We provide a counter-example to Hutchinson's original proof of $C^{1,alpha}$ representation of curvature $m$-varifolds with $L^q$-integrable second fundamental form and $q>m$ in [6]. We also provide an alternative proof of the same result and introduce a method of decomposing varifolds into nested components preserving weakly differentiability of a given function. Furthermore, we prove the structure theorem for curvature varifolds with null second fundamental form which is widely used in the literature.
{"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":null,"pages":null},"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}
The present paper introduces the geometry of screen generic lightlike submanifolds of a locally bronze semi-Riemannian manifolds endowed with an (l,m)-type connection. The characterization theorems on geodesicity of such submanifolds with respect to the integrability and parallelism of the distributions are provided. It is proved that there exists no coisotropic , isotropic or totally proper screen generic lightlike submanifold of a locally bronze semi-Riemannian manifold. Assertions for the smooth transversal vector fields in totally umbilical proper screen generic lightlike submanifold are obtained. The structure of a minimal screen generic lightlike submanifold of a locally bronze semi-Riemannian manifold is detailed with an example.
{"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":null,"pages":null},"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}
Newton Solórzano, Víctor León, Alexandre Henrique, Marcelo Souza
We investigate the travel time in a navigation problem from a geometric perspective. The setting involves an open subset of the Euclidean plane, representing a lake perturbed by a symmetric wind flow proportional to the distance from the origin. The Randers metric derived from this physical problem generalizes the well-known Euclidean metric on the Cartesian plane and the Funk metric on the unit disk. We obtain formulas for distances, or travel times, from point to point, from point to line, and vice-versa
{"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":null,"pages":null},"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}
We prove that the moduli space of all noncongruent linearly full totally real flat minimal immersions from the complex plane C into HP^3 that do not lie in CP^3 has three components, each of which is a manifold of real dimension 6. As an application, we give a description of the moduli space of all noncongruent linearly full totally real flat minimal tori in HP^3 that do not lie in CP^3.
{"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":null,"pages":null},"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}
Twistor spaces are certain compact complex threefolds with an additional real fibre bundle structure. We focus here on twistor spaces over $3mathbb{C}mathbb{P}^2$. Such spaces are either small resolutions of double solids or they can be described as modifications of conic bundles. The last type is the more special one: they deform into double solids. We give an explicit description of this deformation, in a more general context.
{"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":null,"pages":null},"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}
We generalise the notion of a Killing superalgebra which arises in the physics literature on supergravity to general dimension, signature and choice of spinor module and squaring map, and also allowing for Lie algebras as well as superalgebras, capturing a set of examples of such algebras on higher-dimensional spheres. We demonstrate that the definition requires a connection on a spinor bundle -- provided by supersymmetry transformations in the supergravity examples and by the Killing spinor equation on the spheres -- and obtain a set of sufficient conditions on such a connection for the Killing (super)algebra to exist. We show that these (super)algebras are filtered deformations of graded subalgebras of (a generalisation of) the Poincar'e superalgebra and then study such deformations abstractly using Spencer cohomology. In the highly supersymmetric Lorentzian case, we describe the filtered subdeformations which are of the appropriate form to arise as Killing superalgebras, lay out a classification scheme for their odd-generated subalgebras and prove that, under certain technical conditions, there exist homogeneous Lorentzian spin manifolds on which these deformations are realised as Killing superalgebras. Our results generalise previous work in the 11-dimensional supergravity literature.
{"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":null,"pages":null},"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}
In this paper we compute the third coefficient arising from the TYCZ-expansion of the epsilon function associated to a Kaehler-Einstein metric and discuss the consequences of its vanishing.
{"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":null,"pages":null},"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}
We consider the K"ahler-Ricci flow $(X, omega(t))_{t in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K"ahler manifold $X$ to some analytic variety $Y$ which admits a K"ahler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[omega_0] in H^{1,1}(X,mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense.
{"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":null,"pages":null},"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}
In this article, we study space-like and time-like surfaces in a Robertson-Walker space-time,, denoted by $L^4_1(f,c)$, having positive relative nullity. First, we give the necessary and sufficient conditions for such space-like and time-like surfaces in $L^4_1(f,c)$. Then, we obtain the local classification theorems for space-like and time-like surfaces in $L^4_1(f,0)$ with positive relative nullity. Finally, we consider the space-like and time-like surfaces in $mathbb{E}^1_1timesmathbb{S}^3$ and $mathbb{E}^1_1timesmathbb{H}^3$ with positive relative nullity. These are the special spaces of $L^4_1(f,c)$ when the warping function $f$ is a constant function, with $c=1$ for $mathbb{E}^1_1timesmathbb{S}^3$ and $c=-1$ for $mathbb{E}^1_1timesmathbb{H}^3$.
{"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":null,"pages":null},"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}
Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul
We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity.
{"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":null,"pages":null},"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}
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