\dots,S_r\) are closed smooth manifolds, then a sequence of nested embedded submanifolds \(N_1\subseteq\dots\subseteq N_r\subseteq M\) such that \(N_i\) is diffeomorphic to \(S_i\) for all \(i=1,\dots,r\) is called a nonlinear flag of type \(\mathscr{S}=(S_1,\dots,S_r)\) in \(M\).
The space of all nonlinear flags of type \(\mathscr{S}\) in \(M\) can be equipped with the structure of a Fréchet manifold in a natural way and is denoted by \(\mathrm{Flag}_{\mathscr{S}}(M)\).
The main goal of this paper is to study the geometry of this space.
A nonlinear Grassmannian is a special case of a nonlinear flag and corresponds to the case \(r=1\).
The authors present some applications of nonlinear flag manifolds by using them to describe certain coadjoint orbits of the Hamiltonian group.
If \(M\) is a closed symplectic manifold, \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) is the open subset in \(\mathrm{Flag}_{\mathscr{S}}(M)\) consisting of all symplectic flags of type \(\mathscr{S}\), then the symplectic form on \(M\) induces by transgression a symplectic form on the manifold of symplectic nonlinear flags. The Hamiltonian group \(\mathrm{Ham}(M)\) acts on \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) in a Hamiltonian fashion with equivariant moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\).
This moment map is injective and identifies each connected component of \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) with a coadjoint orbit of \(\mathrm{Ham}(M)\).
The main result of the paper states that the restriction of the moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\) to any connected component is one-to-one onto a coadjoint orbit cf the Hamiltonian group \(\mathrm{Ham}(M)\). The Kostant-Kirillov-Souriau symplectic form \(\omega_{\mathrm{KKS}}\) on the coadjoint orbit satisfies \(J^*\omega_{\mathrm{KKS}}=\Omega\), where \(\Omega\) is a natural symplectic form.
Reviewer: Andrew Bucki (Edmond)Notes on functions of hyperbolic type.https://www.zbmath.org/1456.430022021-04-16T16:22:00+00:00"Monod, Nicolas"https://www.zbmath.org/authors/?q=ai:monod.nicolasSummary: Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of \(\mathbf{SU}(1,n)\) and of its infinite-dimensional kin \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\). We further classify all the self-representations of \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\) that satisfy a compatibility condition for the subgroup \(\text{Is}(\mathbf{H}_{\mathbf{R}}^{\infty})\). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.Exterior multiplication with singularities: a Saito theorem in vector bundles.https://www.zbmath.org/1456.580032021-04-16T16:22:00+00:00"Jakubczyk, B."https://www.zbmath.org/authors/?q=ai:jakubczyk.bronislawSummary: Let \(E\) be a vector bundle over a differentiable manifold \(M\) and let \(\bigwedge^pE\) denote the \(p\)th exterior product of \(E\). Given sections \(\omega_1,\dots ,\omega_k\) of \(E\) and a section \(\eta\) of \(\bigwedge^pE\), we consider the problem of whether \(\eta\) can be written in the form \[\eta =\sum \omega_i\wedge \gamma_i,\] where \(\gamma_i\) are sections of \(\bigwedge^{p-1}E\). An obvious necessary condition \(\Omega \wedge \eta =0\), where \(\Omega =\omega_1\wedge \cdots \wedge \omega_k\), has to be supplemented with a condition that the form \(\Omega\) has sufficiently regular singularities at points where \(\Omega (x)=0\). Such a local condition is suggested by an algebraic theorem of K. Saito and is given in terms of the depth of the ideal defined by the coefficients of \(\Omega \). Working in the smooth, real analytic or holomorphic (with \(M\) a Stein manifold) category, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for the existence of a continuous right inverse to the operator defined by \((\gamma_1,\dots ,\gamma_k)\mapsto \sum \omega_i\wedge \gamma_i\). All these results are also proven in the case where \(E\) is a bundle over a suitable closed subset of \(M\).Estimate for evolutionary surfaces of prescribed mean curvature and the convergence.https://www.zbmath.org/1456.350542021-04-16T16:22:00+00:00"Wang, Peihe"https://www.zbmath.org/authors/?q=ai:wang.peihe"Gao, Xinyu"https://www.zbmath.org/authors/?q=ai:gao.xinyuSummary: In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition \(f_\tau\ge -\kappa \), which is the same as the one in the interior estimate by K. Ecker and generalizes the condition \(f_\tau\ge 0\) studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing.https://www.zbmath.org/1456.530582021-04-16T16:22:00+00:00"Huang, Shaosai"https://www.zbmath.org/authors/?q=ai:huang.shaosai"Rong, Xiaochun"https://www.zbmath.org/authors/?q=ai:rong.xiaochun"Wang, Bing"https://www.zbmath.org/authors/?q=ai:wang.bing|wang.bing.1The authors state the following result. If a closed Kähler manifold with a vanishing first Chern class, i.e., a Calabi-Yau manifold, is such that its sectional curvature is upper bounded (by one) and its volume collapses, then it admits a Ricci-flat Kähler metric (Theorem 5.1, 1)). This result is associated to a conjecture stated by \textit{J. Cheeger} et al. [J. Am. Math. Soc. 5, No. 2, 327--372 (1992; Zbl 0758.53022)]. The article is endowed with a review of collapsing Riemannian manifolds with Ricci curvature bounded below.
