Recent zbMATH articles in MSC 53Chttps://www.zbmath.org/atom/cc/53C2021-11-25T18:46:10.358925ZWerkzeugBook review of: S. Alexander et al., An invitation to Alexandrov geometry. CAT(0) spaces.https://www.zbmath.org/1472.000212021-11-25T18:46:10.358925Z"Kunzinger, M."https://www.zbmath.org/authors/?q=ai:kunzinger.michaelReview of [Zbl 1433.53065].Book review of: E. J. Beggs and S. Majid, Quantum Riemannian geometryhttps://www.zbmath.org/1472.000332021-11-25T18:46:10.358925Z"Schenkel, Alexander"https://www.zbmath.org/authors/?q=ai:schenkel.alexanderReview of [Zbl 1436.53001].Inner-outer curvatures, Ollivier-Ricci curvature and volume growth of graphshttps://www.zbmath.org/1472.051162021-11-25T18:46:10.358925Z"Adriani, Andrea"https://www.zbmath.org/authors/?q=ai:adriani.andrea"Setti, Alberto G."https://www.zbmath.org/authors/?q=ai:setti.alberto-gSummary: We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationships of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.Units in number fields satisfying a multiplicative relation with application to Oeljeklaus-Toma manifoldshttps://www.zbmath.org/1472.112752021-11-25T18:46:10.358925Z"Dubickas, Artūras"https://www.zbmath.org/authors/?q=ai:dubickas.arturasLet \(K\) be a real number field with signature \((r,s)\). Dirichlet's unit theorem states that the group \(U_{K}\) of units of the ring of integers of \(K\) is the direct product of the group \(\{-1,1\}\) and a group with rank \(r+s-1\).
Suppose \(s\geq 1\). According to [\textit{K. Oeljeklaus} and \textit{M. Toma}, Ann. Inst. Fourier 55, No. 1, 161--171 (2005; Zbl 1071.32017)], a subgroup \(U\) of \(U_{K}\) is said to be admissible for \(K\) if \(U\) has \(r\) generators, say \( u_{1},\dots,u_{r}\), such that
\[
\sigma_{i}(u_{j})>0,\text{ for all } (i,j)\in \{1,\dots,r\}^{2},\text{ and }\det [\log \sigma_{i}(u_{j})]_{1\leq i,j\leq r}\neq 0,
\]
where \(\sigma_{1},\dots,\sigma_{r}\) denote the embeddings of \(K\) into \( \mathbb{R}\).
In the paper under review, the author shows that for any real normal field \(N\) of degree \(r\) there are a cubic extension \(K\) of \(N\), with signature \( (r,r)\), and a group \(U\), admissible for \(K\), such that
\[
\sigma_{i}(u)\left\vert \sigma_{i+r}(u)\right\vert^{2}=1,\text{ for all } (u,i)\in U\times \{1,\dots,r\},
\]
where the real embedding \(\sigma_{i}\) and the complex embedding \(\sigma_{i+r}\) are extensions to \(K\) of the same automorphism of \(N\). This theorem, together with a recent result of \textit{A. Otiman} [``Special Hermitian metrics on Oeljeklaus-Toma manifolds'', Preprint, \url{arXiv:2009.02599}], implies that the corresponding Oeljeklaus-Toma manifold \(X(K,U)\) has a pluriclosed metric.Hermitian metrics of positive holomorphic sectional curvature on fibrationshttps://www.zbmath.org/1472.140132021-11-25T18:46:10.358925Z"Chaturvedi, Ananya"https://www.zbmath.org/authors/?q=ai:chaturvedi.ananya"Heier, Gordon"https://www.zbmath.org/authors/?q=ai:heier.gordonIn the article under review, the authors adress the construction of Hermitian metrics with positive holomorphic curvature on compact complex manifolds. The ambiant space is actually the total space of a fibration (holomorphic submersion) \(\pi:X\to Y\) and it is rather natural to ask wether the existence of metrics with positive curvature both on \(Y\) and on the fibers of \(\pi\) implies the existence of such a metric on \(X\).
The corresponding question was answered positively by \textit{C.-K. Cheung} [Math. Z. 201, No. 1, 105--119 (1989; Zbl 0648.53037)] in the opposite case of negative curvature. In the positive curvature case, such metrics were constructed by \textit{N. J. Hitchin} [Proc. Symp. Pure Math. 27, Part 2, 65--80 (1975; Zbl 0321.53052)] on Hirzebruch surfaces.
The main result of this article is a positive answer to the above mentioned question. The proof is quite natural although the computations being a bit involved. As explained by the authors, it is not clear if their method can be used either in the semi-positive case or when the map \(\pi\) has singular fibres. Let us make a final remark: the metric cooked up in this article is merely a Hermitian one, even if the data we started with are Kähler.On the constant scalar curvature Kähler metrics. I: A priori estimateshttps://www.zbmath.org/1472.140422021-11-25T18:46:10.358925Z"Chen, Xiuxiong"https://www.zbmath.org/authors/?q=ai:chen.xiuxiong"Cheng, Jingrui"https://www.zbmath.org/authors/?q=ai:cheng.jingruiThis groundbreaking paper is a technical tour de force on the constant scalar curvature Kähler (CSCK) equation. The basic problem is Calabi's dream of finding canonical metric representatives inside positive line bundle classes. A prototype result is Yau's solution to the Calabi conjecture, and the Chen-Donaldson-Sun solution of the Kähler-Einstein equation in the Fano case subject to K-stability. While a formal infinite dimensional GIT framework has long pointed towards CSCK metrics as the natural generality to consider the canonical metric problem, there are significant technical hurdles to go beyond the Kähler-Einstein case, most notably because the CSCK equation is 4th order instead of second order, and because one loses a priori Ricci curvature bounds, which are essential for applying Cheeger-Colding theory. The goal of this paper is to address the central PDE difficulties. Its techniques are relatively classical, involving maximum principles, Moser style iterations, and Alexandrov maximum principles; the presentation is largely self contained, and accessible to those with basic knowledge of Kähler geometry. However, the application of these techniques are often very clever, and exploit a number of rather delicate cancellation effects which can only be appreciated through substantial calculations.
The 4th order CSCK equation on a compact Kähler manifold can be written as a second order coupled system. The authors consider a slightly more general system (needed for their further work on the continuity method)
\[
\log \det(g_{i\bar{j}}+ \phi_{i\bar{j}}) = F + \log \det(g_{i\bar{j}}),\ \Delta_\phi F = -f + Tr_\phi \eta. \tag{1}
\]
When $f = -R$ (the average Ricci scalar) and $\eta = Ric_g$ this is the CSCK equation. The first equation is complex Monge-Ampère, and the second amounts to a prescription of scalar curvature. The main result of this paper is that under an a priori entropy bound $\int e^FFd\mathrm{vol}_g \leq C$, then the solution to this PDE system is bounded to all derivatives. The entropy bound is natural to the problem, because the CSCK equation is the critical point of the Mabuchi functional (also called the K-energy), which can be written as the entropy term plus a well behaved pluripotential term. The stability condition should be thought of as a coercivity condition on the Mabuchi functional, which will essentially force a bound on the entropy as explained in the subsequent works of the authors in the series.
Some highlights of this paper are:
\begin{itemize}
\item In Section 5, the authors prove a $C^0$-estimate on the Kähler potential from the entropy bound. The main ingredient is an ingenious application of the Alexandrov maximum principle, with the additional input of the Skoda inequality. This is inspired by earlier work of Blocki. Even though the method is quite classical, this result is surprisingly strong, especially in the light of Kolodziej's celebrated $L^\infty$-potential estimate from an a priori $L^p$ bound on $F$ with $p > 1$. This part is of significant independent interest in Kähler geometry, especially the analysis of complex Monge-Ampère.
\item In Section 2, they prove among others things an a priori gradient bound $|\nabla \phi |^2e^{-F} \leq C$. This uses a maximum principle argument, which involves a delicate cancellation effect to knock out some bad mixed derivative terms in the Laplacian.
\item In Section 3, they prove a $W^{2,p}$ type estimate $\int e^{-\alpha(p)F}(n+ \Delta \phi )^pd\mathrm{vol}_g \leq C(p)$ for any exponent $p > 0$. This involves integration by part and an iteration argument. The key is that one can gain exponents on $n+ \Delta \phi$ from the nonlinearity, and any derivative terms of $F$ from the Laplacian computation can be either estimated away in complete squares, or absorbed into the equation for $\Delta F$ which is then a priori controlled.
\item In Section 4, they prove simultaneously $|\nabla F| \leq C$ and $n + \Delta \phi \leq C$, whence the metric has $C^2$ bounds, which reduces the CSCK problem to well known higher order estimates. This is proved by a Moser iteration style argument, based on the $W^{2,p}$ estimate established earlier. The reason for $|\nabla F|^2$ to feature in the proof of the metric upper bound $n + \Delta \phi \leq C$, is that one needs the Laplacian of $|\nabla F|^2$ to provide good Hessian terms. The maximum principle quantity involves another subtle cancellation effect to knock out some bad mixed derivative terms. The reason for the Moser iteration to work, is that Sobolev inequality improves the Lebesgue exponent by a definite magnifying factor > 1, while the fact that \(p\) can be arbitrarily large in the $W^{2,p}$ estimate, ultimately ensures that the application of Hölder inequality can only worsen the Lebesgue exponent by a factor which is arbitrarily close to one, so in the end the improvement effect will win over.
\end{itemize}Fano manifolds and stability of tangent bundleshttps://www.zbmath.org/1472.140442021-11-25T18:46:10.358925Z"Kanemitsu, Akihiro"https://www.zbmath.org/authors/?q=ai:kanemitsu.akihiroLet \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. If the Picard number of \(X\) is one, a widely believed folklore conjecture claims that the tangent bundle \(T_X\) is semistable (in the sense of Mumford-Takemoto). In this paper the author gives a series of counterexamples to this conjecture! \newline The counterexamples are obtained by a family of horospherical varieties classified by \textit{B. Pasquier} [Math. Ann. 344, No. 4, 963--987 (2009; Zbl 1173.14028)]: for these manifolds the action of the group \(\mbox{Aut}^0(X)\) on \(X\) has two orbits, the open orbit \(X^0\) and a closed orbit \(Z\). Moreover the action on the blow-up \(\mbox{Bl}_Z X\) again has two orbits, the open orbit \(X^0\) and the exceptional divisor \(E\). The manifold \(\mbox{Bl}_Z X\) admits a smooth fibration onto a lower-dimensional manifold \(Y\), the push-forward of the relative tangent bundle to \(X\) defines an algebraically integrable foliation \(\mathcal F \subset T_X\). The author shows that this foliation is canonical in the sense that it is the unique algebraically integrable foliation on \(X\) that is \(\mbox{Aut}^0(X)\)-invariant. General arguments show that the stability of \(T_X\) can be verified by computing the slope of the foliation \(\mathcal F\). It turns out that for infinitely many manifolds in Pasquier's list, the subsheaf \(\mathcal F \subset T_X\) destabilises the tangent bundle. The reviewer recommends to any complex geometer to read this beautiful paper.Modules with values in the space of all derivations of an algebrahttps://www.zbmath.org/1472.160412021-11-25T18:46:10.358925Z"Abbasi, H."https://www.zbmath.org/authors/?q=ai:abbasi.huzaifa"Haghighatdoost, GH."https://www.zbmath.org/authors/?q=ai:haghighatdoost.ghorbanaliSummary: In this paper, we construct a groupoid associated to a module with values in the space of all derivations of a unital algebra. More precisely, for a pair \((\mathcal{A}, \mathcal{G})\) consisting of an algebra \(\mathcal{A}\) with a unit, a module \(\mathcal{G}\) over the center \(Z(\mathcal{A})\) of \(\mathcal{A}\) together with a homomorphism of \(Z(\mathcal{A})\)-modules from \(\mathcal{G}\) to the space of all derivations \(\operatorname{Der}(\mathcal{A})\) of \(\mathcal{A}\), we associate a groupoid. We discuss on the equivalence relation induced from this groupoid.Nambu structures and associated bialgebroidshttps://www.zbmath.org/1472.170762021-11-25T18:46:10.358925Z"Basu, Samik"https://www.zbmath.org/authors/?q=ai:basu.samik"Basu, Somnath"https://www.zbmath.org/authors/?q=ai:basu.somnath"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurba"Mukherjee, Goutam"https://www.zbmath.org/authors/?q=ai:mukherjee.goutamSummary: We investigate higher-order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that \(n\)-Lie algebroid structures correspond to \(n\)-ary generalization of Gerstenhaber algebras and are implied by \(n\)-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order \(n>2\)) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order \(n\)). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.Polar actions on Damek-Ricci spaceshttps://www.zbmath.org/1472.220062021-11-25T18:46:10.358925Z"Kollross, Andreas"https://www.zbmath.org/authors/?q=ai:kollross.andreasThe author studies polar actions on Damek-Ricci spaces. A Damek-Ricci space is a non-compact harmonic Riemannian manifold. An isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally.
A Damek-Ricci space can be seen as a semi-direct product \(S=A\ltimes N\), where \(A\) is one dimensional and \(N\) is a generalized Heisenberg group. The Lie algebra \(\mathfrak{n}\) of \(N\) decomposes as \(\mathfrak{n}=\mathfrak{v}\oplus \mathfrak{z}\), where \(\mathfrak{v}\) is a vector space and \(\mathfrak{z}\) is the Lie algebra of the center of \(N\). The group \(N\) is equipped with an inner product, and, for \(Z\in \mathfrak{z}\), the endomorphism \(J_Z\) defined by \[\langle J_ZU,V\rangle =\langle [U,V],Z\rangle \quad (U,V\in \mathfrak{v}),\] satisfies \[J_Z^2=-\langle Z,Z\rangle Id_{\mathfrak{v}}.\] The author considers a closed connected subgroup \(\Sigma\) of \(S\) whose Lie algebra decomposes as \(\mathfrak{a}\oplus \mathfrak{v}'\oplus \mathfrak{z}'\), with \(\mathfrak{v}'\subset \mathfrak{v}\), \(\mathfrak{z}'\subset \mathfrak{z}\). Among other results it is shown that \(\Sigma \) is a totally geodesic submanifold of \(S\) if \(J_{\mathfrak{v}'}=\mathfrak{v}'\). Furthermore, if \(\mathfrak{z}'=0\), there exists a closed subgroup \(H\) of \(S\) which acts polarly on \(S\) by left translations such that \(\Sigma \) is a section for the action of \(H\).Counting one-sided simple closed geodesics on Fuchsian thrice punctured projective planeshttps://www.zbmath.org/1472.300192021-11-25T18:46:10.358925Z"Magee, Michael"https://www.zbmath.org/authors/?q=ai:magee.michaelSince the work of \textit{M. Mirzakhani} [Ann. Math. (2) 168, No. 1, 97--125 (2008; Zbl 1177.37036)], it is known that for an orientable surface \(S\) with a hyperbolic metric \(J\) of finite area, the number of \(n_{J}(L)\) of isotopy classes of simple closed curves of length at most \(L\) satisfies the asymptotic formula \(n_{J}(L)=cL^{d}+o(L^{d})\) where \(c\) is a constant depending on \(J\) and \(d\) is the (integer) dimension of a space of measured laminations on \(S\) (\(d\) only depends on the topology of the surface).
This paper provides an analogous asymptotic formula \(cL^{\beta}+o(L^{\beta})\) in the case of the (nonorientable) real projective plane \(\Sigma\) with three punctures. In contrast with the case of orientable hyperbolic surfaces, exponent \(\beta\) is not an integer and is estimated to be in the range \(2.430< \beta < 2.477\).
The proof is obtained by an identification of the Teichmüller space of \(\Sigma\) with an algebraic variety \(V\) defined by a Markoff-Hurwitz equation, drawing on [\textit{Y. Huang} and \textit{P. Norbury}, Geom. Dedicata 186, 113--148 (2017; Zbl 1360.30040)]. Then, the action of the mapping class group on the curve complex of \(\Sigma\) is related to the action of the combinatorial symmetries of \(V\) (the so-called Markoff moves). Finally, the counting of integer points of the variety provides the estimate of the number of isotopy classes of simple closed curves.Orthogonality of divisorial Zariski decompositions for classes with volume zerohttps://www.zbmath.org/1472.320092021-11-25T18:46:10.358925Z"Tosatti, Valentino"https://www.zbmath.org/authors/?q=ai:tosatti.valentinoConsider the following statement:
Conjecture: Let \((X, \omega)\) be a compact Kähler manifold, and \(\alpha\) a pseudoeffective \((1,1)\) class. Then \[ \langle \alpha^{n-1} \rangle \cdot \alpha = \mathrm{Vol}(\alpha), \] where \(\textrm{Vol}(\alpha)\) is the volume of the class \(\alpha\) and \(\langle \cdot \rangle\) is the moving intersection product of classes in the sense of Boucksom.
The above is known as the orthogonality conjecture for divisorial Zariski decompositions, which was observed by \textit{S. Boucksom} et al. [J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); J. Algebr. Geom. 18, No. 2, 279--308 (2009; Zbl 1162.14003)] and is equivalent to the weak transcendental Morse inequalities, the \(C^1\) differentiability of the volume function on the big cone, and the ``cone duality'' conjecture, i.e., \textit{the dual cone of the pseudoeffective cone is the movable cone}.
This was proven for \(X\) projective in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); \textit{D. W. Nyström}, J. Am. Math. Soc. 32, No. 3, 675--689 (2019; Zbl 1429.32031)], and formulated as a conjecture on arbitrary compact Kähler manifolds in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017)]. The main result of this note is a proof of the orthogonality conjecture on arbitrary compact Kähler manifolds for pseudoeffective \((1,1)\) classes that are assumed to have volume zero.Cohomologies on almost complex manifolds and the \(\partial \overline{\partial} \)-lemmahttps://www.zbmath.org/1472.320122021-11-25T18:46:10.358925Z"Chan, Ki Fung"https://www.zbmath.org/authors/?q=ai:chan.ki-fung"Karigiannis, Spiro"https://www.zbmath.org/authors/?q=ai:karigiannis.spiro"Tsang, Chi Cheuk"https://www.zbmath.org/authors/?q=ai:tsang.chi-cheukThe authors define and study three natural chain complexes associated to an almost complex manifold \((M,J)\), and their cohomology.
The basic operator in the construction of the complexes is the \textit{algebraic derivation} \(\iota_K\) associated to a vector-valued \(k\) form \(K\) on \(M\), defined by sending a form \(\alpha\) to \(K^j \wedge (\iota_{e_j} \alpha)\), where \(K = K^j e_j\) in a local frame \(\{e_j\}\), and \(\iota_{e_j}\) is the usual interior product. One then forms the \textit{Nijenhuis-Lie derivation} \(\mathcal{L}_K\) by taking the graded commutator \([\iota_K, d]\) with the exterior derivative operator.
