We study K-stability on smooth complex Fano 4-folds having large Lefschetz defect, that is greater or equal then 3, with a special focus on the case of Lefschetz defect 3. In particular, we determine whether these Fano 4-folds are K-polystable or not, and show that there are 5 families (out of 19) of K-polystable smooth Fano 4-folds with Lefschetz defect 3.
{"title":"K-polystability of Fano 4-folds with large Lefschetz defect","authors":"Eleonora A. Romano, Saverio A. Secci","doi":"arxiv-2409.03571","DOIUrl":"https://doi.org/arxiv-2409.03571","url":null,"abstract":"We study K-stability on smooth complex Fano 4-folds having large Lefschetz\u0000defect, that is greater or equal then 3, with a special focus on the case of\u0000Lefschetz defect 3. In particular, we determine whether these Fano 4-folds are\u0000K-polystable or not, and show that there are 5 families (out of 19) of\u0000K-polystable smooth Fano 4-folds with Lefschetz defect 3.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper proves that the universal covering of a compact K"{a}hler manifold with small positive sectional curvature in a certain sense is contractible.
{"title":"Almost Non-positive Kähler Manifolds","authors":"Yuguang Zhang","doi":"arxiv-2409.03343","DOIUrl":"https://doi.org/arxiv-2409.03343","url":null,"abstract":"This paper proves that the universal covering of a compact K\"{a}hler\u0000manifold with small positive sectional curvature in a certain sense is\u0000contractible.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $Ksubsetmathbb{R}^{n+1}$ $(nge 2)$, for any embedded hypersurface $Sigmasubset K$ with boundary $partialSigmasubset partial K$ satisfying certain contact angle condition, there holds $$frac1{n+1}int_{Sigma}vert{H}vert^n{rm d}Age{rm AVR}(K)vertmathbb{B}^{n+1}vert.$$ Moreover, equality holds if and only if $Sigma$ is a part of a sphere and $KsetminusOmega$ is a part of the solid cone determined by $Sigma$. Here $Omega$ is the bounded domain enclosed by $Sigma$ and $partial K$, $H$ is the normalized mean curvature of $Sigma$, and ${rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.
{"title":"Willmore-type inequality in unbounded convex sets","authors":"Xiaohan Jia, Guofang Wang, Chao Xia, Xuwen Zhang","doi":"arxiv-2409.03321","DOIUrl":"https://doi.org/arxiv-2409.03321","url":null,"abstract":"In this paper we prove the following Willmore-type inequality: On an\u0000unbounded closed convex set $Ksubsetmathbb{R}^{n+1}$ $(nge 2)$, for any\u0000embedded hypersurface $Sigmasubset K$ with boundary $partialSigmasubset\u0000partial K$ satisfying certain contact angle condition, there holds\u0000$$frac1{n+1}int_{Sigma}vert{H}vert^n{rm d}Age{rm\u0000AVR}(K)vertmathbb{B}^{n+1}vert.$$ Moreover, equality holds if and only if\u0000$Sigma$ is a part of a sphere and $KsetminusOmega$ is a part of the solid\u0000cone determined by $Sigma$. Here $Omega$ is the bounded domain enclosed by\u0000$Sigma$ and $partial K$, $H$ is the normalized mean curvature of $Sigma$,\u0000and ${rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an\u0000anisotropic version of this Willmore-type inequality. As a special case, we\u0000obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a\u0000half-space.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonathan CerqueiraFlorida State University, Emmanuel HartmanUniversity of Houston, Eric KlassenFlorida State University, Martin BauerFlorida State University
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.
{"title":"Sobolev Metrics on Spaces of Discrete Regular Curves","authors":"Jonathan CerqueiraFlorida State University, Emmanuel HartmanUniversity of Houston, Eric KlassenFlorida State University, Martin BauerFlorida State University","doi":"arxiv-2409.02351","DOIUrl":"https://doi.org/arxiv-2409.02351","url":null,"abstract":"Reparametrization invariant Sobolev metrics on spaces of regular curves have\u0000been shown to be of importance in the field of mathematical shape analysis. For\u0000practical applications, one usually discretizes the space of smooth curves and\u0000considers the induced Riemannian metric on a finite dimensional approximation\u0000space. Surprisingly, the theoretical properties of the corresponding finite\u0000dimensional Riemannian manifolds have not yet been studied in detail, which is\u0000the content of the present article. Our main theorem concerns metric and\u0000geodesic completeness and mirrors the results of the infinite dimensional\u0000setting as obtained by Bruveris, Michor and Mumford.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we investigate the focal locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. The main goal is to show that the associated normal exponential map is emph{regular} in the sense of F.W. Warner (textit{Am. J. of Math.}, 87, 1965). This leads to the proof of the fact that the normal exponential is non-injective near tangent focal points. As an application, following R.L. Bishop's work (textit{Proc. Amer. Math. Soc.}, 65, 1977), we express the tangent cut locus as a closure of a certain set of points, called separating tangent cut points. This strengthens the results from the present authors' previous work (textit{J. Geom. Anal.}, 34, 2024).