Reviewer: Mohammed El Aïdi (Bogotá)On the use of the rotation minimizing frame for variational systems with Euclidean symmetry.https://www.zbmath.org/1456.829592021-04-16T16:22:00+00:00"Mansfield, E. L."https://www.zbmath.org/authors/?q=ai:mansfield.elizabeth-louise"Rojo-Echeburúa, A."https://www.zbmath.org/authors/?q=ai:rojo-echeburua.aIn this paper, the authors consider variational problems for curves in 3-space for which the Lagrangian is invariant
under the special Euclidean group \(\mathrm{SE}(3)=\mathrm{SO}(3)\ltimes\mathbb{R}^3\) acting linearly in the standard way. They use the rotation minimizing frame, known as the normal, parallel, or Bishop frame. The authors derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants and syzygy operator for variational problems with a Euclidean symmetry. As application the author use variational problems in the study of stands of proteins, nucleid acids, and polymers.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Linear instability for periodic orbits of non-autonomous Lagrangian systems.https://www.zbmath.org/1456.580132021-04-16T16:22:00+00:00"Portaluri, Alessandro"https://www.zbmath.org/authors/?q=ai:portaluri.alessandro"Wu, Li"https://www.zbmath.org/authors/?q=ai:wu.li"Yang, Ran"https://www.zbmath.org/authors/?q=ai:yang.ranClassical motions of infinitesimal rotators on Mylar balloons.https://www.zbmath.org/1456.530102021-04-16T16:22:00+00:00"Kovalchuk, Vasyl"https://www.zbmath.org/authors/?q=ai:kovalchuk.vasyl"Mladenov, Ivaïlo"https://www.zbmath.org/authors/?q=ai:mladenov.ivailo-mSummary: This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.Rotors in triangles and tetrahedra.https://www.zbmath.org/1456.510132021-04-16T16:22:00+00:00"Bracho, Javier"https://www.zbmath.org/authors/?q=ai:bracho.javier"Montejano, Luis"https://www.zbmath.org/authors/?q=ai:montejano.luisSummary: A barycentric formula that involves the curvatures at the contact points of a rotor within a triangle is proved, and the 3-dimensional case of rotors in tetrahedra is considered.The fundamental groups of open manifolds with nonnegative Ricci curvature.https://www.zbmath.org/1456.530072021-04-16T16:22:00+00:00"Pan, Jiayin"https://www.zbmath.org/authors/?q=ai:pan.jiayinSummary: We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.On the geometry of metallic pseudo-Riemannian structures.https://www.zbmath.org/1456.530252021-04-16T16:22:00+00:00"Blaga, Adara M."https://www.zbmath.org/authors/?q=ai:blaga.adara-monica|blaga.adara-m"Nannicini, Antonella"https://www.zbmath.org/authors/?q=ai:nannicini.antonellaA metallic structure on a differentiable manifold is an endomorphism \(J\) of its tangent bundle satisfying the following condition for some nonnegative integers \(p,q\):
\[
J^2=pJ+qI \; ,
\]
where \(I\) is the identity endomorphism.
This notion generalizes the notions of both almost product structure and complex structure. By considering a Riemannian metric \(g\) such that \(J\) is \(g\)-symmetric, one obtains the notion of metallic Riemannian manifold.
In the paper under review, the authors generalize these notions to the pseudo-Riemannian setting, in particular Norden manifolds, they study the integrability conditions for these structures and define natural connections. The authors give some constructions in the case of tangent and cotangent bundles and discuss the same notions for the generalized tangent bundles.
Reviewer: Gianluca Bande (Cagliari)Spectrum of the Laplacian and the Jacobi operator on rotational CMC hypersurfaces of spheres.https://www.zbmath.org/1456.530512021-04-16T16:22:00+00:00"Perdomo, Oscar M."https://www.zbmath.org/authors/?q=ai:perdomo.oscar-marioSummary: Let \(M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}\) be a compact CMC rotational hypersurface of the \((n+1)\)-dimensional Euclidean unit sphere. Denote by \(|A|^2\) the square of the norm of the second fundamental form and \(J(f)=-\Delta f-nf-|A|^2f\) the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index -- the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than \(3 n+4\) and we also prove that there are at least 2 positive eigenvalues of the Laplacian of \(M\) smaller than \(n\). When \(H\) is not zero, we have that every nonflat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. We assume that the coordinates of this plane has been set up so that the axis of rotation goes through the origin. The planar profile curve is made up of \(m\) copies, each one of them is a is rigid motion of a single curve that we will call the fundamental piece. For this reason every nonflat rotational CMC hypersurface has \(Z_m\) in its group of isometries. If \(\theta\) denotes the change of the angle of the fundamental piece when written in polar coordinates, then \(l=\frac{m\theta}{2 \pi}\) is a nonnegative integer. For unduloids (a subfamily of the rotational CMC hypersurfaces that include all the known embedded examples), we show that the number of negative eigenvalues of the operator \(J\) counted with multiplicity is at least \((2l-1)n+(2m-1)\).