On an almost complex manifold, we have two natural vector-valued forms: \(J\) itself, a vector-valued one-form, and its Nijenhuis tensor \(N\), a vector-valued two-form. One notes that \((\mathcal{L}_J)^2 = -\mathcal{L}_N\); by the Newlander-Nirenberg theorem, it follows that \(\mathcal{L}_J\) squares to zero if and only if \(J\) is induced by holomorphic charts. In general, neither \(\mathcal{L}_J\) nor \(\mathcal{L}_N\) square to zero. However, the authors notice that \([d, \mathcal{L}_J] = [d, \mathcal{L}_N] = [\mathcal{L}_J, \mathcal{L}_N]\), and hence any of the three operators \(d, \mathcal{L}_J, \mathcal{L}_N\) maps the kernel of any other of the three operators to itself. Hence we obtain three chain complexes:
\begin{itemize}
\item \((\ker \mathcal{L}_J, d)\), whose cohomology the authors call the \textit{\(J\)-cohomology} of \((M,J)\),
\item \((\ker \mathcal{L}_N, d)\), whose cohomology is called the \textit{\(N\)-cohomology}, and
\item \((\ker \mathcal{L}_N , \mathcal{L}_J)\), whose cohomology is called the \textit{\(J\)-twisted \(N\)-cohomology}.
\end{itemize}
If \(J\) is integrable, then clearly the \(N\)-cohomology reduces to the de Rham cohomology, and furthermore \(\mathcal{L}_J = -d^c = -J^{-1}dJ\), so the \(J\)-twisted \(N\)-cohomology is the \(d^c\)-cohomology, isomorphic to de Rham cohomology.
These complexes are natural with respect to pseudoholomorphic maps of almost complex manifolds; hence these cohomologies give invariants of almost complex manifolds which are furthermore practically computable, as the authors go on to demonstrate. As an example of distinguishing diffeomorphic but not isomorphic almost complex manifolds, the authors employ the \(N\)-cohomology to distinguish several non-isomorphic almost complex structures on \(S^1 \times \mathbb{R}^3\).
The remainder of the paper is spent studying the \(J\)-cohomology, which gives a new invariant even in the integrable case. Among other results, the authors prove the \(J\)-cohomology of a compact complex manifold is finite-dimensional, and that in the nonintegrable compact case, it is finite-dimensional in zeroth, first, and top degrees.
The authors introduce a \(d\mathcal{L}_J\)-lemma for almost complex manifolds, which is equivalent to the much studied \(\partial \bar{\partial}\)-lemma (or equivalently the \(dd^c\)-lemma) in the integrable case. An almost complex manifold satisfies the \(d\mathcal{L}_J\)-lemma if and only if the natural map from \(J\)-cohomology to de Rham cohomology is an isomorphism. As an illustration of the applicability of this notion, the authors give a new proof that Hopf surfaces and the Iwasawa manifold do not satisfy the \(\partial \bar{\partial}\)-lemma (for Hopf surfaces, one would usually argue this by, say, observing that the \(\partial \bar{\partial}\)-lemma implies degeneration of the Hodge-de Rham spectral sequence, which further implies that the first Betti number must be even; for the Iwasawa manifold, one can show it admits a non-trivial triple Massey product, violating the rational homotopy theoretic formality that is guaranteed for \(\partial \bar{\partial}\)-manifolds by work of Deligne-Griffiths-Morgan-Sullivan). The authors compute that the rank of the first \(J\)-cohomology of Hopf surfaces and the Iwasawa manifold is strictly smaller than the corresponding first Betti number, thus showing the \(d\mathcal{L}_J\)-lemma is not satisfied.
The authors then study in detail the \(J\)-cohomology of a family of almost complex structures on the four-dimensional torus, and end with some potential future directions for investigation.On CR-structures and the general quadratic structurehttps://www.zbmath.org/1472.320152021-11-25T18:46:10.358925Z"Khan, Mohammad Nazrul Islam"https://www.zbmath.org/authors/?q=ai:khan.mohd-nazrul-islam"Das, Lovejoy S."https://www.zbmath.org/authors/?q=ai:das.lovejoy-s-kSummary: The object of the present paper is to determine the relationship between CR-structure and the general quadratic structure and find some basic results. We discuss integrability conditions and prove certain theorems on CR-structure and the general quadratic structure.Adiabatic limit and the Frölicher spectral sequencehttps://www.zbmath.org/1472.320192021-11-25T18:46:10.358925Z"Popovici, Dan"https://www.zbmath.org/authors/?q=ai:popovici.danIn complex geometry, it is well known that the Frölicher spectral sequence of a compact Kähler manifold degenerates at \(E_1\) page (in particular it degenerates at \(E_2\) page). Since the Kähler condition is quite restrictive for compact complex manifolds of dimension at least 3, it is natural to seek other metric conditions which ensure the \(E_2\)-degeneration of the Frölicher spectral sequence.
Let \(X\) be a compact complex manifold with a Hermitian metric \(\omega\). In this article, the author gives a sufficient metric condition for degeneration at \(E_2\), which roughly says that the torsion of \(\omega\) is ``small''. One of the new ideas is to consider the rescalings of \(\omega\) and \(\partial\), which is an adaption of the adiabatic limit construction associated with a Riemann foliation (see, e.g., [\textit{E. Witten}, Commun. Math. Phys. 100, 197--229 (1985; Zbl 0581.58038)]) to the case of the splitting \(d=\partial +\overline{\partial}\). It seems interesting to point out that similar ideas also appeared in the setting of non-abelian Hodge theory, see [\textit{C. Simpson}, Mixed twistor structures, arXiv preprint alg-geom/9705006, 1997] and Theorem 2.2.4 in [\textit{C. Sabbah}, Polarizable twistor \(\mathcal{D}\)-modules. Paris: Sociéteé Mathématique de France (2005; Zbl 1085.32014)].
Moreover, using a variant of the Efremov-Shubin variational principle, along with the pesudodifferential Laplacian in [the author, Int. J. Math. 27, No. 14, Article ID 1650111, 31 p. (2016; Zbl 1365.53067)] and Demaily's Bochner-Kodaira-Nakano formula for Hermitian metrics, the author finds a formula for the dimensions of the vector spaces on each page of the Frölicher spectral sequence in terms of of the number of small eigenvalues of the rescaled Laplacian. This formula is of independent interest, and is inspired by the analogous result for foliations proven in [\textit{J. A. Álvarez López} and \textit{Y. A. Kordyukov}, Geom. Funct. Anal. 10, No. 5, 977--1027 (2000; Zbl 0965.57024)].Local regularity for the harmonic map and Yang-Mills heat flowshttps://www.zbmath.org/1472.350752021-11-25T18:46:10.358925Z"Afuni, Ahmad"https://www.zbmath.org/authors/?q=ai:afuni.ahmadSummary: We establish new local regularity results for the harmonic map and Yang-Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by \textit{K. Ecker} [Calc. Var. Partial Differ. Equ. 23, No. 1, 67--81 (2005; Zbl 1119.35026)] and the author [ibid. 55, No. 1, Paper No. 13, 14 p. (2016; Zbl 1338.35020); Adv. Calc. Var. 12, No. 2, 135--156 (2019; Zbl 1415.58016)].Quantitative regularity for \(p\)-minimizing maps through a Reifenberg theoremhttps://www.zbmath.org/1472.350812021-11-25T18:46:10.358925Z"Vedovato, Mattia"https://www.zbmath.org/authors/?q=ai:vedovato.mattiaSummary: In this article we extend to arbitrary \(p\)-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case \(p=2\). We first show that the set of singular points of such a map can be \textit{quantitatively stratified}: we classify singular points based on the number of \textit{almost-symmetries} of the map around them, as done in [\textit{J. Cheeger} and \textit{A. Naber}, Commun. Pure Appl. Math. 66, No. 6, 965--990 (2013; Zbl 1269.53063)]. Then, adapting the work of \textit{A. Naber} and \textit{D. Valtorta} [Ann. Math. (2) 185, No. 1, 131--227 (2017; Zbl 1393.58009)], we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is \(k\)-rectifiable for a \(k\) which only depends on \(p\) and the dimension of the domain.Conformal boundary operators, \(T\)-curvatures, and conformal fractional Laplacians of odd orderhttps://www.zbmath.org/1472.354352021-11-25T18:46:10.358925Z"Gover, A. Rod"https://www.zbmath.org/authors/?q=ai:gover.ashwin-rod"Peterson, Lawrence J."https://www.zbmath.org/authors/?q=ai:peterson.lawrence-jSummary: We construct continuously parametrised families of conformally invariant boundary operators on densities. These generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with conformal Laplacian power operators, the boundary operators yield conformally invariant fractional Laplacian pseudodifferential operators on the boundary of a conformal manifold with boundary. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson \(Q\)-curvature and include higher-order generalisations of the mean curvature and the \(T\)-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We discuss applications of our results and techniques in the setting of Poincaré-Einstein manifolds and also use our constructions to shed light on some conjectures of Juhl.Topological mixing of Weyl chamber flowshttps://www.zbmath.org/1472.370122021-11-25T18:46:10.358925Z"Dang, Nguyen-Thi"https://www.zbmath.org/authors/?q=ai:dang.nguyen-thi"Glorieux, Olivier"https://www.zbmath.org/authors/?q=ai:glorieux.olivierSummary: In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.The index theorem for Toeplitz operators as a corollary of Bott periodicityhttps://www.zbmath.org/1472.470092021-11-25T18:46:10.358925Z"Baum, Paul F."https://www.zbmath.org/authors/?q=ai:baum.paul-f"van Erp, Erik"https://www.zbmath.org/authors/?q=ai:van-erp.erikSummary: This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel's theorem [\textit{L. Boutet de Monvel}, Invent. Math. 50, 249--272 (1979; Zbl 0398.47018)]. We prove Boutet de Monvel's theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.Bernstein-Moser-type results for nonlocal minimal graphshttps://www.zbmath.org/1472.490602021-11-25T18:46:10.358925Z"Cozzi, Matteo"https://www.zbmath.org/authors/?q=ai:cozzi.matteo"Farina, Alberto"https://www.zbmath.org/authors/?q=ai:farina.alberto"Lombardini, Luca"https://www.zbmath.org/authors/?q=ai:lombardini.lucaSummary: We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs.
Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine.
Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.The area minimizing problem in conformal cones. IIhttps://www.zbmath.org/1472.490652021-11-25T18:46:10.358925Z"Gao, Qiang"https://www.zbmath.org/authors/?q=ai:gao.qiang"Zhou, Hengyu"https://www.zbmath.org/authors/?q=ai:zhou.hengyuIn this paper, the authors continue to study the area minimizing problem with prescribed boundary in a class of conformal cones similar to the one published by the authors in [J. Funct. Anal. 280, No. 3, Article ID 108827, 40 p. (2021; Zbl 1461.49056)]. If \(N\) is an \(n\)-dimensional open Riemannian manifold with a metric \(\sigma\), \(\mathbb{R}\) is the real line with the metric \(dr^2\), and \(\varphi(x)\) is a \(C^2\) positive function on \(N\), then \(M_\varphi=(N\times\mathbb{R},\varphi^2(x)(\sigma+dr^2))\) is called a conformal product manifold, and if \(\Omega\) is a \(C^2\) bounded domain with compact closure \(\overline\Omega\) in \(N\), then \(Q_\varphi=\Omega\times\mathbb{R}\) in \(M_\varphi\) is called a conformal cone. If \(\psi(x)\) is a \(C^1\) function on \(\partial\Omega\) and \(\Gamma\) is its graph in \(\partial\Omega\times\mathbb{R}\), then the area minimizing problem in a conformal cone \(Q_\varphi\) is to find an \(n\)-integer multiplicity current in \(\overline Q_\varphi\) to realize
\[
\min\{\mathbb{M}(T);\ T\in\mathcal{G}\ \text{and}\ \partial T=\Gamma\}, \tag{\(*\)}
\]
where \(\mathbb{M}\) is the mass of integer multiplicity currents in \(M_\varphi\), and \(\mathcal{G}\) denotes the set of \(n\)-integer multiplicity currents with compact support in \(\overline{Q}_\varphi\), that is, for any \(T\in\mathcal{G}\), its support \(\text{spt}(T)\) is contained in \(\overline\Omega\times[a,b]\) for some finite numbers \(a < b\). If \(BV(W)\) is he set of all bounded variation functions on any open set \(W\), then a key concept for the study of the problem \((*)\) is an area functional in \(BV(W)\) defined as \( \mathfrak{F}_\varphi(u,W)=\sup\left\{\int_\Omega\{\varphi^n(x)h+u\,\text{div}(\varphi^n(x)X)\}\,d\,\text{vol}\right\} \) for \(h\in C_0(W)\), \(X\in T_0(W)\), and \(h^2+\langle X,X\rangle\le 1\), where \(d\text{vol}\) and \(\text{div}\) are the volume form and the divergence of \(N\), respectively, and \(C_0(W)\) and \(T_0(W)\) denote the set of smooth functions and vector fields with compact support in \(W\), respectively. If \(u\in C^1(W)\), then \(\mathfrak{F}_\varphi(u,W)\) is the area of the graph of \(u(x)\) in \(M_\varphi\). If \(\Omega\) is the \(C^2\) domain, \(\Omega'\) is a \(C^2\) domain in \(N\) satisfying \(\Omega\subset\!\subset\Omega'\), i.e., the closure of \(\Omega\) is a compact set in \(\Omega'\), and \(\psi(x)\in C^1(\Omega'\setminus\Omega)\), then the following minimizing problem:
\[
\min\{\mathfrak{F}_\varphi(v,\Omega');\ v(x)\in BV(\Omega'), v(x)=\psi(x)\ \text{on}\ \Omega'\setminus\Omega\} \tag{\(**\)}
\]
plays an important role to solve \((*)\). If \(\Sigma\) is a minimal graph of \(u(x)\) in \(M_\varphi\) over \(\Omega\) with \(C^1\) boundary \(\psi(x)\) on \(\partial\Omega\), then the Dirichlet problem is defined as
\[
\text{div}\!\left(\frac{D u}{\sqrt{1+|Du|^2}}\right)+n\left\langle D\log\varphi,\frac{D u}{\sqrt{1+|Du|^2}}\right\rangle=0 \tag{\(*\!*\!*\)}
\]
for \(x\in\Omega\), and \(u(x)=\psi(x)\) for \(x\in\partial\Omega\), where \(\psi(x)\) is a continuous function on \(\partial\Omega\) and div is the divergence of \(\Omega\). The key idea to solve \((*)\) is to establish the connection between the problem \((*)\), the area functional minimizing problem \((**)\), and the Dirichlet problem of minimal surface equations in \(M_\varphi\). If the mean curvature \(H\) of \(\partial\Omega\) satisfies \(H_{\partial\Omega}+n\langle\vec\gamma,D\log\varphi\rangle\ge 0\) on \(\partial\Omega\), where \(\vec\gamma\) is the outward normal vector of \(\partial\Omega\) and \(H_{\partial\Omega}= \text{div}(\vec{\gamma})\), then \(\Omega\) is called \(\varphi\)-mean convex. The authors show that if \(u(x)\) is the solution to the problem \((**)\), then \(T=\partial[\![U]\!]_{\overline{Q}_\varphi}\) solves the problem \((*)\) in \(M_\varphi\), where \(U\) is the subgraph of \(u(x)\) and \([\![U]\!]\) is the corresponding integer multiplicity current. As a direct application of this result is the Dirichlet problem of minimal surface equations in \(M_\varphi\). It is shown that if \(\Omega\) is \(\varphi\)-mean convex, then the Dirichlet problem \((*\!*\!*)\) with continuous boundary data has a unique solution in \(C^2(\Omega)\cap C(\overline\Omega)\). Finally, the authors consider the existence and uniqueness of local area minimizing integer multiplicity current in \(M_\varphi\) with infinity boundary \(\Gamma\) when \(\varphi(x)\) can be written as \(\varphi(d(x,\partial N))\) which goes to \(+\infty\) as \(d(x,\partial N)\to 0\) in \(N\), where \(N\) is a compact Riemannian manifold with \(C^2\) boundary and \(d\) is the distance function in \(N\). If \(N_r=\{x\in N;\ d(x,\partial N) > r\}\), then it is shown that if there is \(r_1\) such that for any \(r\in(0,r_1)\) \(N_r\) is \(\varphi\)-mean convex, then for any \(\psi(x)\in C(\partial N)\) and \(\Gamma=(x,\psi(x))\) there is a unique local area minimizing integer multiplicity current \(T\) with infinity boundary \(\Gamma\), and \(T\) is a minimal graph in \(M_\varphi\) over \(N\).Closed subsets of a \(\mathrm{CAT}(0)\) 2-complex are intrinsically \(\mathrm{CAT}(0)\)https://www.zbmath.org/1472.510082021-11-25T18:46:10.358925Z"Ricks, Russell"https://www.zbmath.org/authors/?q=ai:ricks.russellSummary: Let \(\kappa\leq 0\), and let \(X\) be a complete, locally finite \(\mathrm{CAT}(\kappa)\) polyhedral \(2\)-complex \(X\), each face with constant curvature \(\kappa\). Let \(E\) be a closed, rectifiably connected subset of \(X\) with trivial first singular homology. We show that \(E\), under the induced path metric, is a complete \(\mathrm{CAT}(\kappa)\) space.Aspects of differential geometry Vhttps://www.zbmath.org/1472.530012021-11-25T18:46:10.358925Z"Calviño-Louzao, Esteban"https://www.zbmath.org/authors/?q=ai:calvino-louzao.esteban"García-Río, Eduardo"https://www.zbmath.org/authors/?q=ai:garcia-rio.eduardo"Gilkey, Peter"https://www.zbmath.org/authors/?q=ai:gilkey.peter-b"Park, JeongHyeong"https://www.zbmath.org/authors/?q=ai:park.jeonghyeong"Vázquez-Lorenzo, Ramón"https://www.zbmath.org/authors/?q=ai:vazquez-lorenzo.ramonThe content of Volume V of the book under review is divided into four chapters, numbered from 16 to 19 in order to facilitate cross references with the chapters from Volumes I--IV, numbered from 1 to 3, from 4 to 8, from 9 to 11, and from 12 to 15, respectively; see [Zbl1354.53001; Zbl1354.53002; Zbl1371.53001; Zbl1419.53002]. The present volume is devoted to elliptic operator theory and its applications to differential geometry.
Chapter 16 surveys some basic results from functional analysis. The first section provides a brief review of some elementary concepts in geometry and topology. The next two sections are devoted to establishing some standard results concerning Banach and Hilbert spaces. The last section of this chapter treats the spectral theory of compact self-adjoint operators in Hilbert space.
The basics of elliptic operator theory are presented in Chapter 17. The Fourier transform, which is one of the fundamental tools in studying the partial differential equations, is the topic of the first section. The Sobolev norms on the Schwarz space in \(\mathbb{R}^m\) are defined in the second section, and basic properties of these norms are examined. Then, the setting moves to the general case of compact Riemannian manifolds; several norms on the space of smooth sections to a vector bundle are considered and the geometry of operators of Laplace type is discussed. The last section of the chapter is devoted to the spectral theory of a self-adjoint operator of Laplace type.
The aim of Chapter 18 is to investigate the elliptic complexes which naturally arise in the field of differential geometry. The first section introduces Clifford algebras and discusses operators of Dirac type. Next, the de Rham complex, the Dolbeault complex and spinors are discussed. The last section treats the Serre duality and the Kodaira vanishing Theorem for the complex Laplacian.
