Differential Geometry

New submissions

• New submissions
• Cross-lists
• Replacements

New submissions for Wed, 8 May 24

[1]  arXiv:2405.03756 [pdf, ps, other]

Title: Differential spinors and Kundt three-manifolds with skew-torsion

Authors: C. S. Shahbazi

Comments: 29 pages

Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)

We develop the theory of spinorial polyforms associated with bundles of irreducible Clifford modules of non-simple real type, obtaining a precise characterization of the square of an irreducible real spinor in signature $(p-q) = 1\,\mathrm{mod(8)}$ as a polyform belonging to a semi-algebraic real set. We use this formalism to study differential spinors on Lorentzian three-manifolds, proving that in this dimension and signature, every differential spinor is equivalent to an isotropic line preserved in a given direction by a metric connection with prescribed torsion. We apply this result to investigate Lorentzian three-manifolds equipped with a skew-torsion parallel spinor, namely a spinor parallel with respect to a metric connection with totally skew-symmetric torsion. We obtain several structural results about this class of Lorentzian three-manifolds, which are necessarily Kundt, and in the compact case, we obtain an explicit differential condition that guarantees geodesic completeness. We further elaborate on these results to study the supersymmetric solutions of three-dimensional NS-NS supergravity, which involve skew-torsion parallel spinors whose torsion is given by the curvature of a curving on an abelian bundle gerbe. In particular, we obtain a correspondence between NS-NS supersymmetric solutions and null coframes satisfying an explicit exterior differential system that we solve locally.

[2]  arXiv:2405.03839 [pdf, ps, other]

Title: Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below

Authors: Elia Bruè, Alessandro Pigati, Daniele Semola

Subjects: Differential Geometry (math.DG)

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M_i^n,g_i,p_i)\to (X^n,d)$. We confirm a conjecture proposed by Colding and Naber in dimension $n=4$, showing that the cross-sections of tangent cones at a given point $x\in X^4$ are all homeomorphic to a fixed spherical space form $S^3/\Gamma_x$, and $\Gamma_x$ is trivial away from a $0$-dimensional set. In dimensions $n>4$, we show an analogous statement at points where all tangent cones are $(n-4)$-symmetric. Furthermore, we prove that $(n-3)$-symmetric noncollapsed Ricci limits are topological manifolds, thus confirming a particular case of a conjecture due to Cheeger, Colding, and Tian. Our analysis relies on two key results, whose importance goes beyond their applications in the study of cross-sections of noncollapsed Ricci limit spaces: (i) A new manifold recognition theorem for noncollapsed ${\rm RCD}(-2,3)$ spaces. (ii) A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form $\mathbb{R}^{n-3}\times C(\mathbb{RP}^2)$.

[3]  arXiv:2405.03895 [pdf, ps, other]

Title: Quasi-positive curvature and vanishing theorems

Authors: Kai Tang

Comments: 10 pages

Subjects: Differential Geometry (math.DG)

In this paper, we consider mixed curvature $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold M admits a K\"{a}hler metric with quasi-positive mixed curvature and $3a+2b\geq0$, then it is projective. If $a,b\geq0$, then M is rationally connected. As a corollary, the same result holds for k-Ricci curvature. We also show that any compact K\"{a}hler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature have vanishing Hodge number $h^{2,0}$. Furthermore, if it is K\"{a}hlerian, it is projective.

[4]  arXiv:2405.03921 [pdf, ps, other]

Title: Classification of two and three dimensional complete gradient Yamabe solitons

Authors: Shun Maeta

Comments: 11 pages

Subjects: Differential Geometry (math.DG)

In this paper, we completely classify nontrivial nonflat three dimensional complete shrinking and steady gradient Yamabe solitons without any assumptions. We also give examples of complete expanding gradient Yamabe solitons. Furthermore, we give a proof of the classification of nontrivial two dimensional complete gradient Yamabe solitons without any assumptions.

