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A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by $C^{1,α}$ Multiple Valued Functions 用$C^{1,α}$ 多值函数对一类曲率变折的图形表示的修正证明
arXiv - MATH - Differential Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11861
Nicolau S. Aiex
{"title":"A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by $C^{1,α}$ Multiple Valued Functions","authors":"Nicolau S. Aiex","doi":"arxiv-2409.11861","DOIUrl":"https://doi.org/arxiv-2409.11861","url":null,"abstract":"We provide a counter-example to Hutchinson's original proof of $C^{1,alpha}$\u0000representation of curvature $m$-varifolds with $L^q$-integrable second\u0000fundamental form and $q>m$ in [6]. We also provide an alternative proof of the\u0000same result and introduce a method of decomposing varifolds into nested\u0000components preserving weakly differentiability of a given function.\u0000Furthermore, we prove the structure theorem for curvature varifolds with null\u0000second fundamental form which is widely used in the literature.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254084","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}
引用次数: 0
Screen Generic Lightlike Submanifolds of a Locally Bronze Semi-Riemannian Manifold equipped with an (l,m)-type Connection 配备(l,m)型连接的局部青铜半黎曼流形的屏泛光子流形
arXiv - MATH - Differential Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11730
Rajinder Kaur, Jasleen Kaur
{"title":"Screen Generic Lightlike Submanifolds of a Locally Bronze Semi-Riemannian Manifold equipped with an (l,m)-type Connection","authors":"Rajinder Kaur, Jasleen Kaur","doi":"arxiv-2409.11730","DOIUrl":"https://doi.org/arxiv-2409.11730","url":null,"abstract":"The present paper introduces the geometry of screen generic lightlike\u0000submanifolds of a locally bronze semi-Riemannian manifolds endowed with an\u0000(l,m)-type connection. The characterization theorems on geodesicity of such\u0000submanifolds with respect to the integrability and parallelism of the\u0000distributions are provided. It is proved that there exists no coisotropic ,\u0000isotropic or totally proper screen generic lightlike submanifold of a locally\u0000bronze semi-Riemannian manifold. Assertions for the smooth transversal vector\u0000fields in totally umbilical proper screen generic lightlike submanifold are\u0000obtained. The structure of a minimal screen generic lightlike submanifold of a\u0000locally bronze semi-Riemannian manifold is detailed with an example.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254091","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}
引用次数: 0
Navigation problem; $λ-$Funk metric; Finsler metric 导航问题; $λ-$Funk 公设; 芬斯勒公设
arXiv - MATH - Differential Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.12058
Newton Solórzano, Víctor León, Alexandre Henrique, Marcelo Souza
{"title":"Navigation problem; $λ-$Funk metric; Finsler metric","authors":"Newton Solórzano, Víctor León, Alexandre Henrique, Marcelo Souza","doi":"arxiv-2409.12058","DOIUrl":"https://doi.org/arxiv-2409.12058","url":null,"abstract":"We investigate the travel time in a navigation problem from a geometric\u0000perspective. The setting involves an open subset of the Euclidean plane,\u0000representing a lake perturbed by a symmetric wind flow proportional to the\u0000distance from the origin. The Randers metric derived from this physical problem\u0000generalizes the well-known Euclidean metric on the Cartesian plane and the Funk\u0000metric on the unit disk. We obtain formulas for distances, or travel times,\u0000from point to point, from point to line, and vice-versa","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254082","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}
引用次数: 0
The versal deformation of Kurke-LeBrun manifolds 库克-勒布伦流形的 versal 变形
arXiv - MATH - Differential Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.12022
Bernd Kreussler, Jan Stevens
{"title":"The versal deformation of Kurke-LeBrun manifolds","authors":"Bernd Kreussler, Jan Stevens","doi":"arxiv-2409.12022","DOIUrl":"https://doi.org/arxiv-2409.12022","url":null,"abstract":"Twistor spaces are certain compact complex threefolds with an additional real\u0000fibre bundle structure. We focus here on twistor spaces over\u0000$3mathbb{C}mathbb{P}^2$. Such spaces are either small resolutions of double\u0000solids or they can be described as modifications of conic bundles. The last\u0000type is the more special one: they deform into double solids. We give an\u0000explicit description of this deformation, in a more general context.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254086","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}
引用次数: 0
The space of totally real flat minimal surfaces in the Quaternionic projective space HP^3 四元投影空间 HP^3 中的全实平面极小曲面空间
arXiv - MATH - Differential Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11931
Chuzi Duan, Ling He
{"title":"The space of totally real flat minimal surfaces in the Quaternionic projective space HP^3","authors":"Chuzi Duan, Ling He","doi":"arxiv-2409.11931","DOIUrl":"https://doi.org/arxiv-2409.11931","url":null,"abstract":"We prove that the moduli space of all noncongruent linearly full totally real\u0000flat minimal immersions from the complex plane C into HP^3 that do not lie in\u0000CP^3 has three components, each of which is a manifold of real dimension 6. As\u0000an application, we give a description of the moduli space of all noncongruent\u0000linearly full totally real flat minimal tori in HP^3 that do not lie in CP^3.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254083","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}
引用次数: 0
Killing (super)algebras associated to connections on spinors 与旋量上的连接相关的基林（超）代数
arXiv - MATH - Differential Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11306
Andrew D. K. Beckett