Chapter 19 deals with complex geometry. First, the authors provide an introduction to multivariate holomorphic geometry. Then the geometry of complex projective spaces is widely discussed. The last two sections treat Hodge manifolds and Kodaira embedding Theorem. It is shown that any compact holomorphic manifold admits a positive line bundle if and only if it embeds as a compact holomorphic submanifold of a complex projective space of some dimension.
The present volume ends with a bibliography of 78 items, short biographies of the authors, and an index. Like the previous four books, the present volume is very well written, in a clear and precise style. Certainly, this monograph will be a basic reference in the field of differential geometry.Abelian instantons over the Chen-Teo AF geometryhttps://www.zbmath.org/1472.530022021-11-25T18:46:10.358925Z"Baird, Thomas John"https://www.zbmath.org/authors/?q=ai:baird.thomas-john"Kunduri, Hari"https://www.zbmath.org/authors/?q=ai:kunduri.hari-kSummary: We classify finite energy harmonic 2-forms on the asymptotically flat gravitational instanton constructed by Chen and Teo. We prove that every \(U(1)\)-bundle admits a unique anti-self-dual Yang-Mills instanton (up to gauge equivalence) which we describe explicitly in coordinates. As an application, we compute the classical contribution to partition function for Maxwell theory with theta term.On the Laplacian flow and coflow of \(G_2\)-structureshttps://www.zbmath.org/1472.530042021-11-25T18:46:10.358925Z"Manero, Víctor"https://www.zbmath.org/authors/?q=ai:manero.victor"Otal, Antonio"https://www.zbmath.org/authors/?q=ai:otal.antonio"Villacampa, Raquel"https://www.zbmath.org/authors/?q=ai:villacampa.raquelSummary: We review some recent results on the study of the Laplacian flow and coflow of \(G_2\)-structures.
For the entire collection see [Zbl 1448.65007].Submanifolds, holonomy, and homogeneous geometryhttps://www.zbmath.org/1472.530052021-11-25T18:46:10.358925Z"Olmos, Carlos"https://www.zbmath.org/authors/?q=ai:olmos.carlosSummary: This is an expository article. We would like to draw the attention to some problems in submanifold and homogeneous geometry related to the so-called normal holonomy. We will also survey on recent results obtained in cooperation with J. Berndt, A. J. Di Scala, S. Reggiani, J. S. Rodríguez, H. Tamaru, and F. Vittone.The geometry of \(C^1\) regular curves in sphere with constrained curvaturehttps://www.zbmath.org/1472.530082021-11-25T18:46:10.358925Z"Zhou, Cong"https://www.zbmath.org/authors/?q=ai:zhou.congSummary: In this article, we study \(C^1\) regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a \(C^1\) regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in the same interval.Hypersurfaces In pseudo-Euclidean space with condition \(\Delta\mathbf{H}=\lambda\mathbf{H}\)https://www.zbmath.org/1472.530172021-11-25T18:46:10.358925Z"Gupta, Ram Shankar"https://www.zbmath.org/authors/?q=ai:gupta.ram-shankarSummary: We study hypersurfaces in the pseudo-Euclidean space, whose mean curvature vector satisfies the equation: Laplacian of the vector is parallel to the vector (with constant factor), and the second fundamental form has constant norm. We prove that every such hypersurface of diagonalizable shape operator with at most six distinct principal curvatures has constant mean curvature and constant scalar curvature, and if the above factor is zero then the hypersurface is minimal. We classify locally such non-minimal hypersurfaces with extremal value of the norm of the mean curvature vector. Further, we provide some examples of such hypersurfaces.New approach to uniformly quasi circular motion of quasi velocity biharmonic magnetic particles in the Heisenberg spacehttps://www.zbmath.org/1472.530182021-11-25T18:46:10.358925Z"Körpınar, Talat"https://www.zbmath.org/authors/?q=ai:korpinar.talatSummary: In this paper, we define concept of the uniformly quasi circular motion (UQCM) with biharmonicity condition in the Heisenberg space. That is, we aim to define a new class of UQCM in the three-dimensional Heisenberg space. We further improve an alternative method to find uniformly quasi circular potential electric energy of biharmonic velocity magnetic particles in the Heisenberg space. We also give the relationships between physical and geometrical characterizations of uniformly quasi circular potential electric energy. Finally, we illustrate important figures for uniformly quasi circular potential electric energy with respect to its electric field in the radial direction.Riemannian maps whose total manifolds admit a Ricci solitonhttps://www.zbmath.org/1472.530272021-11-25T18:46:10.358925Z"Yadav, Akhilesh"https://www.zbmath.org/authors/?q=ai:yadav.akhilesh-chandra"Meena, Kiran"https://www.zbmath.org/authors/?q=ai:meena.kiranSummary: In this paper, we study Riemannian maps whose total manifolds admit a Ricci soliton and give a non-trivial example of such Riemannian maps. We obtain necessary conditions for any fiber of such Riemannian map to be Ricci soliton, almost Ricci soliton and Einstein. We also obtain necessary conditions for the range space of such Riemannian map to be Ricci soliton and Einstein. Further, we calculate scalar curvature of total manifold and also for any fiber and range space. Moreover, we study the harmonicity and biharmonicity of Riemannian map from Ricci soliton and obtain necessary and sufficient conditions for such a Riemannian map to be harmonic and biharmonic.An inverse problem for a generalized kinetic equation in semi-geodesic coordinateshttps://www.zbmath.org/1472.530282021-11-25T18:46:10.358925Z"Gölgeleyen, İsmet"https://www.zbmath.org/authors/?q=ai:golgeleyen.ismetSummary: The aim of this article is to investigate the uniqueness of solution of an inverse source problem for a generalized kinetic equation on a Riemannian manifold. The problem is related with an integral geometry problem in semi-geodesic coordinates. We prove the uniqueness in a convex domain by the help of Riemannian coordinates.On product minimal Lagrangian submanifolds in complex space formshttps://www.zbmath.org/1472.530292021-11-25T18:46:10.358925Z"Cheng, Xiuxiu"https://www.zbmath.org/authors/?q=ai:cheng.xiuxiu"Hu, Zejun"https://www.zbmath.org/authors/?q=ai:hu.zejun"Moruz, Marilena"https://www.zbmath.org/authors/?q=ai:moruz.marilena"Vrancken, Luc"https://www.zbmath.org/authors/?q=ai:vrancken.lucMinimal Lagrangian submanifolds in \(n\)-dimensional complex space forms are considered. The authors study such submanifolds which, endowed with the induced metrics, become Riemannian products \(M=M_1\times M_2\) of two Riemannian manifolds, \(M_1\) and \(M_2\), of constant sectional curvature \(c_1\) and \(c_2\). The main theorem, extending \textit{N. Ejiri}'s result [Proc. Am. Math. Soc. 84, 243--246 (1982; Zbl 0485.53022)], states that \(c_1 c_2=0\), and gives a complete classification of these submanifolds. For example, if \(c_1 = c_2=0\), then \(M^n\) is equivalent to either the totally geodesic immersion in \(C^n\) or the Lagrangian flat torus in \(\mathbb{C}P^n(4\tilde c)\).Existence and uniqueness of inhomogeneous ruled hypersurfaces with shape operator of constant norm in the complex hyperbolic spacehttps://www.zbmath.org/1472.530302021-11-25T18:46:10.358925Z"Domínguez-Vázquez, Miguel"https://www.zbmath.org/authors/?q=ai:dominguez-vazquez.miguel"Pérez-Barral, Olga"https://www.zbmath.org/authors/?q=ai:perez-barral.olgaEmbeddedness, convexity, and rigidity of hypersurfaces in product spaceshttps://www.zbmath.org/1472.530312021-11-25T18:46:10.358925Z"Freire de Lima, Ronaldo"https://www.zbmath.org/authors/?q=ai:de-lima.ronaldo-freireThe authors establish the following Hadamard-Stoker-type result: Let \(f: M^n\to H^n\times R\), \(n\ge 3\), be a complete connected hypersurface in a Hadamard manifold with positive definite second fundamental form, and let the height function of \(f\) have a critical point, then \(M\) is embedded and homeomorphic to \(S^n\) or \(\mathbb{R}^n\); furthermore, \(f(M)\) bounds a convex set in \(H^n\times \mathbb{R}\). Section 2 contains some notation and results which will be used afterwards. In Section 3, the authors prove Theorems 1--3 and Corollaries 1 and 2. Analogous results (Theorems 4 and 5) for hypersurfaces in warped product spaces \(\mathbb{R}\times_\rho H^n\) and \(\mathbb{R}\times_\rho S^n\) are proved in Section 4.Twisted-austere submanifolds in Euclidean spacehttps://www.zbmath.org/1472.530322021-11-25T18:46:10.358925Z"Ivey, Thomas A."https://www.zbmath.org/authors/?q=ai:ivey.thomas-a"Karigiannis, Spiro"https://www.zbmath.org/authors/?q=ai:karigiannis.spiroThis paper studies so-called \textit{twisted-austere pairs}, given by \((M,\mu)\) with \(M^k \subset \mathbb{R}^n\) a Riemannian submanifold of dimension \(k\), \(\mu\) a 1-form on \(M\), and such that the subbundle \(N^*M + \mu \subset T^* \mathbb{R}^n \cong \mathbb{C}^n\) is a special Lagrangian submanifold. This condition imposes strong constraints on the geometry of the submanifold \(M\). More precisely: for any normal direction \(\nu\) to \(M\), the second fundamental form \(A^{\nu}\) of \(M\) along \(\nu\), and the 1-form \(\mu\) have to satisfy a certain system of coupled non-linear PDE's, called the \textit{twisted-austere equations}.
The case \(\mu=0\) is well known, and the corresponding submanifolds \(M\) are called \textit{austere}; this case is characterized by the condition that the spectrum \(\sigma(A^{\nu})\) has to satisfy \(-\sigma(A^{\nu})=\sigma(A^{\nu})\) as a set. In particular, austere submanifolds are minimal.
After reviewing the PDE's of twisted-austere pairs \((M,\mu)\), a complete classification of these pairs is given firstly for the easy case when \(M\) is totally geodesic, secondly for dimensions \(k=1,2\), and thirdly for dimension \(k=3\). For dimension \(k=1\) it is proved that \(M\) is a straight line, and for \(k=2\) it is shown that \(M\) has to be a minimal surface and \(\mu\) an harmonic \(1\)-form on \(M\).
The case \(k=3\) is the bulk of the paper, and its classification consists of a careful analysis of the twisted-austere equations. The main conclusion (Theorem 3.1.) is that either \(M^3 \subset \mathbb{R}^n\) is ruled by lines, or \(n=5\) and \(M^3 \subset \mathbb{R}^5\) is a generalized helicoid ruled by planes.Einstein hypersurfaces of \(\mathbb{S}^n\times\mathbb{R}\) and \(\mathbb{H}^n\times\mathbb{R}\)https://www.zbmath.org/1472.530332021-11-25T18:46:10.358925Z"Leandro, Benedito"https://www.zbmath.org/authors/?q=ai:leandro.benedito"Pina, Romildo"https://www.zbmath.org/authors/?q=ai:pina.romildo-s|pina.romildo-da-silva"dos Santos, João Paulo"https://www.zbmath.org/authors/?q=ai:dos-santos.joao-pauloSummary: In this paper, we classify the Einstein hypersurfaces of \(\mathbb{S}^n\times\mathbb{R}\) and \(\mathbb{H}^n\times\mathbb{R}\). We use the characterization of the hypersurfaces of \(\mathbb{S}^n\times\mathbb{R}\) and \(\mathbb{H}^n\times\mathbb{R}\) whose tangent component of the unit vector field spanning the factor \(\mathbb{R}\) is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of \(\mathbb{S}^n\times\mathbb{R}\) and \(\mathbb{H}^n\times\mathbb{R}\) must have constant sectional curvature.Gerbes in geometry, field theory, and quantisationhttps://www.zbmath.org/1472.530342021-11-25T18:46:10.358925Z"Bunk, Severin"https://www.zbmath.org/authors/?q=ai:bunk.severinSummary: This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.Harmonic \(SU(3)\)- and \(G_2\)-structures via spinorshttps://www.zbmath.org/1472.530352021-11-25T18:46:10.358925Z"Niedziałomski, Kamil"https://www.zbmath.org/authors/?q=ai:niedzialomski.kamilLet \((M,g)\) be an oriented Riemannian manifold equipped with a \(G\)-structure, \(G\subset \operatorname{SO}(n)\), where \(n=\dim M\). Each \(G\)-structure \(P\subset \operatorname{SO}(M)\) defines a unique section \(\sigma _{P}\) of the associated bundle \(\operatorname{SO}(M)\times _{G}(\operatorname{SO}(M)/G)\). If \(M\) is compact, the \(G\)-structure \(P\) is said to be harmonic if \(\sigma _{P}\) is a harmonic section. In [\textit{J. C. Gonzalez-Davila} and \textit{F. M. Cabrera}, Math. Proc. Camb. Philos. Soc. 146, No. 2, 435--459 (2009; Zbl 1165.53043)] the authors show that harmonicity is equivalent to a differential equation involving the intrinsic torsion.
In the paper under review the author, using the spinorial description of \(\operatorname{SU}(3)\) and \(G_{2}\)-structures obtained in [\textit{I. Agricola} et al., J. Geom. Phys. 98, 535--555 (2015; Zbl 1333.53037)], gives necessary and sufficient conditions for harmonicity of \(\operatorname{SU}(3)\)- and \(G_{2}\)-structures. The author also describes the results on appropiate homogeneous spaces.On a class of almost Kenmotsu manifolds admitting an Einstein like structurehttps://www.zbmath.org/1472.530362021-11-25T18:46:10.358925Z"Dey, Dibakar"https://www.zbmath.org/authors/?q=ai:dey.dibakar"Majhi, Pradip"https://www.zbmath.org/authors/?q=ai:majhi.pradipSummary: In the present paper, we introduce the notion of \(*\)-gradient \(\rho \)-Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a \((2n+1)\)-dimensional \((k,\mu )'\)-almost Kenmotsu manifold \(M\) admits \(*\)-gradient \(\rho \)-Einstein soliton with Einstein potential \(f\), then (1) the manifold \(M\) is locally isometric to \(\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n\), (2) the manifold \(M\) is \(*\)-Ricci flat and (3) the Einstein potential \(f\) is harmonic or satisfies a physical Poisson's equation. Finally, an illustrative example is presented.Erratum to: ``Almost complex structures in 6D with non-degenerate Nijenhuis tensors and large symmetry groups''https://www.zbmath.org/1472.530372021-11-25T18:46:10.358925Z"Kruglikov, B."https://www.zbmath.org/authors/?q=ai:kruglikov.boris-s"Winther, H."https://www.zbmath.org/authors/?q=ai:winther.henrikSummary: We correct an error in the second part of Theorem 3 of our original paper [ibid. 50, No. 3, 297--314 (2016; Zbl 1362.53039)].Locally conformally balanced metrics on almost abelian Lie algebrashttps://www.zbmath.org/1472.530382021-11-25T18:46:10.358925Z"Paradiso, Fabio"https://www.zbmath.org/authors/?q=ai:paradiso.fabioSummary: We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.\(K\)-contact metrics as Ricci almost solitonshttps://www.zbmath.org/1472.530392021-11-25T18:46:10.358925Z"Patra, Dhriti Sundar"https://www.zbmath.org/authors/?q=ai:patra.dhriti-sundarSummary: In this paper, we prove two fundamental results of Ricci almost soliton on \(K\)-contact manifold. First we prove that if a complete \(K\)-contact metric represents a gradient Ricci almost soliton, then it is compact Einstein Sasakian and isometric to a unit sphere. Next we prove that if a \(K\)-contact metric represents a Ricci almost soliton whose potential vector field \(V\) is contact and the Ricci operator commutes with the structure tensor \(\varphi\), then it is Einstein with Einstein constant \(2n\).Erratum to: ``Ricci solitons in 3-dimensional normal almost paracontact metric manifolds''https://www.zbmath.org/1472.530402021-11-25T18:46:10.358925Z"Perktaş, Selcen Yüksel"https://www.zbmath.org/authors/?q=ai:perktas.selcen-yuksel"Keleş, Sadık"https://www.zbmath.org/authors/?q=ai:keles.sadikSummary: The authors would like to correct some errors which appear in the original publication of their article [ibid. 8, No. 2, 34--45 (2015; Zbl 1328.53038)].Results related to the Chern-Yamabe flowhttps://www.zbmath.org/1472.530412021-11-25T18:46:10.358925Z"Ho, Pak Tung"https://www.zbmath.org/authors/?q=ai:ho.pak-tungThe author studies, for a compact complex manifold \(X\) of complex dimension \(n\), endowed with a Hermitian metric \(\omega_0\), the Chern-Yamabe problem, i.e., to find a conformal metric of \(\omega_0\) such that its Chern scalar curvature is constant. In this paper, as a generalisation of the Chern-Yamabe problem, the author focuses on the problem of prescribing Chern scalar curvature. The main results, with proofs (using geometric flows related to the Chern-Yamabe flow), are:
\begin{itemize}
\item[1] the estimation of the first non-zero eigenvalue of Hodge-de Rham Laplacian of \((X, \omega_0)\),
\item[2] a proof of a version of conformal Schwarz lemma on \((X, \omega_0)\),
\item[3] a proof of the uniqueness of the Chern-Yamabe flow.