[5]  arXiv:2405.03993 [pdf, other]

Title: Capillary Surfaces in Manifolds with Nonnegative Scalar Curvature and Strictly Mean Convex Boundary

Authors: Yujie Wu

Comments: 19 pages, 1 figure, comments welcome!

Subjects: Differential Geometry (math.DG)

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble construction) to study manifolds with strictly mean convex boundary and nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by such 3-manifolds based on the Urysohn width. We also obtain a bandwidth estimate and establish other geometric properties of such manifolds.

[6]  arXiv:2405.04001 [pdf, ps, other]

Title: Volume growth and positive scalar curvature

Authors: Guodong Wei, Guoyi Xu, Shuai Zhang

Comments: submitted to some journal

Subjects: Differential Geometry (math.DG)

For three dimensional complete, non-compact Riemannian manifolds with non-negative Ricci curvature and uniformly positive scalar curvature, we obtain the sharp linear volume growth ratio and the corresponding rigidity.

[7]  arXiv:2405.04162 [pdf, ps, other]

Title: Vaisman's theorem and local reducibility

Authors: Ovidiu Preda, Miron Stanciu

Comments: 8 pages

Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

As proven in a celebrated theorem due to Vaisman, pure locally conformally K\"ahler metrics do not exist on compact K\"ahler manifolds. In a previous paper, we extended this result to the singular setting, more precisely to K\"ahler spaces which are locally irreducible. Without the additional assumption of local irreducibility, there are counterexamples for which Vaisman's theorem does not hold. In this article, we give a much broader sufficient condition under which Vaisman's theorem still holds for compact K\"ahler spaces which are locally reducible.

[8]  arXiv:2405.04208 [pdf, ps, other]

Title: Collapsing immortal Kähler-Ricci flows

Authors: Hans-Joachim Hein, Man-Chun Lee, Valentino Tosatti

Comments: 89 pages

Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)

We consider the K\"ahler-Ricci flow on compact K\"ahler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.

[9]  arXiv:2405.04227 [pdf, ps, other]

Title: The Penrose Inequality for Metrics with Singular Sets

Authors: Huaiyu Zhang

Subjects: Differential Geometry (math.DG)

We study the Penrose inequality and its rigidity for metrics with singular sets. Our result could be viewed as a complement of Theorem 1.1 of Lu and Miao (J. Funct. Anal. 281, 2021) and Theorem 1.2 of Shi, Wang and Yu (Math. Z. 291, 2019), in which they assume the singular set is a hypersurface and assume an additional condition on the mean curvature. As a complement, this paper study the case of singular set of dimensional less than n-1, without any additional conditions.

[10]  arXiv:2405.04255 [pdf, other]

Title: Ruled Ricci surfaces and curves of constant torsion

Authors: Alcides de Carvalho, Iury Domingos, Roney Santos

Comments: 12 pages and 11 figures. All comments are welcome

Subjects: Differential Geometry (math.DG)

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the helicoid as the only surface of this kind that admits a parametrization with plane line of striction, and as the only with constant mean curvature.

[11]  arXiv:2405.04301 [pdf, ps, other]

Title: Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane

Authors: Haizhong Li, Yao Wan

Comments: 19 pages, 2 figures. All comments are welcome

Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Metric Geometry (math.MG)

In \cite{LX}, the first author and Xu introduced and studied the horospherical $p$-Minkowski problem in hyperbolic space $\mathbb{H}^{n+1}$. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and $p\ge -n$. This paper focuses on the isotropic horospherical $p$-Minkowski problem in hyperbolic plane $\mathbb{H}^{2}$, which corresponds to the equation \begin{equation}\label{0}
\varphi^{-p}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^2}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma\quad\text{on}\ \mathbb{S}^1, \end{equation} where $\gamma$ is a positive constant. We provide a classification of solutions to the above equation for $p\ge -7$, as well as a nonuniqueness result of solutions for $p<-7$. Furthermore, we extend this problem to the isotropic horospherical $q$-weighted $p$-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.