在本文中，我们研究了前向完整芬斯勒流形中封闭（不一定紧凑）子流形的焦点位置。我们的主要目标是证明相关的法向指数图在华纳（F.W. Warner）的意义上是（emph{regular}的（textit{Am. J. of Math.}，87，1965）。这就证明了法向指数在切焦点附近是非注入的这一事实。作为应用，根据毕晓普的工作 (textit{Proc.Amer.Math.Soc.}，65，1977)，我们把切线切点表达为某一组点的闭包，称为分离切点。这加强了本文作者之前的研究成果 (textit{J. Geom. Anal.}，34，2024)。
{"title":"On the Focal Locus of Submanifold of a Finsler Manifold","authors":"Aritra Bhowmick, Sachchidanand Prasad","doi":"arxiv-2409.02643","DOIUrl":"https://doi.org/arxiv-2409.02643","url":null,"abstract":"In this article, we investigate the focal locus of closed (not necessarily\u0000compact) submanifolds in a forward complete Finsler manifold. The main goal is\u0000to show that the associated normal exponential map is emph{regular} in the\u0000sense of F.W. Warner (textit{Am. J. of Math.}, 87, 1965). This leads to the\u0000proof of the fact that the normal exponential is non-injective near tangent\u0000focal points. As an application, following R.L. Bishop's work (textit{Proc.\u0000Amer. Math. Soc.}, 65, 1977), we express the tangent cut locus as a closure of\u0000a certain set of points, called separating tangent cut points. This strengthens\u0000the results from the present authors' previous work (textit{J. Geom. Anal.},\u000034, 2024).","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main purpose of this article is to study conditions for a curve on a submanifold $Msubsetmathbb{R}^n$, constructed in a particular way involving the Euclidean distance to $M$, to be a geodesic. We also present the naturally arising generalization of Clairaut's formula needed for the generalization of the main result to higher dimensions.
{"title":"A generalization of Clairaut's formula and its applications","authors":"Vadym Koval","doi":"arxiv-2409.02895","DOIUrl":"https://doi.org/arxiv-2409.02895","url":null,"abstract":"The main purpose of this article is to study conditions for a curve on a\u0000submanifold $Msubsetmathbb{R}^n$, constructed in a particular way involving\u0000the Euclidean distance to $M$, to be a geodesic. We also present the naturally\u0000arising generalization of Clairaut's formula needed for the generalization of\u0000the main result to higher dimensions.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, we show that the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.
{"title":"Monotonicity of the modulus under curve shortening flow","authors":"Arjun Sobnack, Peter M. Topping","doi":"arxiv-2409.03098","DOIUrl":"https://doi.org/arxiv-2409.03098","url":null,"abstract":"Given two disjoint nested embedded closed curves in the plane, both evolving\u0000under curve shortening flow, we show that the modulus of the enclosed annulus\u0000is monotonically increasing in time. An analogous result holds within any\u0000ambient surface satisfying a lower curvature bound.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $mathbb{H}^n$ for $n geq 4$. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [7] for the hyperbolic space $mathbb{H}^n$ when $n geq 4$.
{"title":"Weinstock inequality in hyperbolic space","authors":"Pingxin Gu, Haizhong Li, Yao Wan","doi":"arxiv-2409.02766","DOIUrl":"https://doi.org/arxiv-2409.02766","url":null,"abstract":"In this paper, we establish the Weinstock inequality for the first non-zero\u0000Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space\u0000$mathbb{H}^n$ for $n geq 4$. In particular, when the domain is convex, our\u0000result gives an affirmative answer to Open Question 4.27 in [7] for the\u0000hyperbolic space $mathbb{H}^n$ when $n geq 4$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein's models. The aim of this short survey is to explain carefully the basic concepts and algebraic tools built over several recent decades. We focus on the direct link between the jets of sections of homogeneous bundles and the associated induced modules, allowing us to understand the overall structure of invariant linear differential operators in purely algebraic terms. This allows us to extend essential parts of the concepts and procedures to the curved cases.
{"title":"Semiholonomic jets and induced modules in Cartan geometry calculus","authors":"Jan Slovák, Vladimír Souček","doi":"arxiv-2409.01844","DOIUrl":"https://doi.org/arxiv-2409.01844","url":null,"abstract":"The famous Erlangen Programme was coined by Felix Klein in 1872 as an\u0000algebraic approach allowing to incorporate fixed symmetry groups as the core\u0000ingredient for geometric analysis, seeing the chosen symmetries as intrinsic\u0000invariance of all objects and tools. This idea was broadened essentially by\u0000Elie Cartan in the beginning of the last century, and we may consider (curved)\u0000geometries as modelled over certain (flat) Klein's models. The aim of this\u0000short survey is to explain carefully the basic concepts and algebraic tools\u0000built over several recent decades. We focus on the direct link between the jets\u0000of sections of homogeneous bundles and the associated induced modules, allowing\u0000us to understand the overall structure of invariant linear differential\u0000operators in purely algebraic terms. This allows us to extend essential parts\u0000of the concepts and procedures to the curved cases.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that $operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $dgeq5$. Furthermore, we show that $operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $ngeq11$ and $nequiv 1pmod 5$, or $ngeq7$ and $nequiv 2pmod 5$, or $ngeq3$ and $nequiv 3pmod 5$.
{"title":"Isospectral spherical space forms and orbifolds of highest volume","authors":"Alfredo Álzaga, Emilio A. Lauret","doi":"arxiv-2409.02213","DOIUrl":"https://doi.org/arxiv-2409.02213","url":null,"abstract":"We prove that $operatorname{vol}(S^{d})/8$ is the highest volume of a pair\u0000of $d$-dimensional isospectral and non-isometric spherical orbifolds for any\u0000$dgeq5$. Furthermore, we show that $operatorname{vol}(S^{2n-1})/11$ is the\u0000highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric\u0000spherical space forms if either $ngeq11$ and $nequiv 1pmod 5$, or $ngeq7$\u0000and $nequiv 2pmod 5$, or $ngeq3$ and $nequiv 3pmod 5$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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