\end{itemize}Boundary expansions of complete conformal metrics with negative Ricci curvatureshttps://www.zbmath.org/1472.530422021-11-25T18:46:10.358925Z"Wang, Yue"https://www.zbmath.org/authors/?q=ai:wang.yue.6Summary: We study the boundary behaviors of a complete conformal metric which solves the \(\sigma_k\)-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also \(C^1\) and \(C^2\) estimates for this metric multiplied by the square of the distance in a small neighborhood of the boundary.Some conformally invariant gap theorems for Bach-flat 4-manifoldshttps://www.zbmath.org/1472.530432021-11-25T18:46:10.358925Z"Zhang, Siyi"https://www.zbmath.org/authors/?q=ai:zhang.siyiSummary: \textit{S.-Y. Chang} et al. [Geom. Funct. Anal. 17, No. 2, 404--434 (2007; Zbl 1124.53020)] proved an important gap theorem for Bach-flat metrics with round sphere as model case. In this article, we generalize this result by establishing conformally invariant gap theorems for Bach-flat 4-manifolds with \((\mathbb{CP}^2, g_{FS})\) and \((S^2 \times S^2,g_{prod})\) as model cases. An iteration argument plays an important role in the case of \((\mathbb{CP}^2, g_{FS})\) and the convergence theory of Bach-flat metrics is of particular importance in the case of \((S^2 \times S^2,g_{prod})\). The latter result provides an interesting way to distinguish \((S^2 \times S^2,g_{prod})\) from \((\mathbb{CP}^2\#\bar{\mathbb{CP}}^2,g_{Page})\).Minimal hypersurfaces in the product of two spheres with index onehttps://www.zbmath.org/1472.530442021-11-25T18:46:10.358925Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: Given \(n_1\ge n_2\ge 2\), let \(\Sigma\) be an orientable, minimal hypersurface of \(\mathbb{S}^{n_1}(1)\times \mathbb{S}^{n_2}(a)\) with index one. Assume \(\Sigma\) is closed and either smooth or singular with a singular set \(\mathrm{sing}(\Sigma)\) satisfying \(\mathcal{H}^{n-2}(\mathrm{sing}(\Sigma))=0\). By using the almost product structure, we prove that, when \(a^2\ge \frac{n_2}{n_1-1}\) or \(a^2\le \frac{n_2-1}{n_1} \), such \(\Sigma\) must be totally geodesic. As an application, combining with the results of [\textit{X. Zhou}, J. Differ. Geom. 105, No. 2, 291--343 (2017; Zbl 1367.53054)], we compute the width of \(\mathbb{S}^{n_1}(1)\times \mathbb{S}^{n_2}(a)\).Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifoldshttps://www.zbmath.org/1472.530452021-11-25T18:46:10.358925Z"Eltzner, Benjamin"https://www.zbmath.org/authors/?q=ai:eltzner.benjamin"Galaz-García, Fernando"https://www.zbmath.org/authors/?q=ai:galaz-garcia.fernando"Huckemann, Stephan F."https://www.zbmath.org/authors/?q=ai:huckemann.stephan-f"Tuschmann, Wilderich"https://www.zbmath.org/authors/?q=ai:tuschmann.wilderichSummary: We obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus central limit theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvaturehttps://www.zbmath.org/1472.530462021-11-25T18:46:10.358925Z"Pan, Jiayin"https://www.zbmath.org/authors/?q=ai:pan.jiayinThe escape rate \(E(M,x)\) of a noncompact and complete Riemannian manifold \((M,x)\) with an infinite fundamental group \(\pi_1(M,x)\) measures how fast the minimizing geodesic loops representing elements of the fundamental group escape from bounded balls centered at \(x\). It is defined as follows: \[ E(M,x) = \limsup_{|\gamma| \rightarrow \infty} \frac{d_H(x,c_\gamma)}{|\gamma|}\] where \(\gamma \in \pi_1(M,x)\), \(c_\gamma\) is the shortest geodesic loop representing \(\gamma\), \(|\gamma|\) is the length of \(c_\gamma\), and \(d_H\) is Hausdorff distance.
In this paper it is shown that if \((M,x)\) is a complete open manifold with nonnegative Ricci curvature and \(E(M,x) =0\), then \(\pi_1(M,x)\) is virtually abelian, that is, it contains an abelian subgroup of finite index.
The author also notes that one can apply the Cheeger-Gromoll splitting theorem [\textit{J. Cheeger} and \textit{D. Gromoll}, Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] to prove that the fundamental group of a complete open manifold with nonnegative Ricci curvature is virtually abelian if one assumes the stronger hypothesis that the minimizing geodesic loops representing elements of the fundamental group are contained in a bounded set. This proof is contained in an appendix.
A second appendix contains estimates of the escape rates of several manifolds.The Obata equation with Robin boundary conditionhttps://www.zbmath.org/1472.530472021-11-25T18:46:10.358925Z"Chen, Xuezhang"https://www.zbmath.org/authors/?q=ai:chen.xuezhang"Lai, Mijia"https://www.zbmath.org/authors/?q=ai:lai.mijia"Wang, Fang"https://www.zbmath.org/authors/?q=ai:wang.fangFor an $n$-dimensional closed Riemannian manifold $(M,g)$ with $\operatorname{Ric}(g)\geq (n-1)g$, \textit{A. Lichnérowicz} [Géométrie des groupes de transformations. Paris: Dunod (1958; Zbl 0096.16001)] proved that the first eigenvalue of the Laplace-Beltrami operator satisfies $\lambda_1\geq n$. Moreover, Obata has shown that the equality $\lambda_1=n$ holds if and only if $(M,g)$ is isometric to the round sphere, as a consequence of the following rigidity result: $(M,g)$ admits a non-constat function $f$ satisfying $$ \nabla^2 g+fg=0 $$ if and only if $(M,g)$ is isometric to the standard sphere. The above displayed equation is known as the Obata equation.
We now assume that $M$ has non-empty boundary $\partial M$. The paper under review studies the Obata equation under the Robin boundary condition:
\[ \begin{cases} \nabla^2 g+fg=0 & \text{ in }M,\\
\frac{\partial f}{\partial \nu} + af=0&\text{ on }\partial M,
\end{cases} \]
where $\nu$ is the outward unit normal on $\partial M$ and $a$ is a non-zero constant. As a consequence, the authors obtain that, under certain conditions including $\operatorname{Ric}(g)\geq (n-1)g$, the equality $\lambda_1=n$ holds if and only if $(M,g)$ is a spherical cup (i.e., a geodesic ball in a round sphere).Remarks on scalar curvature of gradient Ricci-Bourguignon sollitonshttps://www.zbmath.org/1472.530482021-11-25T18:46:10.358925Z"Mi, Rong"https://www.zbmath.org/authors/?q=ai:mi.rongSummary: In this paper, we work on the scalar curvature of complete non-compact gradient Ricci-Bourguignon solitons and prove several results. In the first part, we prove that, with some conditions, any complete non-compact gradient Ricci-Bourguignon soliton has nonnegative scalar curvature. In the second part, we show that the quadratic decay at infinity of the Ricci curvature of complete non-compact gradient Ricci-Bourguignon soliton has constant scalar curvature.Nonsmooth convexity and monotonicity in terms of a bifunction on Riemannian manifoldshttps://www.zbmath.org/1472.530492021-11-25T18:46:10.358925Z"Ansari, Qamrul Hasan"https://www.zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Islam, Monirul"https://www.zbmath.org/authors/?q=ai:islam.monirul"Yao, Jen-Chih"https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, we introduce geodesic \(h\)-convexity, geodesic \(h\)-pseudoconvexity and geodesic \(h\)-quasiconvexity of a real-valued function defined on a geodesic convex subset of a Riemannian manifold in terms of a bifunction \(h\). We extend Diewart's mean value theorem for Dini directional derivatives to the Riemannian manifolds. By using this mean value theorem, we present some relations between geodesic convexity and geodesic \(h\)-convexity, geodesic pseudoconvexity and geodesic \(h\)-pseudoconvexity, and geodesic quasiconvexity and geodesic \(h\)-quasiconvexity. We also introduce monotonicity, quasimonotonicity and pseudomonotonicity for the bifunction \(h\). We investigate the relations between geodesic \(h\)-convexity of a real-valued function and monotonicity of \(h\), geodesic \(h\)-pseudoconvexity of a real-valued function and pseudomonotonicity of \(h\), and geodesic \(h\)-quasiconvexity of a real-valued function and quasimonotonicity of \(h\). We introduce the geodesic \(h\)-pseudolinearity of a real-valued function defined on geodesic convex subset of a Riemannian manifold. We provide some characterizations of geodesic \(h\)-pseudolinearity, and give some relations between geodesic \(h\)-pseudolinearity and geodesic pseudolinearity. The pseudoaffiness of a bifunction \(h\) is introduced and some of its characterizations are also presented.The minimal length product over homology bases of manifoldshttps://www.zbmath.org/1472.530502021-11-25T18:46:10.358925Z"Balacheff, Florent"https://www.zbmath.org/authors/?q=ai:balacheff.florent"Karam, Steve"https://www.zbmath.org/authors/?q=ai:karam.steve"Parlier, Hugo"https://www.zbmath.org/authors/?q=ai:parlier.hugoMinkowski's second theorem asserts that the product of the successive minima of a symmetric convex body in Euclidean space is bounded in terms of its volume. It can be reinterpreted as an inequality for flat Finsler tori, i.e., any flat \(n\)-dimensional Finsler torus with Busemann-Hausdorff volume \(V\) admits a family of closed geodesics \(\gamma_1, \ldots, \gamma_n\) inducing a basis of its first real homology group whose product of lengths satisfies \[\prod_{k=1}^{n}\ell(\gamma_k) \leq \frac{2^n}{b_n}V,\] where \(b_n\) denotes the volume of the Euclidean unit ball.
This paper generalises the inequality to a larger class of Riemannian and Finsler manifolds including surfaces. The main theorem shows that if a closed Riemannian \(n\)-manifold \(M\) with first \(\mathbb{Z}_2\)-Betti number \(b>0\) satisfies a non-vanishing condition for the hyperdeterminant reduced modulo 2 of the multilinear map induced by the fundamental class on the first \(\mathbb{Z}_2\)-cohomology group via the cup product, then there exists a \(\mathbb{Z}_2\)-homology basis induced by closed geodesics \(\gamma_1, \ldots, \gamma_b\) whose length product satisfies \[\prod_{k=1}^{b}\ell(\gamma_k) \leq n^b\mathrm{Vol}(M)^{\frac{b}{n}}.\] A corollary is Minkowski's second principle for closed Riemannian surfaces with non-vanishing first Betti number.
The authors also prove a result for the case of dimension 2 via \(\mathbb{Z}\)-homology basis. It asserts that if \(S\) is an orientable closed Riemannian surface of genus \(g\), then there exists a \(\mathbb{Z}\)-homology basis induced by closed geodesics \(\gamma_1, \ldots, \gamma_{2g}\) whose length product satisfies \[\prod_{k=1}^{2g}\ell(\gamma_k) \leq 2^{31g}\mathrm{Area}(S)^{g}.\] Furthermore, many analogous inequalities are obtained in the paper under various conditions.The \(L^p\)-Calderón-Zygmund inequality on non-compact manifolds of positive curvaturehttps://www.zbmath.org/1472.530512021-11-25T18:46:10.358925Z"Marini, Ludovico"https://www.zbmath.org/authors/?q=ai:marini.ludovico"Veronelli, Giona"https://www.zbmath.org/authors/?q=ai:veronelli.gionaSummary: We construct, for \(p> n\), a concrete example of a complete non-compact \(n\)-dimensional Riemannian manifold of positive sectional curvature which does not support any \(L^p\)-Calderón-Zygmund inequality:
\[
\begin{aligned}\Vert\mathrm{Hess}\varphi\Vert_{L^p}\le C(\Vert\varphi\Vert_{L^p}+\Vert\Delta\varphi\Vert_{L^p}),\qquad\forall\varphi\in C^{\infty}_c(M).\end{aligned}
\]
The proof proceeds by local deformations of an initial metric which (locally) Gromov-Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez-Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the \(L^p\)-gradient estimates and the \(L^p\)-Calderón-Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.Erratum to: ``The degree theorem in higher rank''https://www.zbmath.org/1472.530522021-11-25T18:46:10.358925Z"Connell, Chris"https://www.zbmath.org/authors/?q=ai:connell.chris"Farb, Benson"https://www.zbmath.org/authors/?q=ai:farb.bensonSummary: The purpose of this erratum is to correct a mistake in the proof of Theorem 4.1 of the authors' article [ibid. 65, No. 1, 19--59 (2003; Zbl 1067.53032)].Embedding surfaces inside small domains with minimal distortionhttps://www.zbmath.org/1472.530532021-11-25T18:46:10.358925Z"Shachar, Asaf"https://www.zbmath.org/authors/?q=ai:shachar.asafSummary: Given two-dimensional Riemannian manifolds \(\mathcal{M},\mathcal{N} \), we prove a lower bound on the distortion of embeddings \(\mathcal{M}\rightarrow \mathcal{N} \), in terms of the areas' discrepancy \(V_{\mathcal{N}}/V_{\mathcal{M}} \), for a certain class of distortion functionals. For \(V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4\), homotheties, provided they exist, are the unique energy minimizing maps attaining the bound, while for \(V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4\), there are non-homothetic minimizers. We characterize the maps attaining the bound, and construct explicit non-homothetic minimizers between disks. We then prove stability results for the two regimes. We end by analyzing other families of distortion functionals. In particular we characterize a family of functionals where no phase transition in the minimizers occurs; homotheties are the energy minimizers for all values of \(V_{\mathcal{N}}/V_{\mathcal{M}} \), provided they exist.Codimension one Ricci soliton subgroups of solvable Iwasawa groupshttps://www.zbmath.org/1472.530542021-11-25T18:46:10.358925Z"Domínguez-Vázquez, Miguel"https://www.zbmath.org/authors/?q=ai:dominguez-vazquez.miguel"Sanmartín-López, Víctor"https://www.zbmath.org/authors/?q=ai:sanmartin-lopez.victor"Tamaru, Hiroshi"https://www.zbmath.org/authors/?q=ai:tamaru.hiroshiSummary: Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.Almost Ricci-like solitons with torse-forming vertical potential of constant length on almost contact B-metric manifoldshttps://www.zbmath.org/1472.530552021-11-25T18:46:10.358925Z"Manev, Mancho"https://www.zbmath.org/authors/?q=ai:manev.manchoSummary: A generalization of Ricci-like solitons with torse-forming potential, which is constant multiple of the Reeb vector field, is studied. The conditions under which these solitons are equivalent to almost Einstein-like metrics are given. Some results are obtained for a parallel symmetric second-order covariant tensor. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.Correction to: ``Gradient shrinking Ricci solitons of half harmonic Weyl curvature''https://www.zbmath.org/1472.530562021-11-25T18:46:10.358925Z"Wu, Jia-Yong"https://www.zbmath.org/authors/?q=ai:wu.jiayong"Wu, Peng"https://www.zbmath.org/authors/?q=ai:wu.peng"Wylie, William"https://www.zbmath.org/authors/?q=ai:wylie.william-cCorrection to the authors' paper [ibid. 57, No. 5, Paper No. 141, 15 p. (2018; Zbl 1401.53037)].A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponenthttps://www.zbmath.org/1472.530572021-11-25T18:46:10.358925Z"Bartsch, Thomas"https://www.zbmath.org/authors/?q=ai:bartsch.thomas.2|bartsch.thomas.1"Xu, Tian"https://www.zbmath.org/authors/?q=ai:xu.tianOn a compact spin manifold \((M^m,g)\), the authors study solutions of the nonlinear Dirac equation \(D\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi\) where \(\lambda\in\mathbb{R}\) and \(f=o(s^{\frac{2}{m-1}})\) as \(s\to\infty\). Such an equation is called in the paper (NLD) and is the Euler-Lagrange equation associated to some functional \(\mathcal{L}_\lambda(\psi)\). The authors show in Theorem 2.1 that if \(f\) satisfies some conditions (called \(f_1\),\(f_2\) and \(f_3\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda>0\). Also the map \(\mathbb{R}^+\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_k,\lambda_{k+1})\). On the other hand, if \(f\) satisfies some other conditions (called \(f_1\),\(f_4\) and \(f_5\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda\in \mathbb{R}\setminus \{\lambda_k:k\leq 0\}\). The map \(\mathbb{R}\setminus \{\lambda_k:k\leq 0\}\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_{k-1},\lambda_{k})\), if \(k\geq 2\), respectively \((\lambda_{k-1},\lambda_{k})\), if \(k\leq 1\).Some properties of Dirac-Einstein bubbleshttps://www.zbmath.org/1472.530582021-11-25T18:46:10.358925Z"Borrelli, William"https://www.zbmath.org/authors/?q=ai:borrelli.william"Maalaoui, Ali"https://www.zbmath.org/authors/?q=ai:maalaoui.aliSummary: We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on \(\mathbb{R}^3\), which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of \(- \frac{1}{2}\)-Killing spinors on the round sphere \(\mathbb{S}^3\).\(\text{String}^c\) structures and modular invariantshttps://www.zbmath.org/1472.530592021-11-25T18:46:10.358925Z"Huang, Ruizhi"https://www.zbmath.org/authors/?q=ai:huang.ruizhi"Han, Fei"https://www.zbmath.org/authors/?q=ai:han.fei.1|han.fei"Duan, Haibao"https://www.zbmath.org/authors/?q=ai:duan.haibaoIn the paper the authors study algebraic topology concepts revolving around String\(^c\), strong String\(^c\)-structures, weak String\(^c\)-structures, relations between strong and weak String\(^c\) manifolds, modular invariants and group actions on String\(^c\) manifolds; a string structure is a higher version of a Spin structure related to quantum anomalies in physics. They look at string structures from the perspective of Whitehead tower and also of the free loop space \(LM\) (lifting the structure group of the looped spin frame bundle from the loop group of Spin\(^c(n)\) to its universal central extension). These two approaches of looking at string structures are equivalent when the manifold considered is 2-connected. The authors also extend the generalised Witten genera constructed for the first time by Chen.Examples of singularity models for \(\mathbb{Z}/2\) harmonic 1-forms and spinors in dimension threehttps://www.zbmath.org/1472.530602021-11-25T18:46:10.358925Z"Taubes, C. H."https://www.zbmath.org/authors/?q=ai:taubes.clifford-henry"Wu, Y."https://www.zbmath.org/authors/?q=ai:wu.yuanhao|wu.yinting|wu.yihao|wu.yangyong|wu.yurui|wu.yingdong|wu.yakui|wu.yijing|wu.yuqia|wu.yabo|wu.yunhua|wu.yating|wu.yunfang|wu.yandong|wu.yueliang|wu.yuwen|wu.yadi|wu.yongqi|wu.yuching|wu.yuancui|wu.yujia|wu.yungao|wu.yingyi|wu.yizhong|wu.yanxue|wu.yingfei|wu.yasha|wu.yanqing|wu.yinkai|wu.yanjie|wu.yuejian|wu.yundong|wu.yizhou|wu.yilei|wu.yuexiang|wu.yuankai|wu.yunna|wu.yihu|wu.yuwei|wu.yuhang|wu.yajuan|wu.yusen|wu.yushan|wu.yun|wu.yinghui|wu.yijia|wu.yanke|wu.yuesheng|wu.yunxi|wu.yiyun|wu.yongge|wu.yunxing|wu.yining|wu.yueping|wu.yiqian|wu.yi|wu.yanlan|wu.yanyan|wu.yuezhu|wu.yirui|wu.yongle|wu.yongfen|wu.yuesong|wu.yaofa|wu.yilin|wu.yatao|wu.yuliang|wu.yingbo|wu.yue|wu.yimin|wu.yumo|wu.yusheng|wu.ying|wu.yaohao|wu.yanlin|wu.yanmi|wu.yongyong|wu.yanliang|wu.yuxin|wu.yuxi|wu.yuanying|wu.ya|wu.yangbing|wu.yushi|wu.yinan|wu.yinghua|wu.yongwu|wu.yushu|wu.youping|wu.yonghon|wu.yongjian|wu.yuhai|wu.yangqiang|wu.yakun|wu.yutran|wu.yuqi|wu.yunshun|wu.yuhua|wu.yonghong|wu.yunyi|wu.yuhou|wu.yong|wu.yingxiang|wu.yingcai|wu.yadong|wu.yingxin|wu.yile|wu.yunhai|wu.yiqiong|wu.yunli|wu.yuanqi|wu.yuyuan|wu.yongpeng|wu.yulian|wu.yunpin|wu.yongtang|wu.yarong|wu.yixuan|wu.yanhong|wu.yan|wu.yinhu|wu.youwei|wu.yuanming|wu.yuanqing|wu.yuting|wu.youshou|wu.yezhou|wu.yuchen|wu.yumin|wu.yixin|wu.yicheng|wu.yunfeng|wu.yanzhi|wu.yongsheng|wu.yangru|wu.yaoguo|wu.yingzhu|wu.yingtao|wu.yemo|wu.youshi|wu.yifei.1|wu.yanwen|wu.yayun|wu.yiting|wu.yueqin|wu.yuqin|wu.yuanlan|wu.yonghui|wu.yougui|wu.yinyin|wu.yunchao|wu.yuhuai|wu.yijie|wu.yingyan|wu.yongdong|wu.yuxia|wu.yihren|wu.yingjie|wu.yichuan|wu.yuanbing|wu.yuhu|wu.yuzheng|wu.yongfeng|wu.yuefeng|wu.yahao|wu.yonghong.2|wu.yingxue|wu.yuzhen|wu.yuyan|wu.youli|wu.yongwei|wu.yunzhu|wu.yudie|wu.yanan|wu.yingqing|wu.yichong|wu.yuntian|wu.yonggang|wu.yunfei|wu.yuanxiao|wu.yuehui|wu.yuefei|wu.yanhui|wu.yuhui|wu.yuanyin|wu.yichao|wu.yongke|wu.yajun|wu.yudong|wu.you|wu.youlin|wu.yinglong|wu.yanyun|wu.yizhao|wu.yijun|wu.yuanxin|wu.yueshi|wu.yanrui|wu.yanxia|wu.yixiang|wu.yunkai|wu.yahui|wu.yuheng|wu.yiyan|wu.yanmei|wu.yaohua|wu.yunhao|wu.yuqiang|wu.yiming|wu.yiyue|wu.yanghui|wu.yongshen|wu.yana|wu.yongbin|wu.yelei|wu.yuanze|wu.yadong.1|wu.yongxiang|wu.yihong|wu.yongxian|wu.yunnan|wu.yifei|wu.yingnian|wu.yanlei|wu.yongli|wu.yunbing|wu.yuanpeng|wu.yueming|wu.yinlin|wu.yuhao|wu.yihua|wu.yihui|wu.yuan|wu.yejun|wu.yingyu|wu.yanfang|wu.yuchi|wu.yanyu|wu.yaochen|wu.youxun|wu.yanping|wu.yubin|wu.yang|wu.yangyang|wu.yadan|wu.yewei|wu.yongbao|wu.youchang|wu.yaokun|wu.yajing|wu.yufei|wu.yingjiang|wu.yanbin|wu.yonghai|wu.yuin|wu.yali|wu.yuanfang|wu.yansheng.1|wu.yunhui|wu.yonglan|wu.yunbiao|wu.yipeng|wu.yungchao|wu.ye|wu.yujue|wu.yingchun|wu.yuanshan|wu.yueqing|wu.yufeng|wu.yanfeng|wu.yanhua|wu.yongcheng|wu.yifeng|wu.yanli|wu.yuqiu|wu.yipin|wu.yunjian|wu.yongwen|wu.yongren|wu.yiwei|wu.yunping|wu.yongjun|wu.yinli|wu.yanpeng|wu.yingchuan|wu.yuxiao|wu.yaoqiang|wu.yaqiong|wu.yuanheng|wu.yanxian|wu.yingying|wu.yongshi|wu.yongchao|wu.yaozhi|wu.yunjie|wu.yunan|wu.yuling|wu.yanzheng|wu.yinyuan|wu.yunqing|wu.yangcheng|wu.yujing|wu.yuebin|wu.yanghong|wu.yongfei|wu.yunwen|wu.yanmin|wu.yuankang|wu.yu|wu.yanqiu|wu.yimeng|wu.yushuang|wu.yunjiang|wu.yueyu|wu.yingli|wu.yuening|wu.yefan|wu.yaqi|wu.yulun|wu.yinjun|wu.yongqing|wu.yujie|wu.yanming|wu.youlun|wu.yidong|wu.yirong|wu.yuqing|wu.yuelei|wu.yuanyong|wu.yilong|wu.yiqing|wu.yunlong|wu.yafei|wu.yurong|wu.yinzhong|wu.youming|wu.yihong.1|wu.yaping|wu.yuehua|wu.yonghong.1|wu.yonghan|wu.yulin|wu.yingyuan|wu.yuanyuan|wu.yongyan|wu.yanchun|wu.yinshu|wu.yanguo|wu.yongan|wu.yuezhong|wu.yaozhong|wu.yingquan|wu.yuchun|wu.yanchen|wu.yunjin|wu.yulai|wu.youcai|wu.yilun|wu.yuning|wu.yiling|wu.yujiang|wu.yujun|wu.yiquan|wu.yicai|wu.yudan|wu.yuping|wu.yujuan|wu.youlong|wu.yongxin|wu.yunshuang|wu.yanru|wu.yansui|wu.yanling|wu.yanjun|wu.yafeng|wu.yuntao|wu.yibo|wu.yunqiang|wu.yifan|wu.yunpei|wu.yinghwa|wu.yuhan|wu.yongping|wu.yin|wu.yitian|wu.yazhou|wu.yao|wu.yili|wu.yazhen|wu.yuncheng|wu.yaojun|wu.yaoyao|wu.yongzhong|wu.yanting|wu.yousheng|wu.youfeng|wu.yutian|wu.yumei|wu.yansheng|wu.yanqiang|wu.yina|wu.yuxiang|wu.yongfu|wu.youfuSummary: We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a \(\mathbb{Z}/2\) harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are \(\mathbb{Z}/2\) harmonic 1-forms or spinors on \(\mathbb{R}^3\) that are homogeneous with respect to the rescaling of \(\mathbb{R}^3\) with their zero loci consisting of 4 or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example, and to those of a centered octahedron and at centered icosahedron in two others.