[12]  arXiv:2405.04444 [pdf, ps, other]

Title: Immortal solutions of the Kähler-Ricci flow

Authors: Valentino Tosatti

Comments: 18 pages; submitted to the AMS Contemporary Mathematics volume in memory of Steve Zelditch

Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

We survey some recent developments on solutions of the K\"ahler-Ricci flow on compact K\"ahler manifolds which exist for all positive times.

[13]  arXiv:2405.04492 [pdf, other]

Title: Geometric Structures for the $G_2'$-Hitchin Component

Authors: Parker Evans

Comments: Comments welcome!

Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We give an explicit geometric structures interpretation of the $G_2'$-Hitchin component $Hit(S, G_2') \subset \chi(\pi_1S,G_2')$ of a closed oriented surface $S$ of genus $g \geq 2$. In particular, we prove $Hit(S, G_2')$ is naturally homeomorphic to a moduli space $\mathscr{M}$ of $(G,X)$-structures for $G = G_2'$ and $X = Ein^{2,3}$ on a fiber bundle $\mathscr{C}$ over $S$ via the descended holonomy map. Explicitly, $\mathscr{C}$ is the direct sum of fiber bundles $\mathscr{C} = UTS \oplus UTS \oplus \underline{\mathbb{R}_+}$ with fiber $\mathscr{C}_p = UT_p S \times UT_p S \times \mathbb{R}_+$, where $UT S$ denotes the unit tangent bundle.
The geometric structure associated to a $G_2'$-Hitchin representation $\rho$ is explicitly constructed from the unique associated $\rho$-equivariant alternating almost-complex curve $\hat{\nu}: \tilde{S} \rightarrow \hat{\mathbb{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2'$-Fuchsian case and shown to be unrelated to the $(G_2', Ein^{2,3})$-structures of Guichard-Wienhard.

Cross-lists for Wed, 8 May 24

[14]  arXiv:2405.03982 (cross-list from math.AP) [pdf, ps, other]

Title: Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds

Authors: Kazuhiro Ishige, Asuka Takatsu, Haruto Tokunaga

Comments: 18 pages. Comments welcome!

Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)

We prove that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain.

[15]  arXiv:2405.04002 (cross-list from math.AG) [pdf, ps, other]

Title: Quadratic varieties of small codimension

Authors: Kiwamu Watanabe

Comments: 14 pages

Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG)

Let $X \subset \mathbb P^{n+c}$ be a nondegenerate smooth projective variety of dimension $n$ defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that X is a complete intersection provided that $n\geq 2c+1$. As the extremal case, they also classified $X$ with $n=2c$. In this paper, we classify $X$ with $n=2c-1$.

[16]  arXiv:2405.04348 (cross-list from math.AP) [pdf, other]

Title: Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

Authors: Guowei Dai, Pieralberto Sicbaldi, Yong Zhang

Comments: 26 pages

Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha}\left(\Omega\right) \cap H^1\left(\Omega\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.

[17]  arXiv:2405.04384 (cross-list from math.CV) [pdf, ps, other]

Title: Geodesic connectivity and rooftop envelopes in the Cegrell classes

Authors: Per Åhag, Rafał Czyż, Chinh H. Lu, Alexander Rashkovskii

Subjects: Complex Variables (math.CV); Analysis of PDEs (math.AP); Differential Geometry (math.DG)

This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Amp\`ere equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford-Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge-Amp\`ere equation.