{"title":"Killing (super)algebras associated to connections on spinors","authors":"Andrew D. K. Beckett","doi":"arxiv-2409.11306","DOIUrl":"https://doi.org/arxiv-2409.11306","url":null,"abstract":"We generalise the notion of a Killing superalgebra which arises in the\u0000physics literature on supergravity to general dimension, signature and choice\u0000of spinor module and squaring map, and also allowing for Lie algebras as well\u0000as superalgebras, capturing a set of examples of such algebras on\u0000higher-dimensional spheres. We demonstrate that the definition requires a\u0000connection on a spinor bundle -- provided by supersymmetry transformations in\u0000the supergravity examples and by the Killing spinor equation on the spheres --\u0000and obtain a set of sufficient conditions on such a connection for the Killing\u0000(super)algebra to exist. We show that these (super)algebras are filtered\u0000deformations of graded subalgebras of (a generalisation of) the Poincar'e\u0000superalgebra and then study such deformations abstractly using Spencer\u0000cohomology. In the highly supersymmetric Lorentzian case, we describe the\u0000filtered subdeformations which are of the appropriate form to arise as Killing\u0000superalgebras, lay out a classification scheme for their odd-generated\u0000subalgebras and prove that, under certain technical conditions, there exist\u0000homogeneous Lorentzian spin manifolds on which these deformations are realised\u0000as Killing superalgebras. Our results generalise previous work in the\u000011-dimensional supergravity literature.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254087","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}
引用次数: 0
On the third coefficient in the TYCZ-expansion of the epsilon function of Kaehler-Einstein manifolds 论凯勒-爱因斯坦流形ε函数 TYCZ 展开中的第三个系数
arXiv - MATH - Differential Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11137
Simone Cristofori, Michela Zedda
{"title":"On the third coefficient in the TYCZ-expansion of the epsilon function of Kaehler-Einstein manifolds","authors":"Simone Cristofori, Michela Zedda","doi":"arxiv-2409.11137","DOIUrl":"https://doi.org/arxiv-2409.11137","url":null,"abstract":"In this paper we compute the third coefficient arising from the\u0000TYCZ-expansion of the epsilon function associated to a Kaehler-Einstein metric\u0000and discuss the consequences of its vanishing.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254088","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}
引用次数: 0
Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities 具有有限时间奇点的凯勒-里奇流的全局里奇曲率行为
arXiv - MATH - Differential Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11608
Alexander Bednarek
{"title":"Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities","authors":"Alexander Bednarek","doi":"arxiv-2409.11608","DOIUrl":"https://doi.org/arxiv-2409.11608","url":null,"abstract":"We consider the K\"ahler-Ricci flow $(X, omega(t))_{t in [0,T)}$ on a\u0000compact manifold where the time of singularity, $T$, is finite. We assume the\u0000existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic\u0000variety $Y$ which admits a K\"ahler metric on a neighbourhood of the image of\u0000$X$ and that the pullback of this metric yields the limiting cohomology class\u0000along the flow. This is satisfied, for instance, by the assumption that the\u0000initial cohomology class is rational, i.e., $[omega_0] in\u0000H^{1,1}(X,mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate\u0000on the behaviour of the Ricci curvature and that the Riemannian curvature is\u0000Type $I$ in the $L^2$-sense.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254085","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}
引用次数: 0
Surfaces in Robertson-Walker Space-Times with Positive Relative Nullity 具有正相对无效性的罗伯逊-沃克时空曲面
arXiv - MATH - Differential Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11050
Burcu Bektaş Demirci, Nurettin Cenk Turgay
{"title":"Surfaces in Robertson-Walker Space-Times with Positive Relative Nullity","authors":"Burcu Bektaş Demirci, Nurettin Cenk Turgay","doi":"arxiv-2409.11050","DOIUrl":"https://doi.org/arxiv-2409.11050","url":null,"abstract":"In this article, we study space-like and time-like surfaces in a\u0000Robertson-Walker space-time,, denoted by $L^4_1(f,c)$, having positive relative\u0000nullity. First, we give the necessary and sufficient conditions for such\u0000space-like and time-like surfaces in $L^4_1(f,c)$. Then, we obtain the local\u0000classification theorems for space-like and time-like surfaces in $L^4_1(f,0)$\u0000with positive relative nullity. Finally, we consider the space-like and\u0000time-like surfaces in $mathbb{E}^1_1timesmathbb{S}^3$ and\u0000$mathbb{E}^1_1timesmathbb{H}^3$ with positive relative nullity. These are\u0000the special spaces of $L^4_1(f,c)$ when the warping function $f$ is a constant\u0000function, with $c=1$ for $mathbb{E}^1_1timesmathbb{S}^3$ and $c=-1$ for\u0000$mathbb{E}^1_1timesmathbb{H}^3$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254089","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}
引用次数: 0
Enumeration of Rational Cuspidal Curves via the WDVV equation 通过 WDVV 方程枚举有理无顶曲线
arXiv - MATH - Differential Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10238
Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul
{"title":"Enumeration of Rational Cuspidal Curves via the WDVV equation","authors":"Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul","doi":"arxiv-2409.10238","DOIUrl":"https://doi.org/arxiv-2409.10238","url":null,"abstract":"We give a conjectural formula for the characteristic number of rational\u0000cuspidal curves in the projective plane by extending the idea of Kontsevich's\u0000recursion formula (namely, pulling back the equality of two divisors in the\u0000four pointed moduli space). The key geometric input that is needed here is that\u0000in the closure of rational cuspidal curves, there are two component rational\u0000curves which are tangent to each other at the nodal point. While this fact is\u0000geometrically quite believable, we haven't as yet proved it; hence our formula\u0000is for the moment conjectural. The answers that we obtain agree with what has\u0000been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We\u0000extend this technique (modulo another conjecture) to obtain the characteristic\u0000number of rational quartics with an E6 singularity.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254096","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}
引用次数: 0
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