For the entire collection see [Zbl 1458.55002].A compact \(\mathrm G_2\)-calibrated manifold with first Betti number \(b_1 = 1\)https://www.zbmath.org/1472.530612021-11-25T18:46:10.358925Z"Fernández, Marisa"https://www.zbmath.org/authors/?q=ai:fernandez.marisa"Fino, Anna"https://www.zbmath.org/authors/?q=ai:fino.anna"Kovalev, Alexei"https://www.zbmath.org/authors/?q=ai:kovalev.aleksei-viktorovich|kovalev.alexei-g"Muñoz, Vicente"https://www.zbmath.org/authors/?q=ai:munoz.vicenteThe authors construct a compact formal 7-manifold with a closed \(\mathrm{G}_2\)-structure and with first Betti number \(b_1 = 1\) not admitting any torsion-free \(\mathrm{G}_2\)-structure. This manifold is not a product. To construct such a manifold, they start with a compact 7-manifold \(M\) equipped with a closed \(\mathrm{G}_2\) form \(\varphi\) and with first Betti number \(b_1(M) = 3\). In fact \(M\) is a nilmanifold, that is, the coset space of a nilpotent Lie group by a cocompact lattice. Then they quotient \(M\) by \(\mathbb Z_2\) preserving \(\varphi\) to obtain an orbifold \(\widehat{M}\) with a closed orbifold \(\mathrm{G}_2\) form \(\widehat{\varphi}\) and with first Betti number \(b_1(\widehat{M}) = 1\). The authors resolve the singularities of the 7-orbifold \(\widehat{M}\) to produce a smooth 7-manifold \(\widetilde{M}\) with a closed \(\mathrm{G}_2\) form \(\widetilde{\varphi}\), with first Betti number \(b_1(\widetilde{M}) = 1\) and such that \((\widetilde{M},\widetilde{\varphi})\) is isomorphic to \((\widehat{M},\widehat{\varphi})\) outside the singular locus of \(\widehat{M}\). Then they prove the properties following: the 7-manifold \(\widetilde{M}\) is formal, with fundamental group \(\pi_1(\widetilde{M}) = \mathbb Z\) and \(\widetilde{M}\) does not admit any torsion-free \(\mathrm{G}_2\)-structure. For the compact 7-manifold \(M\) with the closed \(\mathrm{G}_2\) form \(\varphi\) mentioned above, the authors consider a non-trivial involution of \(M\) preserving \(\varphi\), and they construct an example of a 3-dimensional family of associative volume-minimizing 3-tori in \(\widetilde{M}\). This deformation family is ``maximal''. Finally the authors construct a smooth fibration map \(\widetilde{M}\to S^2 \times S^1\) with generic fiber a coassociative torus and some singular fibers, with both smooth and singular fibers forming maximal deformation families.Local foliation of manifolds by surfaces of Willmore-typehttps://www.zbmath.org/1472.530622021-11-25T18:46:10.358925Z"Lamm, Tobias"https://www.zbmath.org/authors/?q=ai:lamm.tobias"Metzger, Jan"https://www.zbmath.org/authors/?q=ai:metzger.jan"Schulze, Felix"https://www.zbmath.org/authors/?q=ai:schulze.felixThe authors obtain the Euler-Lagrange equation \(\Delta H + H|\tilde A|^2 + H\,\mathrm{Ric}(\nu,\nu)=\lambda\,H\) for the Willmore functional \(J(\Sigma_a)=\frac14\int_{\Sigma_a} H^2 d\mu\) for surfaces \(\Sigma_a\) immersed in a 3-dimensional Riemannian manifold \((M,g)\) and having area \(a>0\). Here, \(H\) denotes the sum of the principal curvatures, \(\tilde A\) is the trace free part of the second fundamental form \(A\), and \(\nu\) is a unit normal of \(\Sigma_a\). The first two authors have shown in [the first two authors, Int. Math. Res. Not. 2010, No. 19, 3786--3813 (2010; Zbl 1202.53056); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 3, 497--518 (2013; Zbl 1290.49090); the first author et al., Math. Ann. 350, No. 1, 1--78 (2011; Zbl 1222.53028)] that if \((M, g)\) is compact then there exists a small \(a_0>0\) depending only on \((M, g)\) such that the infimum in \(J(\Sigma_a)\) is attained for all \(a\in(0, a_0)\) on smooth surfaces \(\Sigma_a\). In Section 2, they find (Proposition~2.1) the expansion of the above Willmore equation on small geodesic spheres. In Section 3, the authors use the implicit function theorem to solve the Willmore equation around non-degenerate critical points of the scalar curvature (Theorem 1.1) in a similar manner to [\textit{R. Ye}, Pac. J. Math. 147, No. 2, 381--396 (1991; Zbl 0722.53022)]. In Section 5, they prove (Corollaries 5.1 and 5.2) a local uniqueness result for the \(\Sigma_a\) as solutions to the Willmore equation.Correction to: ``The total intrinsic curvature of curves in Riemannian surfaces''https://www.zbmath.org/1472.530632021-11-25T18:46:10.358925Z"Mucci, Domenico"https://www.zbmath.org/authors/?q=ai:mucci.domenico"Saracco, Alberto"https://www.zbmath.org/authors/?q=ai:saracco.albertoFrom the text: In the authors' paper [ibid. 70, No. 1, 521--557 (2021; Zbl 1466.53064)], in the statements of the main results, Theorems 1--9 and Proposition 3, one has to assume in addition that the curve \(\mathbf{c}\) is rectifiable.A rigidity result of spacelike self-shrinkers in pseudo-Euclidean spaceshttps://www.zbmath.org/1472.530642021-11-25T18:46:10.358925Z"Qiu, Hongbing"https://www.zbmath.org/authors/?q=ai:qiu.hongbingSummary: In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.Generalized Killing-Ricci tensor for real hypersurfaces in complex hyperbolic two-plane Grassmannianshttps://www.zbmath.org/1472.530652021-11-25T18:46:10.358925Z"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinThis paper classifies the Hopf real hypersurfaces in the complex hyperbolic two-plane Grassmannians \(\operatorname{SU}_{2,m}/\operatorname{S}(\operatorname{U}_2\cdot \operatorname{U}_m)\) with generalized Killing-Ricci tensor. A real hypersurface \(M\) in \(\operatorname{SU}_{2,m}/\operatorname{S}(\operatorname{U}_2\cdot \operatorname{U}_m)\) is said to be \textit{Hopf} if the shape operator \(A\) of \(M\) satisfies \(A\xi = g(A\xi,\xi)\xi\), for the Reeb vector field \(\xi= -JN\), where \(N\) denotes a unit normal vector field, and is said to be \textit{with generalized Killing-Ricci tensor} if the symmetric Ricci tensor \(\operatorname{Ric}\) of \(M\) satisfies
\[ (\nabla_X\operatorname{Ric})(X,X) =0\]
for all vector fields \(X\) on \(M\).A Björling representation for Jacobi fields on minimal surfaces and soap film instabilitieshttps://www.zbmath.org/1472.530662021-11-25T18:46:10.358925Z"Alexander, Gareth P."https://www.zbmath.org/authors/?q=ai:alexander.gareth-p"Machon, Thomas"https://www.zbmath.org/authors/?q=ai:machon.thomasSummary: We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.LW-surfaces with higher codimension and Liebmann's theorem in the hyperbolic spacehttps://www.zbmath.org/1472.530672021-11-25T18:46:10.358925Z"Araújo, Jogli G."https://www.zbmath.org/authors/?q=ai:araujo.jogli-g"de Lima, Henrique F."https://www.zbmath.org/authors/?q=ai:fernandes-de-lima.henriqueA surface \(M\) is said to be \textit{LW} if its Gaussian and mean curvatures \(K\) and \(H\) satisfy a \textit{linear Weingarten} condition \(K = aH + b\) for some real constants \(a\) and \(b\). Assuming that such an \(M\) is complete and immersed in the hyperbolic space \(\mathbb{H}^n\) with flat normal bundle and parallel normalized mean curvature vector field the authors prove that either \(M\) is umbilical or it is isometric to one of the following (flat) surfaces: \(S^1(r)\times\mathbb{R}\), \(S^1(r)\times S^1(\rho)\), \(S^1(r)\times\mathbb{H}^1(\tilde\rho)\) with suitable \(r,\, \rho\) and \(\tilde\rho\). In the proof, they show that \(M\) is an isoparametric surface. Hence, the result follows from the classification of isoparametric submanifolds of hyperbolic spaces due to \textit{B. Wu} [Trans. Am. Math. Soc. 331, No. 2, 609--626 (1992; Zbl 0760.53035)].Minimal \(n\)-noids in hyperbolic and anti-De Sitter 3-spacehttps://www.zbmath.org/1472.530682021-11-25T18:46:10.358925Z"Bobenko, Alexander I."https://www.zbmath.org/authors/?q=ai:bobenko.alexander-ivanovich"Heller, Sebastian"https://www.zbmath.org/authors/?q=ai:heller.sebastian-gregor"Schmitt, Nicholas"https://www.zbmath.org/authors/?q=ai:schmitt.nicholasSummary: We construct minimal surfaces in hyperbolic and anti-de Sitter 3-space with the topology of a \(n\)-punctured sphere by loop group factorization methods. The end behaviour of the surfaces is based on the asymptotics of Delaunay-type surfaces, i.e. rotational symmetric minimal cylinders. The minimal surfaces in \(H^3\) extend to Willmore surfaces in the conformal 3-sphere \(S^3 = H^3 \cup S^2 \cup H^3\).Curvature estimates for graphs over Riemannian domainshttps://www.zbmath.org/1472.530692021-11-25T18:46:10.358925Z"Coswosck, Fabiani Aguiar"https://www.zbmath.org/authors/?q=ai:coswosck.fabiani-aguiar"Fontenele, Francisco"https://www.zbmath.org/authors/?q=ai:fontenele.francisco-xSummary: Let \(M^n\) be a complete \(n\)-dimensional Riemannian manifold and \(\Gamma_f\) the graph of a \(C^2\)-function \(f\) defined on a metric ball of \(M^n\). In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in \(\mathbb{R}^3\) which is a graph over an open disk in the plane, we obtain in this work upper estimates for \(\inf |R|, \inf |A|\) and \(\inf |H_k|\), where \(R\), \(|A|\) and \(H_k\) are, respectively, the scalar curvature, the norm of the second fundamental form and the \(k\)-th mean curvature of \(\Gamma_f\). From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if \(M^n, \; n\geq 3,\) is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant \(c\), and \(\Gamma_f\) is a graph over \(M\) with Ricci curvature less than \(c\), then \(\inf |A| \leq 3(n-2)\sqrt{-c} \). This result generalizes and improves a theorem of Chern for entire graphs in \(\mathbb R^{n+1} \).Rigidity and nonexistence results for \(r\)-trapped submanifold in GRW spacetimeshttps://www.zbmath.org/1472.530702021-11-25T18:46:10.358925Z"Cruz, F. C. Jr."https://www.zbmath.org/authors/?q=ai:cruz.f-c-jun"Lima, E. A. Jr."https://www.zbmath.org/authors/?q=ai:lima.eraldo-almeida-jun"Santos, M. S."https://www.zbmath.org/authors/?q=ai:santos.marcio-sSummary: In this paper, we introduce the notion of \(r\)-trapped submanifolds immersed in generalized Robertson-Walker spacetimes as generalization of the trapped submanifolds introduced by Penrose. Considering some properties such as parabolicity and stochastic completeness, we prove rigidity and nonexistence results for \(r\)-trapped in some configurations of GRW spacetimes and, lastly, we provide examples of \(r\)-trapped submanifolds, some of them are also simultaneously trapped, but we provided examples proving that the notion of \(r\)-trapped submanifolds is different accordingly to the number \(r\).On constant curvature submanifolds of space formshttps://www.zbmath.org/1472.530712021-11-25T18:46:10.358925Z"Dajczer, M."https://www.zbmath.org/authors/?q=ai:dajczer.marcos"Onti, C.-R."https://www.zbmath.org/authors/?q=ai:onti.christos-raent"Vlachos, Th."https://www.zbmath.org/authors/?q=ai:vlachos.theodorosLet \(\mathbb{Q}_{\tilde{c}}^{n+p}\) be a simply connected space form. A well-known result of \textit{E. Cartan} [Bull. Soc. Math. Fr. 47, 125--160 (1920; JFM 47.0692.03)] states that an isometric immersion \(f:M_c^n\to \mathbb{Q}_{\tilde{c}}^{n+p}\) of a connected Riemannian manifold \(M_c^n\) of constant sectional curvature \(c\) has codimension \(p\geq n-1\) and if \(p=n-1\) the normal bundle is flat, provided that \(n\geq 3\) and \(c<\tilde{c}\). The dual case \(c>\tilde{c}\) was considered by \textit{J. D. Moore} [Duke Math. J. 44, 449--484 (1977; Zbl 0361.53050)] and the same conclusion was obtained under the extra assumption that \(f\) is free of weak-umbilic points.