Replacements for Wed, 8 May 24

[18]  arXiv:2005.02055 (replaced) [pdf, other]

Title: On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Authors: Bingxiao Liu

Comments: 61 pages; this version will appear at Analysis & PDE

Subjects: Differential Geometry (math.DG)

[19]  arXiv:2210.05292 (replaced) [pdf, ps, other]

Title: Thurston's asymmetric metrics for Anosov representations

Authors: León Carvajales, Xian Dai, Beatrice Pozzetti, Anna Wienhard

Comments: Final version. Accepted by Groups, Geometry, and Dynamics

Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

[20]  arXiv:2303.08807 (replaced) [pdf, ps, other]

Title: A characterization of chains in dimension three

Authors: Wojciech Kryński, Omid Makhmali

Comments: 23 pages; final version; minor corrections are made; Section 4 in the previous version is removed and will be extended into a separate article titled "Lewy curves in para-CR geometry"; To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)

Subjects: Differential Geometry (math.DG)

[21]  arXiv:2307.07232 (replaced) [pdf, other]

Title: Envelopes of straight line families in the plane

Authors: Takashi Nishimura

Comments: Slightly modified version. To appear in Hokkaido Mathematical Journal

Subjects: Differential Geometry (math.DG)

[22]  arXiv:2401.12712 (replaced) [pdf, ps, other]

Title: Moutard hyperquadrics and generalized Darboux directions

Authors: Fernanda Py Silva Cordeiro, Marcos Craizer

Comments: 10 pages

Subjects: Differential Geometry (math.DG)

[23]  arXiv:2404.03221 (replaced) [pdf, other]

Title: Generalized double bracket vector fields

Authors: Petre Birtea, Zohreh Ravanpak, Cornelia Vizman

Comments: 24 pages, 25 figures

Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph)

[24]  arXiv:2404.10417 (replaced) [pdf, ps, other]

Title: Gradient estimates for positive eigenfunctions of $ \mathcal{L} $-operator on conformal solitons and its applications

Authors: Guangwen Zhao

Comments: This version revises the statements of Corollary 1.2 and Corollary 1.6 in the version dated April 16, 2024, making them clearer, and also corrects some typographical errors

Subjects: Differential Geometry (math.DG)

[25]  arXiv:2404.12937 (replaced) [pdf, ps, other]

Title: Coupled $\operatorname{G}_2$-instantons

Authors: Agnaldo A. da Silva Jr., Mario Garcia-Fernandez, Jason D. Lotay, Henrique N. Sá Earp

Comments: 45 pages, new Theorem 2.32 with complete solution to Problem 1, in arbitrary dimensions. Improved version of Theorem 4.9. New Remark 4.10 about the Spin(7) case. Submitted for consideration in the International Journal of Mathematics special issue "At the interface of complex geometry and string theory"

Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th)

[26]  arXiv:2004.01871 (replaced) [pdf, ps, other]

Title: Good Basic Invariants and Frobenius Structures

Authors: Ikuo Satake

Comments: 24 pages, added section 4 examples

Subjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG); Representation Theory (math.RT)

[27]  arXiv:2004.03587 (replaced) [pdf, ps, other]

Title: Good Basic Invariants for Elliptic Weyl Groups and Frobenius Structures

Authors: Ikuo Satake

Comments: 32 pages. arXiv admin note: text overlap with arXiv:2004.01871. Rewrote section 6;added appendices

Subjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG); Representation Theory (math.RT)

[28]  arXiv:2008.13728 (replaced) [pdf, other]

Title: Dynamical instability of minimal surfaces at flat singular points

Authors: Salvatore Stuvard, Yoshihiro Tonegawa

Comments: 31 pages, 3 figures. In v2 we discuss more extensively the range of applicability of our result in the context of the state of the art on the analysis of branched minimal surfaces. This is the final version, to appear on the Journal of Differential Geometry

Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)

[29]  arXiv:2308.06643 (replaced) [pdf, other]

Title: Geometry of fundamental shadow link complements and applications to the 1-loop conjecture

Authors: Tushar Pandey, Ka Ho Wong

Comments: 61 pages, 25 figures

Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

[30]  arXiv:2403.10090 (replaced) [pdf, other]

Title: Convex co-compact hyperbolic manifolds are determined by their pleating lamination

Authors: Bruno Dular, Jean-Marc Schlenker

Comments: 13 pages, 1 figure. v2: argument expanded to include manifolds with compressible boundary. Added some references and explanations

Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

• New submissions
• Cross-lists
• Replacements