In this short paper, the authors consider the converse of the above-mentioned results. The main result is that if the dimension of the first normal space of the immersion \(f:M_c^n\to \mathbb{Q}_{\tilde{c}}^{n+p}, n\geq 2, c\neq \tilde{c}\) is \(n-1\) (if \(c>\tilde{c}\), it is also assumed that \(f\) is free of weak-umbilic points.), then the substantial codimension of the immersion is \(p=n-1\). Examples are also provided to show that any substantial codimension is possible if the first normal space has the highest possible rank \(n\).Helicoids and catenoids in \(M\times \mathbb{R} \)https://www.zbmath.org/1472.530722021-11-25T18:46:10.358925Z"de Lima, Ronaldo F."https://www.zbmath.org/authors/?q=ai:de-lima.ronaldo-freire"Roitman, Pedro"https://www.zbmath.org/authors/?q=ai:roitman.pedroSummary: Given an arbitrary \(C^\infty\) Riemannian manifold \(M^n\), we consider the problem of introducing and constructing minimal hypersurfaces in \(M\times \mathbb{R}\) which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space \(\mathbb{R}^3=\mathbb{R}^2\times \mathbb{R} \). Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections and then called \textit{vertical helicoids} and \textit{vertical catenoids}. We establish that vertical helicoids in \(M\times \mathbb{R}\) have the same fundamental uniqueness properties of the helicoids in \(\mathbb{R}^3\). We provide several examples of properly embedded vertical helicoids in the case where \(M\) is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on \(\text{Nil}_3\) and \(\text{Sol}_3\) are also presented. We show that vertical helicoids of \(M\times \mathbb{R}\) whose horizontal sections are totally geodesic in \(M\) are locally given by a ``twisting'' of a fixed totally geodesic hypersurface of \(M\). We give a local characterization of hypersurfaces of \(M\times \mathbb{R}\) which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in \(M\times \mathbb{R}\) if and only if \(M\) admits families of isoparametric hypersurfaces. If so, properly embedded vertical catenoids can be constructed through the solutions of a certain first-order linear differential equation. Finally, we give a complete classification of the hypersurfaces of \(M\times \mathbb{R}\) whose angle function is constant.Rigidity of surfaces with constant extrinsic curvature in Riemannian product spaceshttps://www.zbmath.org/1472.530732021-11-25T18:46:10.358925Z"dos Santos, Fábio R."https://www.zbmath.org/authors/?q=ai:dos-santos.fabio-reisSummary: The present paper deals with complete surfaces having constant extrinsic curvature in a Riemannian product space \(M^2(c)\times\mathbb{R}\), where \(M^2(c)\) is a space form with constant sectional curvature \(c\in\{-1,1\}\). In such setting, we find a Simons-type formula for Cheng-Yau's operator which is used to prove that such surfaces are isometric to a cylinder \(\mathbb{H}^1\times\mathbb{R}\), when \(c=-1\) or isometric to a slice \(\mathbb{S}^2\times\{t_0\}\) for some \(t_0\in\mathbb{R}\) when \(c=1\). Finally, we extend the result, when \(c=-1\), for the Weingarten linear case.Semi Riemannian hypersurfaces with a canonical principal directionhttps://www.zbmath.org/1472.530742021-11-25T18:46:10.358925Z"Garcia Dinorin, Adrian"https://www.zbmath.org/authors/?q=ai:garcia-dinorin.adrian"Ruiz-Hernández, Gabriel"https://www.zbmath.org/authors/?q=ai:ruiz-hernandez.gabrielSummary: We study semi-Riemannian hypersurfaces with a canonical principal direction (CPD) with respect to a nondegenerate closed conformal vector field on a semi-Riemannian ambient manifold. We give a characterization of such hypersurfaces. In the case when such hypersurface is a surface with constant mean curvature in a semi-Riemannian space form, we prove that it has an intrinsic Killing vector field. A special case of hypersurfaces with a CPD are those with constant angle with respect to a parallel vector field in the semi-Riemannian ambient. We prove that a surface with zero mean curvature and constant angle, in a Loretzian ambient of arbitrary dimension, is necessarily flat. When the surface is timelike and the ambient has non positive curvature then the surface is totally geodesic. When the surface is spacelike and the ambient has non negative curvature then the surface is totally geodesic. In general when the ambient is of dimension three then the surface is always totally geodesic.Half-space type theorem for translating solitons of the mean curvature flow in Euclidean spacehttps://www.zbmath.org/1472.530752021-11-25T18:46:10.358925Z"Kim, Daehwan"https://www.zbmath.org/authors/?q=ai:kim.daehwan"Pyo, Juncheol"https://www.zbmath.org/authors/?q=ai:pyo.juncheolSummary: In this paper, we determine which half-space contains a complete translating soliton of the mean curvature flow and it is related to the well-known half-space theorem for minimal surfaces. We prove that a complete translating soliton does not exist with respect to the velocity \(\mathrm{v}\) in a closed half-space \(\mathcal{H}_{\widetilde{\mathrm{v}}}= \{ x \in \mathbb{R}^{n+1} \mid \langle x, \widetilde{\mathrm{v}}\rangle \leq 0 \}\) for \(\langle\mathrm{v}, \widetilde{\mathrm{v}} \rangle > 0\), whereas in a half-space \(\mathcal{H}_{\widetilde{\mathrm{v}}}\), \(\langle\mathrm{v}, \widetilde{\mathrm{v}} \rangle \leq 0\), a complete translating soliton can be found. In addition, we extend this property to cones: there are no complete translating solitons with respect to \(\mathrm{v}\) in a right circular cone \(C_{ {\mathrm{v}}, a}=\{ x \in \mathbb{R}^{n+1} \mid \langle \frac{x}{\Vert x\Vert}, \mathrm{v} \rangle \leq a < 1 \} \).Rotational surfaces of constant astigmatism in space formshttps://www.zbmath.org/1472.530762021-11-25T18:46:10.358925Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez-camino.rafael"Pámpano, Álvaro"https://www.zbmath.org/authors/?q=ai:pampano.alvaroA surface in space form \(\mathbb{M}^3\) is said to have \textit{constant astigmatism} if the difference \(\rho_2 - \rho_1\) of the principal radii \(\rho_1\) and \(\rho_2\) of curvature is a constant function. Here, the authors classify constant astigmatism surfaces of revolution. To this end, they provide a variational characterisation of generating curves of such surfaces and classify solutions of the corresponding Euler-Lagrange equation.Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energieshttps://www.zbmath.org/1472.530772021-11-25T18:46:10.358925Z"Wei, Guodong"https://www.zbmath.org/authors/?q=ai:wei.guodongThe author extends \textit{F. Hélein}'s convergence theorem [Harmonic maps, conservation laws and moving frames. Transl. from the French. Cambridge: Cambridge University Press (2002; Zbl 1010.58010)] to a sequence of rescaled branched conformal immersions. He applies this result to study the blow-up behavior of a sequence of branched conformal immersions of a closed Riemann surface into \(\mathbb{R}^n\) whose areas and Wilmore energies are uniformly bounded. In this situation he gives a bubble tree construction and proves an integral identity for the Gauss curvature of the limit.CR-harmonic mapshttps://www.zbmath.org/1472.530782021-11-25T18:46:10.358925Z"Dietrich, Gautier"https://www.zbmath.org/authors/?q=ai:dietrich.gautierSummary: We develop the notion of renormalized energy in Cauchy-Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.Energy identity and necklessness for \(\alpha\)-Dirac-harmonic maps into a spherehttps://www.zbmath.org/1472.530792021-11-25T18:46:10.358925Z"Li, Jiayu"https://www.zbmath.org/authors/?q=ai:li.jiayu"Liu, Lei"https://www.zbmath.org/authors/?q=ai:liu.lei.1"Zhu, Chaona"https://www.zbmath.org/authors/?q=ai:zhu.chaona"Zhu, Miaomiao"https://www.zbmath.org/authors/?q=ai:zhu.miaomiaoSummary: Let \((\phi_{\alpha}, \psi_{\alpha})\) be a sequence of \(\alpha \)-Dirac-harmonic maps from a Riemann surface \(M\) to a compact Riemannian manifold \(N\) with uniformly bounded energy. If the target \(N\) is a sphere \(S^{K-1}\), we show that the energy identity and necklessness hold during the interior blow-up process as \(\alpha \searrow 1\) for such a sequence .Boundary value problems for Dirac-harmonic maps and their heat flowshttps://www.zbmath.org/1472.530802021-11-25T18:46:10.358925Z"Liu, Lei"https://www.zbmath.org/authors/?q=ai:liu.lei.1"Zhu, Miaomiao"https://www.zbmath.org/authors/?q=ai:zhu.miaomiaoSummary: Dirac-harmonic maps are critical points of an action functional that is motivated from the nonlinear \(\sigma\)-model of quantum field theory. They couple a harmonic map like field with a nonlinear spinor field. In this article, we shall discuss the latest progress on heat flow approaches for the existence of Dirac-harmonic maps under appropriate boundary conditions. Also, we discuss the refined blow-up analysis for two types of approximating Dirac-harmonic maps arising from those heat flow approaches.Compact null hypersurfaces in Lorentzian manifoldshttps://www.zbmath.org/1472.530812021-11-25T18:46:10.358925Z"Atindogbé, C."https://www.zbmath.org/authors/?q=ai:atindogbe.cyriaque"Gutiérrez, M."https://www.zbmath.org/authors/?q=ai:gutierrez.manuel"Hounnonkpe, R."https://www.zbmath.org/authors/?q=ai:hounnonkpe.raymondIn this paper the authors exhibit new results on the existence and various geometric and causal properties of compact null hypersurfaces in Lorentzian manifolds.
Recall that in a Lorentzian manifold \((M,g)\) a submanifold \(N\) is called \textit{null} if the Lorentzian metric \(g\vert_N\) restricted to this submanifold is degenerate. This degeneracy, among other things, implies that the normal bundle of \(N\) is tangent to \(N\) hence the usual projection techniques applicable to submanifolds (to obtain induced connection, etc.) do not work here; consequently studying null submanifolds in Lorentzian manifolds requires special techniques. Despite this difficulty, from the physical viewpoint, null hypersurfaces are important because they describe black hole event horizons and their properties contain information about the global causal structure of the ambient spacetime, too.
In this paper the authors prove several results on compact null hypersurfaces like their (non)-existence under various symmetry assumptions on \((M,g)\) (see, e.g., Theorem 4) as well as geometric properties such as when a compact null hypersurface must be totally geodesic (see Theorem 7). In the last section elements of dynamical system theory are used to obtain some implications in causality theory with special attention to the imprisoning property (see Theorems 30, 33 and 35). Recall that the imprisoning property helps one to decide whether or not a certain null hypersurface describes a black hole event horizon in the ambient spacetime.The light ray transform in stationary and static Lorentzian geometrieshttps://www.zbmath.org/1472.530822021-11-25T18:46:10.358925Z"Feizmohammadi, Ali"https://www.zbmath.org/authors/?q=ai:feizmohammadi.ali"Ilmavirta, Joonas"https://www.zbmath.org/authors/?q=ai:ilmavirta.joonas"Oksanen, Lauri"https://www.zbmath.org/authors/?q=ai:oksanen.lauriSummary: Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.A new proof of a conjecture on nonpositive Ricci curved compact Kähler-Einstein surfaceshttps://www.zbmath.org/1472.530832021-11-25T18:46:10.358925Z"Guan, Zhuang-Dan Daniel"https://www.zbmath.org/authors/?q=ai:guan.zhuang-dan-danielSummary: In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of \textit{Y. Hong} et al. [Acta Math. Sin. 31, No. 5, 595--602 (1988; Zbl 0678.53060); Sci. China, Math. 54, No. 12, 2627--2634 (2011; Zbl 1259.53067)]. Moreover, we proved that any compact Kähler-Einstein surface \(M\) is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) \(M\) has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction.Construction of projective special Kähler manifoldshttps://www.zbmath.org/1472.530842021-11-25T18:46:10.358925Z"Mantegazza, Mauro"https://www.zbmath.org/authors/?q=ai:mantegazza.mauroSummary: In this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on \(\text{SU}(n,1)/\text{S}(\text{U}(n)\text{U}(1))\). We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groupshttps://www.zbmath.org/1472.530852021-11-25T18:46:10.358925Z"Berestovskii, V. N."https://www.zbmath.org/authors/?q=ai:berestovskii.valerii-nikolaevich"Zubareva, I. A."https://www.zbmath.org/authors/?q=ai:zubareva.irina-aleksandrovnaSummary: We find the abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra. In terms of the structure constants of a Lie algebra and the Minkowski support function of the unit ball of the seminorm on the two-dimensional subspace of a Lie algebra which defines a quasimetric, we establish a criterion for the strict abnormality of these extremals.The dual Bonahon-Schläfli formulahttps://www.zbmath.org/1472.530862021-11-25T18:46:10.358925Z"Mazzoli, Filippo"https://www.zbmath.org/authors/?q=ai:mazzoli.filippoIn the paper, for a differentiable deformation of geometrically finite 3-manifold the Bonahon-Schläfli Formula gives the derivative of the volume of the convex cones in terms of the variation of the geometry of their boundaries (the classical Schläfli Formula is applicable for determining the volume of the hyperbolic polyhedra). Here the author studies the analogous problem for the dual volume and gives a self-contained proof of the dual Bonahon-Schläfli Formula (without using Bonahon's results).Evolution equations for non-degenerate 2-formshttps://www.zbmath.org/1472.530882021-11-25T18:46:10.358925Z"He, Weiyong"https://www.zbmath.org/authors/?q=ai:he.weiyongSummary: In this paper we introduce several geometric flows that evolve primarily non-degenerate 2-forms, with the motivation to develop a geometric flow to approach the existence of the symplectic forms on a compact manifold that supports a non-degenerate 2-form. In particular, we introduce \(\text{d}^{\ast }\text{d} \)-flow and \(\text{d}^{\ast }\text{d} \)-Ricci flow for a compatible pair \((\omega , J)\) of an almost Hermitian structure. We prove the short time existence and uniqueness of these flows with smooth initial data, and give some examples of long time existence and convergence.Contact metric structures with the typical contact form on the 3-dimensional manifoldhttps://www.zbmath.org/1472.530892021-11-25T18:46:10.358925Z"Yamamoto, Akio"https://www.zbmath.org/authors/?q=ai:yamamoto.akioSummary: Let \(\eta\) be a typical contact form on the manifold \(M^3=S^3\), \(\mathbb{R}^3\) and \(T^3\). We determine contact metric structures \((\varphi,\xi,\eta,g)\) on \(M^3\). And then we consider the cases that \((\varphi,\xi,\eta,g)\) is \(\eta\)-Einstein, or it is Sasakian, or it is K-contact, respectively.Almost contact Riemannian three-manifolds with Reeb flow symmetryhttps://www.zbmath.org/1472.530912021-11-25T18:46:10.358925Z"Perrone, Domenico"https://www.zbmath.org/authors/?q=ai:perrone.domenico\textit{J. T. Cho} and \textit{M. Kimura} [Differ. Geom. Appl. 35, 266--273 (2014; Zbl 1319.53094)] studied almost Kenmotsu three-manifolds whose Ricci operator is invariant along the Reeb flow. They claim to obtain a classification result for such manifolds, but unfortunately the proof presents some problems (cf. Remark 4.1). The aim of the paper under review is to correct the classification. Therefore, using the canonical foliation on such spaces, the author obtains the complete classification of simply connected homogeneous almost \(\alpha\)-Kenmotsu three-manifolds whose Ricci operator is invariant along the Reeb flow (cf. Theorem 1.2).Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold \(M^2\times\mathbb{R}\)https://www.zbmath.org/1472.530982021-11-25T18:46:10.358925Z"Chen, Li"https://www.zbmath.org/authors/?q=ai:chen.li.2|chen.li.3|chen.li.4|chen.li.1|chen.li.6|chen.li.7|chen.li.5"Hu, Dan-Dan"https://www.zbmath.org/authors/?q=ai:hu.dandan"Mao, Jing"https://www.zbmath.org/authors/?q=ai:mao.jing"Xiang, Ni"https://www.zbmath.org/authors/?q=ai:xiang.niSummary: In this paper, for the Lorentz manifold \(M^2\times\mathbb{R}\) with \(M^2\) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in \(M^2\), which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.Mean curvature flow in asymptotically flat product spacetimeshttps://www.zbmath.org/1472.530992021-11-25T18:46:10.358925Z"Kröncke, Klaus"https://www.zbmath.org/authors/?q=ai:kroncke.klaus"Petersen, Oliver Lindblad"https://www.zbmath.org/authors/?q=ai:petersen.oliver-lindblad"Lubbe, Felix"https://www.zbmath.org/authors/?q=ai:lubbe.felix"Marxen, Tobias"https://www.zbmath.org/authors/?q=ai:marxen.tobias"Maurer, Wolfgang"https://www.zbmath.org/authors/?q=ai:maurer.wolfgang.1"Meiser, Wolfgang"https://www.zbmath.org/authors/?q=ai:meiser.wolfgang"Schnürer, Oliver C."https://www.zbmath.org/authors/?q=ai:schnurer.oliver-christian"Szabó, Áron"https://www.zbmath.org/authors/?q=ai:szabo.aron"Vertman, Boris"https://www.zbmath.org/authors/?q=ai:vertman.borisSummary: We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M \times \mathbb{R} \), where \(M\) is asymptotically flat. If the initial hypersurface \(F_0 \subset M \times \mathbb{R}\) is uniformly spacelike and asymptotic to \(M \times \{ s \}\) for some \(s \in \mathbb{R}\) at infinity, we show that a mean curvature flow starting at \(F_0\) exists for all times and converges uniformly to \(M \times \{ s\}\) as \(t \rightarrow \infty \).Type II ancient compact solutions to the Yamabe flowhttps://www.zbmath.org/1472.531022021-11-25T18:46:10.358925Z"Daskalopoulos, Panagiota"https://www.zbmath.org/authors/?q=ai:daskalopoulos.panagiota"del Pino, Manuel"https://www.zbmath.org/authors/?q=ai:del-pino.manuel-a"Sesum, Natasa"https://www.zbmath.org/authors/?q=ai:sesum.natasaThe authors construct new type II ancient compact solutions to the Yamabe flow. These solutions are rotationally symmetric and converge to a tower of two spheres as \(t\to -\infty\). The ancient solutions to the Yamabe flow are constructed by gluing two exact solutions to the rescaled equations, that is spheres, with narrow cylindrical necks. They use perturbation theory via fixed points arguments based on sharp estimates on ancient solutions of the approximated linear equation and a careful estimation of the error terms. This result can be generalized to the gluing of \(k\) spheres for any \(k\ge 2\) in such a way that the configuration of radii of the spheres was driven by a first-order Toda system as \(t\to -\infty\).Complete Ricci solitons via estimates on the soliton potentialhttps://www.zbmath.org/1472.531042021-11-25T18:46:10.358925Z"Wink, Matthias"https://www.zbmath.org/authors/?q=ai:wink.matthiasSummary: In this paper, a growth estimate on the soliton potential is shown for a large class of cohomogeneity one manifolds. This is used to construct continuous families of complete steady and expanding Ricci solitons in the setups of
\textit{H. Lü} et al. [Phys. Lett., B 593, No. 1--4, 218--226 (2004; Zbl 1247.53054)]
and
\textit{A. S. Dancer} and \textit{M. Y. Wang} [Ann. Global Anal. Geom. 39, No. 3, 259--292 (2011; Zbl 1215.53040)].
It also provides a different approach to the two summands system
[the author, ``Cohomogeneity one Ricci solitons from Hopf fibrations'', Preprint, \url{arXiv:1706.09712}]
that applies to all known geometric examples.Toward a quasi-Möbius characterization of invertible homogeneous metric spaceshttps://www.zbmath.org/1472.540142021-11-25T18:46:10.358925Z"Freeman, David"https://www.zbmath.org/authors/?q=ai:freeman.david-mandell|freeman.david-f|freeman.david-l|freeman.david-j"Le Donne, Enrico"https://www.zbmath.org/authors/?q=ai:le-donne.enricoThe paper under review contributes to the ongoing metric characterization of boundaries of rank-one symmetric spaces of non-compact type. These spaces posses boundaries at infinity, metric spaces equipped with visual distances. In the following a boundary of a rank-one symmetric space is abbreviated BROSS.
In the present paper it is conjectured that:
Conjecture 1.1. A metric space is bi-Lipschitz equivalent to some BROSS if and only if it is locally compact, connected, uniformly bi-Lipschitz homogeneous and quasi-invertible.
The article presents a series of theorems that work towards resolving this conjecture. First in terms of Möbius homogenity, then coarse versions in terms of uniformly strongly quasi-Möbius homogenity, and finally two results pertaining to unbounded, proper and disconnected metric spaces are presented.
The following terminology is used to state the theorems. For a set \(X\) and \(p \in X\) let
\[
X_p := X \setminus \{p\} \quad\text{and}\quad \hat{X}:=X\cup\{\infty\}.
\]
A metric space \((X,d)\) is called \emph{invertible} if it is unbounded and admits a homeomorphism \(\tau_p:X_p\to X_p\) (called \emph{inversion at p}) such that for \(x,y \in X_p\):
\[
d(\tau_p(x), \tau_p(y)) = \frac{d(x,y)}{d(x,p)d(y,p)},
\]
and \(\tau_p\) admits a continuous extension to \(\infty \in \hat{X}\). Furthermore define \(\text{Inv}_p(X) := (\hat{X},i_p)\) and \(\text{Sph}_p(X):=(\hat{X},s_p)\), where
\[
i_p(x,y):= \frac{d(x,y)}{d(x,p)d(y,p)},\quad i_p(x,\infty):= \frac{1}{d(x,p)},
\]
\[
s_p(x,y):= \frac{d(x,y)}{(1+d(x,p))(1+d(y,p))},\quad s_p(x,\infty):= \frac{1}{1+d(x,p)}.
\]
\(i_p\) respectively \(s_p\) are metrics if and only if \(X\) is a Ptolemy space. Furthermore a space admits a metric inversion if and only if \(\text{Inv}_p(X)\) is isometric to \(X\).
A homeomorphism \(f:X \to Y\) between (quasi-)metric spaces is called \emph{Möbius} if it preserves the cross-ratio for any quadruples \((a,b,c,d)\) of distinct points in \(X\):
\[
\frac{d(f(a),f(c))d(f(b),f(d))}{d(f(a),f(d))d(f(b),f(c))} = \frac{d(a,c)d(b,d)}{d(a,d)d(b,c)}.
\]
The group of all Möbius self-homeomorphisms of \(X\) is denoted \(\text{Möb}(X)\). A metric space \(X\) is called \emph{\(2\)-point Möbius homogeneous} if for every two pairs \(\{x,y\}, \{a,b\}\) of distinct points in \(X\), there exists an \(f \in \text{Möb}(X)\) with \(f(x) = a\) and \(f(y) = b\).
Coarse versions in the definitions (quasi-invertible, quasi-Möbius, etc.) are established by requiring only bi-Lipschitz equivalence instead of equality in the defining statements.
Given a metric space \((X,d)\) and \(\alpha \in \ ]0,1]\), \((X,d^\alpha)\) is called the \emph{\(\alpha\)-snowflake of \((X,d)\)}.
The main results then are:
Theorem 1.2. Suppose \(X\) is an unbounded, locally compact, complete, and connected metric space. The following statements are equivalent:
\begin{enumerate}
\item \(X\) is Möbius homeomorphic to some BROSS.
\item \(X\) is isometric to some BROSS.
\item \(X\) is isometrically homogeneous and invertible.
\item The sphericalized space \(\text{Sph}_p(X)\) is \(2\)-point Möbius homogeneous, for some (and hence any) \(p \in X\).
\end{enumerate}
This is similar to the main result in [\textit{S. Buyalo} and \textit{V. Schroeder}, Geom. Dedicata 172, 1--45 (2014; Zbl 1362.53061)] but in the present theorem the involution is not required to be fixed point free and does not assume the presence of a Ptolemy circle.
Theorem 1.2. results in
Corollary 1.3. Suppose \(X\) is an unbounded, locally compact, and connected metric space. There exists \(n \in \mathbb{N}\) and \(\alpha \in \ ]0,1]\) such that \(X\) is isometric to \((\mathbb{R}^n, |\cdot|^\alpha)\) if and only if the space \(\text{Sph}_p(X)\) is \(3\)-point Möbius homogeneous, for some/any \(p \in X\).
The consequences of (\(1\)-point) Möbius homogenity are also explored:
Theorem 1.4. Let \(X\) be a compact and quasi-convex metric space of finite topological dimension. If \(X\) is Möbius homogeneous, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Corollary 1.5. Let \(X\) be the boundary of a CAT(\(-1\))-space. If \(X\) is Möbius homogeneous, of finite topological dimension, and connected by Möbius circles, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Coarse versions of the results are then presented:
Proposition 1.6. A proper and unbounded metric space \(X\) is uniformly bi-Lipschitz homogeneous and quasi-invertible if and only if, for some \(p \in X\), the quasi-sphericalized space \(\text{sph}_p(X)\) (the metric space bi-Lipschitz to \(\text{Sph}_p(X)\)) is \(2\)-point uniformly strongly quasi-Möbius homogeneous.
Proposition 1.7. A homeomorphism \(f:X \to Y\) between proper and unbounded metric spaces is strongly quasi-Möbius if and only if it is bi-Lipschitz. Furthermore, \(f\) is Möbius if and only if \(f\) is a similarity.
The article's main contribution towards Conjecture 1.1. is the following
Theorem 1.8. If \(X\) is an unbounded locally compact metric space that is uniformly bi-Lipschitz homogeneous, quasi-invertible, and contains an non-degenerate curve, then \(X\) is path connected, locally path connected, proper, and Ahlfors regular. Furthermore,
\begin{enumerate}
\item if in addition \(X\) contains a cut point, then \(X\) is bi-Lipschitz homeomorphic to \((\mathbb{R}, |\cdot|^\alpha)\), for some \(\alpha \in \ ]0,1]\);
\item if instead \(X\) contains no cut points, then \(X\) is linearly locally connected. Moreover,
\begin{enumerate}
\item if in addition \(X\) contains a non-degenerate rectifiable curve, then \(X\) is annularly quasi-convex;
\item if instead all rectifiable curves in \(X\) are degenerate, then, for some \(\alpha \in \ ]0,1[\), the space \(X\) is bi-Lipschitz homeomorphic to an \(\alpha\)-snowflake.
\end{enumerate}
\end{enumerate}
There is no quasi-Möbius analogue to Corollary 1.3.:
Proposition 1.9. The sphericalized Heisenberg group \(\text{Sph}_e(\mathbb{H}_{\mathbb{C}}^1)\) is \(3\)-point uniformly strongly quasi-Möbius homogeneous. Equivalently, there exists \(L \geq 1\) such that, given any \(x,y \in \mathbb{H}_{\mathbb{C}}^1 \setminus \{e\}\), there exists a \((\lambda, L)\)-quasi-dilation \(f: \mathbb{H}_{\mathbb{C}}^1 \to \mathbb{H}_{\mathbb{C}}^1\) such that \(f(e) = e, f(x)=y\), and \(\lambda = \rho(e,y)/\rho(e,x)\).
Results pertaining to unbounded, proper, and disconnected metric spaces are presented:
Theorem 1.10. Suppose \(X\) is disconnected, unbounded, locally compact, and isometrically homogeneous. If \(X\) is invertible, then there exists \(s>1\) and a positive integer \(N \geq 2\) such that \(X\) is bi-Lipschitz homeomorphic to \((C_N, \rho_s)\).
Here \(C_N\) is the parabolic visual boundary of the \((N+1)\)-regular tree equipped with the path distance with edge length \(1\). \(\rho_s\) is the parabolic visual distance with parameter \(s\).
Theorem 1.11. Suppose \(X\) is a disconnected, unbounded, and locally compact metric space. There exists \(s > 1\) and a positive integer \(N \geq 2\) such that \(X\) is isometric to \((C_N, \rho_s)\) if and only if \(\hat{X}\) is \(3\)-point Möbius homogeneous.
Theorem 1.12. Suppose \(X\) is a disconnected, unbounded, locally compact, and uniformly bi-Lipschitz homogeneous metric space. If \(X\) is quasi-invertible, then \(X\) is quasi-Möbius homeomorphic to \((C_2, \rho_2)\).On the realization of symplectic algebras and rational homotopy types by closed symplectic manifoldshttps://www.zbmath.org/1472.550102021-11-25T18:46:10.358925Z"Milivojević, Aleksandar"https://www.zbmath.org/authors/?q=ai:milivojevic.aleksandarConcerning a classic conjecture of Thurston about symplectic structures on almost complex manifolds, Lupton, Oprea and Tralle proposed certain variations of it and related open questions [\textit{A. Tralle} and \textit{J. Oprea}, Symplectic manifolds with no Kähler structure. Berlin: Springer (1997; Zbl 0891.53001); \textit{G. Lupton} and \textit{J. Oprea}, J. Pure Appl. Algebra 91, No. 1--3, 193--207 (1994; Zbl 0789.55010)]. In this paper the author gives negative answers to some of them.
The author shows that there are symplectic real algebras \(H\) in dimension \(4k\) for every \(k\geq 1\), such that there is no closed symplectic manifold \(M\) with \(H^*(M; \mathbb R )\cong H\). This in particular provides a negative answer, in dimension divisible by four, to the conjecture proposed by Oprea and Tralle on the realizability of symplectic algebras by symplectic manifolds.
The author proves that there are cohomologically symplectic manifolds in dimension \(2k\) for each \(k\geq 2\), that do not admit a symplectic structure. In particular this gives a negative answer to another related question proposed by Lupton and Oprea about manifolds such that their rational cohomology algebra is a symplectic algebra in all even dimensions.
The author also proves that there are no algebraic conditions on the minimal model of a manifold that imply the existence of a symplectic structure on the given manifold for dimensions six or greater. This gives a negative answer to the question proposed by Oprea and Tralle about the possibility of such algebraic conditions on the minimal model of the manifold.Fake 13-projective spaces with cohomogeneity one actionshttps://www.zbmath.org/1472.570302021-11-25T18:46:10.358925Z"He, Chenxu"https://www.zbmath.org/authors/?q=ai:he.chenxu"Rajan, Priyanka"https://www.zbmath.org/authors/?q=ai:rajan.priyankaFake real projective spaces are manifolds that are homotopy equivalent but not diffeomorphic to standard real projective spaces. First examples thereof were discovered by \textit{M. W. Hirsch} and \textit{J. W. Milnor} [Bull. Am. Math. Soc. 70, 372--377 (1964; Zbl 0201.25601)] as quotients of standard 5- and 6-spheres embedded in Milnor's exotic 7-spheres under certain free involutions. Similarly, \textit{P. Rajan} and \textit{F. Wilhelm} [Bull. Aust. Math. Soc. 94, No. 2, 304--315 (2016; Zbl 1364.53040)] detected some standard 13- and 14-spheres embedded in Shimada's exotic 15-spheres [\textit{N. Shimada}, Nagoya Math. J. 12, 59--69 (1957; Zbl 0145.20303)] with quotients that are homotopy equivalent to \(\mathbb{R}\mathrm{P}^{13}\) and \(\mathbb{R}\mathrm{P}^{14}\), respectively, and they also showed that some of the quotients in the 14-dimensional examples are not diffeomorphic to \(\mathbb{R}\mathrm{P}^{14}\), i.\,e.\ they are fake projective spaces.
The first main result of the publication under review is that some of the 13-dimensional homotopy projective spaces \(P^{13}\) found by Rajan and Wilhelm are fake projective spaces as well.
Further main results are concerned with invariant metrics of non-negative sectional curvature on fake projective spaces. It is known that the 5-dimensional Hirsch-Milnor fake projective spaces \(P^5\) admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\) [\textit{M. W. Davis}, Am. J. Math. 104, 59--90 (1982; Zbl 0509.57029)] and that all of the \(P^5\)'s carry \(\mathsf{SO}(2)\times \mathsf{SO}(3)\)-invariant metrics of non-negative sectional curvature [\textit{K. Grove} and \textit{W. Ziller}, Ann. Math. (2) 152, No. 1, 331--367 (2000; Zbl 0991.53016), p.334]. Similarly, it is shown here that all Rajan-Wilhelm \(P^{13}\)'s admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{G}_2\), but in contrast to the 5-dimensional case the authors now prove the following alternative fact: None of the \(P^{13}\)'s supports an \(\mathsf{SO}(2)\times \mathsf{G}_2\)-invariant metric of non-negative sectional curvature.
In fact, it is shown that the 5-dimensional case is rather special in this regard: If a homotopy sphere admits a non-negatively curved metric that is invariant under a cohomogeneity one action, then the sphere is a standard sphere \(\mathbb{S}^n\) and either the action is linear, i.\,e.\ a sub-action of the standard action of \(\mathsf{SO}(n+1)\), or \(n=5\) and the action is nonlinear by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\).
Moreover, the authors show that the 13-dimensional Rajan-Wilhelm fake projective spaces are \(\mathsf{SO}(2)\times \mathsf{G}_2\)-equivariantly diffeomorphic to quotients of Brieskorn varieties by involutions. This reformulation is the key technical resource in the proofs of the main results.Monopole Floer homology and the spectral geometry of three-manifoldshttps://www.zbmath.org/1472.570322021-11-25T18:46:10.358925Z"Lin, Francesco"https://www.zbmath.org/authors/?q=ai:lin.francescoThere is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result.On cohomogeneity one linear actions on pseudo-Euclidean space \(\mathbb{R}^{p , q} \)https://www.zbmath.org/1472.570352021-11-25T18:46:10.358925Z"Ahmadi, P."https://www.zbmath.org/authors/?q=ai:ahmadi.parviz"Safari, S."https://www.zbmath.org/authors/?q=ai:safari.salimIn this article, the authors study the natural actions of some (noncompact) subgroups of \( SO(p,q) \) (matrices preserving the quadratic form of signature \( p,q \ (p\geq q)\)) on \( \mathbb{R}^{p+q} \). The case \( q=1 \) has been investigated in [\textit{J. Berndt} et al., Monatsh. Math. 184, No. 2, 185--200 (2017; Zbl 1379.53091)]. In this article, similar results are obtained for \( q>1 \). All orbits of this cohomogeniety one action are found. A decomposition of the subgroup \( Q \) leaving a certain subspace invariant is determined. Restricted actions to a certain class of subgroups of \( Q \) are studied in detail. The authors prove that the actions of these subgroups are also of cohomogeniety one and all their orbits are determined. It is shown that, when \( p>q+1\geq2 \), the orbits outside a certain subspace (\( \mathbb{W} \) of dimension \( p \)) are independent of the acting group (in this class) but the orbits of points in \( \mathbb{W} \) depend on the acting group as in [loc. cit.].Harmonic maps with torsionhttps://www.zbmath.org/1472.580092021-11-25T18:46:10.358925Z"Branding, Volker"https://www.zbmath.org/authors/?q=ai:branding.volkerThis interesting paper under review is devoted to a first study of harmonic maps that are coupled to a torsion endomorphism on the target manifold. Such maps are called ``harmonic maps with torsion'' and they appear as solutions of an equation obtained by taking the standard harmonic map equation and then changing to a connection with metric torsion. The author investigates various geometric aspects of harmonic maps with torsion. First, some Bochner type formulas are established and the effects of conformal transformations on harmonic maps with torsion are studied. Next, the stability of harmonic maps with torsion is discussed. The last part of the article investigates analytic aspects of harmonic maps with torsion. It is showed that they satisfy the unique continuation property, a removable singularity theorem is proved and a Liouville type result under a small energy assumption is stated.Weakly biharmonic maps from the ball to the spherehttps://www.zbmath.org/1472.580102021-11-25T18:46:10.358925Z"Fardoun, Ali"https://www.zbmath.org/authors/?q=ai:fardoun.ali"Montaldo, S."https://www.zbmath.org/authors/?q=ai:montaldo.stefano"Ratto, A."https://www.zbmath.org/authors/?q=ai:ratto.andreaBiharmonic maps are critical point of the \textit{bienergy functional}
\[
E_2(u)=\frac{1}{2}\int_M|\tau(u)|^2 d\,v_g,
\]
where \(\tau(u)=\textrm{trace}\,\nabla d u\). In particular, harmonic maps are trivially biharmonic maps, as harmonic maps are critical points of the \textit{energy functional}
\[
E(u)=\frac{1}{2}\int_M| d u |^2 d\,v_g.
\]
Biharmonic maps that are not harmonic are called proper biharmonic maps.
Let \(B^n\) and \(S^n\) denote the \(n\)-dimensional Euclidean unit ball and sphere respectively. The authors study the family of rotationally symmetric maps \(u_a:B^n\rightarrow S^n\subset \mathbb{R}^n\times \mathbb{R}\) given by
\[
u_a(x)=\left(\sin a \,\,\frac{x}{|x|},\cos a\right),
\]
where \(a\) is a constant in \((0,\pi/2)\). In particular, the authors show (Theorem 1.1) that such maps are proper weakly biharmonic if and only if if either \( n = 5\) and \(a = \pi/3\) or \(n = 6\) and \(a = 1/2 \arccos(-4/5)\). In any of these two cases, \(u_a\) is unstable (Theorem 1.2).
The paper has three sections and it is quite self-contained, with a recollection of the necessary results on Sobolev spaces and weak solutions presented in section 2. Proofs of the main results are given in section 3.Triharmonic Riemannian submersions from 3-dimensional space formshttps://www.zbmath.org/1472.580122021-11-25T18:46:10.358925Z"Miura, Tomoya"https://www.zbmath.org/authors/?q=ai:miura.tomoya"Maeta, Shun"https://www.zbmath.org/authors/?q=ai:maeta.shun\(k\)-polyharmonic maps, as a generalization of harmonic maps, are maps between Riermannian manifolds which are critical points of the \(k\) energy \(\frac{1}{2}\int_M |(d+\delta)^k\phi|^2dv_g\). \(k\)-polyharmonic maps for \(k=2, 3\) are called biharmonic and triharmonic maps respectively. Harmonic maps are always biharmonic and triharmonic but a biharmonic map need not be triharmonic. For some recent study and progress on biharmonic maps and biharmonic submanifolds see a recent book [\textit{Y.-L. Ou} and \textit{B.-Y. Chen}, Biharmonic submanifolds and biharmonic maps in Riemannian geometry. Hackensack, NJ: World Scientific (2020; Zbl 1455.53002)] and the references therein.
For biharmonic Riemannian submersions, it was proved in [\textit{Z.-P. Wang} and \textit{Y.-L. Ou}, Math. Z. 269, No. 3--4, 917--925 (2011; Zbl 1235.53065)] that any biharmonic Riemannian submersion from a \(3\)-dimensional space form onto a surface is harmonic. The paper under review proves that this result can be generalized to the cases of triharmonic Riemannian submersions and \(f\)-biharmonic Riemannian submersions.Biharmonic maps on principal \(G\)-bundles over complete Riemannian manifolds of nonpositive Ricci curvaturehttps://www.zbmath.org/1472.580132021-11-25T18:46:10.358925Z"Urakawa, Hajime"https://www.zbmath.org/authors/?q=ai:urakawa.hajimeThis article considers principal \(G\)-bundles, equipped with a Sasaki-type metric, over a Riemannian manifold and the canonical projection \(\pi\), which is then a Riemannian submersion.
The problem investigated is to find conditions such that \(\pi\) biharmonic implies \(\pi\) harmonic.
The first theorem proved is that if the principal \(G\)-bundle is compact and has non-positive Ricci curvature then \(\pi\) biharmonic implies \(\pi\) harmonic. The reader will notice in the proof that the one-form \(\alpha\) defined by Equation (3.7) is not quite well-defined on the base manifold unless the tension field of \(\pi\) is actually basic. This should then be compared with [\textit{C. Oniciuc}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 48, No. 2, 237--248 (2002; Zbl 1061.58015)].
The second set of conditions is to assume the principal \(G\)-bundle is non-compact complete with non-positive Ricci curvature and the energy and bienergy of \(\pi\) are finite. Then \(\pi\) biharmonic implies \(\pi\) harmonic. The proof of this second theorem is really only a rehash of \textit{N. Nakauchi} et al. [Geom. Dedicata 169, 263--272 (2014; Zbl 1316.58012)].Foliated manifolds, algebraic \(K\)-theory, and a secondary invarianthttps://www.zbmath.org/1472.580162021-11-25T18:46:10.358925Z"Bunke, Ulrich"https://www.zbmath.org/authors/?q=ai:bunke.ulrichIn this paper, Bunke introduces a numerical invariant (taking values in \({\mathbb{C}}/{\mathbb{Z}}\)) associated to foliations over odd-dimensional closed spin manifolds by means of a certain transgression of the \(\hat{A}\) form. The paper is carefully written and contains a wealth of examples making it a very pleasurable read.
The foliation \(\mathcal{F}\), defined as an involutive \(\mathbb{C}\)-vector subbundle in the complexification of the tangent bundle, is assumed to be stably trivial and equipped with a stable framing \(s\). Additionally, one fixes a complex vector bundle \(V\) over the base manifold \(M\), together with a partial connection in the direction of the foliation (a partial connection means the restriction of a connection to those covariant derivatives in the direction of the foliation). The partial connection \(\nabla^I\) on \(V\) is assumed to be flat. The normal bundle to the foliation \(T_{\mathbb{C}}M/{\mathcal{F}}\) is equipped with a natural such flat partial connection, given by the Lie bracket.
Under these hypothesis, the invariant \(\rho(M,\mathcal{F},\nabla^I,s)\in {\mathbb{C}}/{\mathbb{Z}}\) is constructed by fixing some additional geometric data: an extension of the flat partial connection to an actual connection on \(V\), a Riemannian metric on \(M\), and also an extension of the natural flat partial connection on the transverse bundle to the foliation to an actual connection over \(M\). However, if the codimension of the foliation is small enough in the sense that \(2\mathrm{codim}(\mathcal{F})<\dim(M)\), then the invariant is shown to be independent of the above additional data.
The definition of invariant involves the so-called Umkehr map in differential \(K\)-theory. In particular cases, it reduces to known invariants like the (reduced) eta invariant of twisted Dirac operators on spin manifolds, the Godbillon-Vey invariant of a foliation of codimension \(1\), or Adams' \(e\)-invariant. In the last sections of the paper, the invariant is linked to a regulator map in algebraic \(K\)-theory.Small-time asymptotics for subelliptic Hermite functions on \(SU(2)\) and the CR spherehttps://www.zbmath.org/1472.580182021-11-25T18:46:10.358925Z"Campbell, Joshua"https://www.zbmath.org/authors/?q=ai:campbell.joshua"Melcher, Tai"https://www.zbmath.org/authors/?q=ai:melcher.tai\textit{J. J. Mitchell} [J. Funct. Anal. 164, No. 2, 209--248 (1999; Zbl 0928.22010)] studied the small-time behavior of Hermite functions on compact Lie groups. In particular, he demonstrated that, when written in exponential coordinates with a natural rescaling, these functions converge to the classical Euclidean Hermite polynomials. In a subsequent work [\textit{J. J. Mitchell}, J. Math. Anal. Appl. 263, No. 1, 165--181 (2001; Zbl 0997.43007)], he proved that Hermite functions on compact Riemannian manifolds, again written in exponential coordinates with appropriate rescaling, admit asymptotic expansions with a classical Hermite polynomial as the leading coefficient.
In the paper under review, the authors investigate heat kernels related to the natural subRiemannian structure on \(\mathrm{SU}(2)\simeq \S^3\) and, more generally, on higher-order odd-dimensional spheres.
More specifically, they prove that under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on \(\mathrm{SU}(2)\) converges to their analogues on the Heisenberg group at time 1. Next, they generalize these results to the CR sphere \(\S^{2d+1} \) equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups. In other words, they obtain similar results as in [Mitchell, loc. cit.] for the Hermite functions on the CR spheres.Points on nodal lines with given directionhttps://www.zbmath.org/1472.580202021-11-25T18:46:10.358925Z"Rudnick, Zeév"https://www.zbmath.org/authors/?q=ai:rudnick.zeev"Wigman, Igor"https://www.zbmath.org/authors/?q=ai:wigman.igorThis paper treats the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. Furthermore, the authors give upper bounds for the flat torus, and a computation of the expected number for arithmetic random waves is executed.
More precisely, let $\Omega$ be a planar domain with piecewise smooth boundary, and let $f$ be an eigenfunction of the Dirichlet Laplacian with eigenvalue $E$ such that $- \varDelta f = E f$. Given a direction $\zeta \in S^1$, let $N_{\zeta}(f)$ be the number of points $x$ on the nodal line $\{ x \in \Omega: \, f(x) = 0 \}$ with normal pointing in the direction $\pm \zeta$, i.e.,
\[
N_{\zeta}(f) := \# \left\{ x \in \Omega: \, f(x) =0, \,\, \frac{ \nabla f(x)}{ \Vert \nabla f(x) \Vert } = \pm \zeta \right\}.
\]
The first result below asserts an upper bound for $N_{\zeta}(f)$ with the only exceptions being when the nodal line contains a closed geodesic. It will follow as a particular case of a structure result on the set
\[
A_{\zeta}(f) := \{ x \in \Omega: \, f(x) =0, \, \langle \nabla f(x), \zeta^{\perp} \rangle = 0 \}
\]
of nodal directional points, i.e., the set of nodal points where $\nabla f$ is orthogonal to $\zeta^{\perp}$, thus it is co-linear to $\zeta$, $A_{\zeta}(f)$ contains all the singular nodal points of $f^{-1}(0)$, and could also contain certain closed geodesics in direction orthogonal to $\zeta$.
Theorem 1. Let $\zeta \in S^1$ be a direction, and $f$ be a toral eigenfunction such that $- \varDelta f = E f$ for some $E > 0$.
(i) If $\zeta$ is rational, then the set $A_{\zeta}(f)$ consists of at most $\sqrt{E}/ \pi h(\zeta)$ closed geodesics orthogonal to $\zeta$, at most $\frac{2}{\pi^2} E$ nonsingular points not lying on the geodesics, and possibly, singular points of the nodal set, where $h(\zeta)$ is the height for a rational vector.
(ii) If $\zeta$ is not rational, then the set $A_{\zeta}(f)$ consists of at most $\frac{2}{\pi^2} E$ nonsingular points, and possibly, singular points of the nodal set.
(iii) In particular, if $A_{\zeta}(f)$ does not contain a closed geodesics, then
\[
N_{\zeta}(f) \leqslant \frac{2}{\pi^2} \cdot E
\]
holds.
Next the authors compute the expected value of $N_{\zeta}$ for arithmetic random waves, cf. [\textit{F. Oravecz} et al., Ann. Inst. Fourier (Grenoble) 58, No.1, 299--335 (2008; Zbl 1153.35058)]. There are random eigenfunctions on the torus,
\[
f(x) = f_n(x) = \sum_{ \lambda \in {\mathcal E}_n } c_{\lambda} \cdot e( \langle \lambda, x \rangle )
\]
where $e(z) = e^{ 2 \pi i z}$ and $ {\mathcal E} := \{ \lambda = ( \lambda_1, \lambda_2) \in{\mathbb Z}: \, \Vert \lambda \Vert^2 = n \}$ is the set of all representations of the integer $n = \lambda_1^2 + \lambda_2^2$ as a sum of two integer squares, and $c_{\lambda}$ are standard Gaussian random variables, identically distributed and independent wave for the constraint $c_{- \lambda} = \bar{ c}_{\lambda}$, making $f_n$ real valued eigenfunctions of the Laplacian with eigenvalue $E = 4 \pi^2 n$ for every choice of the coefficients $\{ a_{\lambda} \}, \lambda \in {\mathcal E}_{\lambda}$. Let $\mu_n$ be the atomic measure on the unit circle given by
\[
\mu_n = \frac{1}{ r_2(n)} \sum_{ \lambda \in {\mathcal E}_n }\delta_{ \lambda / \sqrt{n} },
\]
where $r_2(n) := \# {\mathcal E}_n$, and let
\[
\hat{\mu}_n (k) = \frac{1}{ r_2(n)} \sum_{ \lambda = ( \lambda_1, \lambda_2) \in {\mathcal E}_n } \left( \frac{ \lambda_1 + i \lambda_2}{\sqrt{n} } \right)^k \in {\mathbb R}
\]
be its Fourier coefficients.
Theorem 2. For $\zeta = e^{ i \theta} \in S^1$, the expected value of $N_{\zeta}(f)$ for the arithmetic random wave (1.4) is
\[
{\mathbb E} [ N_{\zeta} ] = \frac{1}{ \sqrt{2} } n ( 1 + \hat{\mu}_n(4) \cdot \cos( 4 \theta) )^{1/2}.
\]
For other related works, see e.g. [\textit{M. Krishnapur} et al., Ann. Math. (2) 177, No. 2, 699--737 (2013; Zbl 1314.60101)] as to nodal length fluctuations for arithmetic random waves, and [\textit{A. Logunov}, Ann. Math. (2) 187, No. 1, 221--239 (2018; Zbl 1384.58020)] for nodal sets of Laplace eigenfunctions.Erratum to: ``Constraint algorithm for singular field theories in the \(k\)-cosymplectic framework''https://www.zbmath.org/1472.700422021-11-25T18:46:10.358925Z"Gràcia, Xavier"https://www.zbmath.org/authors/?q=ai:gracia.xavier"Rivas, Xavier"https://www.zbmath.org/authors/?q=ai:rivas.xavier"Román-Roy, Narciso"https://www.zbmath.org/authors/?q=ai:roman-roy.narcisoFrom the text: A simple geometric description of singular autonomous field theories is provided
by \(k\)-presymplectic geometry. Consistency of field equations can be analyzed by
means of a constraint algorithm. In our recent paper [ibid. 12, No. 1, 1--23 (2020; Zbl 1434.70047)] we extended this analysis
to the non-autonomous case. In this case the geometric setting is provided by the
notion of \(k\)-precosymplectic structure. However, to ensure the existence of Reeb vector fields and Darboux coordinates, we restricted our attention to \(k\)-precosymplectic
manifolds of the form \(\mathbb{R}^k \times P\), with \(P\) a \(k\)-presymplectic manifold.
As a typical example, we analized the case of affine Lagrangians of the type
\[
L(x^\alpha,q^i,v^i_\alpha) = f^\mu_j(x^\alpha,q^i)v^j_\mu + g(x^\alpha,q^i)
\]
on the manifold \(\mathbb{R}^k \times T^1_k Q\), and a particular academic example (sections 6.1 and
6.2). Nevertheless, such Lagrangians do not result in \(k\)-precosymplectic structures
of the above mentioned type, and their analysis as presented in the paper is not
correct (for instance, Reeb vector fields may not be well defined).
In this note we correct this mistake by restricting our study to the family of
affine Lagrangians of the type \(L(x^\alpha,q^i,v^i_\alpha) = f(q^i)v^j_\mu + g(x^\alpha,q^i)\), which lead to
\(k\)-precosymplectic structures as previously indicated. We also analyze a particular
example in this class that replaces the one in section 6.2.Curvature of space and time, with an introduction to geometric analysishttps://www.zbmath.org/1472.830012021-11-25T18:46:10.358925Z"Stavrov, Iva"https://www.zbmath.org/authors/?q=ai:stavrov.ivaThe book grew out of a summer program at the Park City Math institute (PCMI) in 2013 and a class at Lewis and Clark college in 2015. It was the aim of the courses to introduce students to general relativity and geometric analysis at an undergraduate level.
The book follows an unusual track, concepts from Riemannian geometry are introduced without refering to results from differential topology. Hence the standard definitions of manifolds and tangent vectors are not presented. This is not by accident but follows the belief of the author stated in the preface that \textit{the practice of organizing information in a deductive manner, which as mathematicians we are committed to, is not always conducive to learning mathematics and developing intuition.}
The book is divided into five chapters, the topic of the first chapter is \textit{Introduction to Riemannian geometry.} It starts with a quotation from Riemann's habilitation lecture presented in 1854 in Göttingen. The concept of a Riemannian metric is introduced by discussing the examples of the euclidean metric, the round metric on a sphere and the standard metric of the real projective plane. The concept of a manifold is introduced by using different coordinate systems with certain transition maps. Geodesics are introduced using their variational characterization. The geodesic equation then leads to the introduction of Christoffel symbols and geodesically completness is discussed shortly.
The title of chapter 2 is \textit{Differential calculus with tensors.} Here directional derivatives of functions, vector fields and the Levi-Civita connection are discussed and motivated. Then tensor fields and their coordinate expression are presented. The gradient of a function, the divergence of a vector field and the Laplacian of a function are defined, Green's identities as well as the maximum principle are formulated. The next section deals with the differentiation of tensors and motivates the definition of the curvature tensor. Then the symmetries of the curvature tensor and the Bianchi identities are given.
The topic of the third chapter is \textit{Curvature.} Here the connection with Jacobi fields as variational vector fields of variations by geodesics is emphasized. Ricci curvature and scalar curvature are introduced and their geometric meaning is explained.
The fourth chapter is named \textit{General relativity.} Concepts from special relativity are presented in the first section, whereas the next section deals with \textit{Gravity and general relativity}. It motivates the Einstein equation and also presents its initial value formulation. In the next two sections the geometry of the Schwarzschild spacetime and its Kruskal Szekeres extension is discussed. In this chapter the relativistic Poisson equation occurs.
In the last chapter titled \textit{Introduction to geometric analysis} ideas of the proof of the existence of a solution of the relativistic Poisson equation are given. This is meant to be an introduction to geometric analysis. In the final section the concept of the ADM-mass of an asymptotically Euclidean metric is motivated.
Without doubt it is a remarkable book. It deals with an enormous amount of mathematical contents, it provides excellent motivations and insights starting from typical examples. Often proofs are not presented, but there are a lot of excercises following each section with many useful explainations. The book gives an impressing overview on Riemannian geometry, general relativity and geometric analysis. It supplements standard books or courses on the topics of the book. And it may invite readers to study parts of the material in detail. It is likely that readers agree with the author's statement from the preface: \textit{An approach to teaching and learning mathematics which relies on several passes through a subject, each at a higher level of mathematical rigor, is in a sense historically proven to be pedagogically optimal.}Optical functions in De Sitterhttps://www.zbmath.org/1472.830242021-11-25T18:46:10.358925Z"Schlue, Volker"https://www.zbmath.org/authors/?q=ai:schlue.volkerSummary: This paper addresses pure gauge questions in the study of asymptotically de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results that are relevant for the foliations used by the author in the context of the stability problem of the expanding region of Schwarzschild de Sitter spacetimes: (i) Small perturbations of round spheres on the cosmological horizons lead to spheres that pinch off at infinity. (ii) Globally well-behaved double null foliations can be constructed from infinity using a choice of spheres related to the level sets of a new \textit{mass aspect function}. While (i) shows that in the above stability problem a \textit{final gauge choice} is necessary, the proof of (ii) already outlines a strategy for the case of spacetimes with decaying, instead of vanishing, conformal Weyl curvature.
{\copyright 2021 American Institute of Physics}Edge modes and surface-preserving symmetries in Einstein-Maxwell theoryhttps://www.zbmath.org/1472.830262021-11-25T18:46:10.358925Z"Setare, Mohammad Reza"https://www.zbmath.org/authors/?q=ai:setare.mohammad-reza"Adami, Hamed"https://www.zbmath.org/authors/?q=ai:adami.hamedSummary: Einstein-Maxwell theory is not only covariant under diffeomorphisms but also is under \(U(1)\) gauge transformations. We introduce a combined transformation constructed out of diffeomorphism and \(U(1)\) gauge transformation. We show that symplectic potential, which is defined in covariant phase space method, is not invariant under combined transformations. In order to deal with that problem, following \textit{W. Donnelly} and \textit{L. Freidel} proposal [J. High Energy Phys. 2016, No. 9, Paper No. 102, 45 p. (2016; Zbl 1390.83016)], we introduce new fields. In this way, phase space and consequently symplectic potential will be extended. We show that new fields produce edge modes. We consider surface-preserving symmetries and we show that the group of surface-preserving symmetries is semi-direct sum of 2-dimensional diffeomorphism group on a spacelike codimension two surface with \(\mathrm{SL}(2, \mathbb{R})\) and \(U(1)\). Eventually, we deduce that the Casimir of \(\mathrm{SL}(2, \mathbb{R})\) is the area element, similar to the pure gravity case [loc. cit.].Natural vs. artificial topologies on a relativistic spacetimehttps://www.zbmath.org/1472.830782021-11-25T18:46:10.358925Z"Papadopoulos, Kyriakos"https://www.zbmath.org/authors/?q=ai:papadopoulos.kyriakosSummary: Consider a set \(M\) equipped with a structure \(\ast \). We call a natural topology \(T_\ast \), on (M, \( \ast )\), the topology induced by \(\ast \). For example, a natural topology for a metric space \((X, d)\) is a topology \(T_d\) induced by the metric \(d\), and for a linearly ordered set \((X, <)\), a natural topology should be the topology \(T_<\) that is induced by the order \(<\). This fundamental property, for a topology to be called ``natural,'' has been largely ignored while studying topological properties of spacetime manifolds \((M, g)\), where \(g\) is the Lorentz ``metric,'' and the manifold topology \(T_M\) has been used as a natural topology, ignoring the spacetime ``metric'' \(g\). In this survey, we review critically candidate topologies for a relativistic spacetime manifold, and we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term ``natural topology'' for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Gao-Wald type of ``time dilation'' are such examples.
For the entire collection see [Zbl 1470.49002].Complete model of cosmological evolution of a classical scalar field with the Higgs potential. I: Analysis of the modelhttps://www.zbmath.org/1472.831162021-11-25T18:46:10.358925Z"Ignat'ev, Yu. G."https://www.zbmath.org/authors/?q=ai:ignatev.yu-g"Samigullina, A. R."https://www.zbmath.org/authors/?q=ai:samigullina.a-rSummary: A complete model of cosmological evolution of a classical scalar field with the Higgs potential is studied and simulated on a computer without assumption that the Hubble constant is nonnegative. The corresponding dynamical system is qualitatively analyzed and the Einstein-Higgs hypersurfaces the topology of which determines the global properties of the phase trajectories of the cosmological model are classified.