This paper establishes the conditions under which minimal and stable minimal hypersurfaces are characterized as hyperplanes in Euclidean spaces and as totally geodesic submanifolds in Riemannian manifolds.
{"title":"On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds","authors":"Josef Mikes, Sergey Stepanov, Irina Tsyganok","doi":"arxiv-2409.04426","DOIUrl":"https://doi.org/arxiv-2409.04426","url":null,"abstract":"This paper establishes the conditions under which minimal and stable minimal\u0000hypersurfaces are characterized as hyperplanes in Euclidean spaces and as\u0000totally geodesic submanifolds in Riemannian manifolds.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199026","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 a smooth $s$-dimensional submanifold $S$ of $mathbb{R}^{m+c}$ and a smooth distribution $mathcal{D}supset TS$ of rank $m$ along $S$, we study the following geometric Cauchy problem: to find an $m$-dimensional rank-$s$ submanifold $M$ of $mathbb{R}^{m+c}$ (that is, an $m$-submanifold with constant index of relative nullity $m-s$) such that $M supset S$ and $TM |_{S} = mathcal{D}$. In particular, under some reasonable assumption and using a constructive approach, we show that a solution exists and is unique in a neighborhood of $S$.
{"title":"The geometric Cauchy problem for constant-rank submanifolds","authors":"Matteo Raffaelli","doi":"arxiv-2409.04358","DOIUrl":"https://doi.org/arxiv-2409.04358","url":null,"abstract":"Given a smooth $s$-dimensional submanifold $S$ of $mathbb{R}^{m+c}$ and a\u0000smooth distribution $mathcal{D}supset TS$ of rank $m$ along $S$, we study the\u0000following geometric Cauchy problem: to find an $m$-dimensional rank-$s$\u0000submanifold $M$ of $mathbb{R}^{m+c}$ (that is, an $m$-submanifold with\u0000constant index of relative nullity $m-s$) such that $M supset S$ and $TM |_{S}\u0000= mathcal{D}$. In particular, under some reasonable assumption and using a\u0000constructive approach, we show that a solution exists and is unique in a\u0000neighborhood of $S$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199036","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 construct a compact, simply connected manifold with holonomy $mathrm{G}_2$ that is non-formal. We use the construction method of compact torsion-free $mathrm{G}_2$ manifolds developed by D.D. Joyce and S. Karigiannis. A non-vanishing triple Massey product is obtained by arranging the singular locus in a particular configuration.
{"title":"Compact holonomy $mathrm{G}_2$ manifolds need not be formal","authors":"Lucía Martín-Merchán","doi":"arxiv-2409.04362","DOIUrl":"https://doi.org/arxiv-2409.04362","url":null,"abstract":"We construct a compact, simply connected manifold with holonomy\u0000$mathrm{G}_2$ that is non-formal. We use the construction method of compact\u0000torsion-free $mathrm{G}_2$ manifolds developed by D.D. Joyce and S.\u0000Karigiannis. A non-vanishing triple Massey product is obtained by arranging the\u0000singular locus in a particular configuration.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199030","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 study the Dirichlet problem for a class of prescribed curvature equations in Minkowski space. We prove the existence of smooth spacelike hypersurfaces with a class of prescribed curvature and general boundary data based on establishing the emph{a priori} $C^2$ estimates.
{"title":"The Dirichlet problem for a class of curvature equations in Minkowski space","authors":"Mengru Guo, Heming Jiao","doi":"arxiv-2409.03308","DOIUrl":"https://doi.org/arxiv-2409.03308","url":null,"abstract":"In this paper, we study the Dirichlet problem for a class of prescribed\u0000curvature equations in Minkowski space. We prove the existence of smooth\u0000spacelike hypersurfaces with a class of prescribed curvature and general\u0000boundary data based on establishing the emph{a priori} $C^2$ estimates.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199044","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 monotonicity formulas for capillary surfaces in the half-space $mathbb{R}^3_+$ and in the unit ball $mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).
{"title":"Monotonicity Formulas for Capillary Surfaces","authors":"Guofang Wang, Chao Xia, Xuwen Zhang","doi":"arxiv-2409.03314","DOIUrl":"https://doi.org/arxiv-2409.03314","url":null,"abstract":"In this paper, we establish monotonicity formulas for capillary surfaces in\u0000the half-space $mathbb{R}^3_+$ and in the unit ball $mathbb{B}^3$ and extend\u0000the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221.\u0000href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7})\u0000for surfaces with free boundary. As applications, we obtain Li-Yau-type\u0000inequalities for the Willmore energy of capillary surfaces, and extend\u0000Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in\u0000$mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030.\u0000href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007})\u0000to the capillary setting, which is different to another optimal area estimate\u0000proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201.\u0000href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).","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":"142199040","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}
Alcides de Carvalho, Roney Santos, Federico Trinca
Given a $n$-dimensional Riemannian manifold with non-negative sectional curvatures and convex boundary, that is conformal to an Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension $2leq kleq n-2$, provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvatures are strictly positive.
{"title":"On the stability of free boundary minimal submanifolds in conformal domains","authors":"Alcides de Carvalho, Roney Santos, Federico Trinca","doi":"arxiv-2409.03943","DOIUrl":"https://doi.org/arxiv-2409.03943","url":null,"abstract":"Given a $n$-dimensional Riemannian manifold with non-negative sectional\u0000curvatures and convex boundary, that is conformal to an Euclidean convex\u0000bounded domain, we show that it does not contain any compact stable free\u0000boundary minimal submanifold of dimension $2leq kleq n-2$, provided that\u0000either the boundary is strictly convex with respect to any of the two metrics\u0000or the sectional curvatures are strictly positive.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"186 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199031","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 start by revisiting the derivation of the variational formulae for the functional assigning to a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains in certain weighted manifolds. This is further extended to other functionals defined by certain Dirichlet energy integrals, with a Morse index formula for the corresponding critical domains being established. We complement these infinitesimal results by proving a couple of global rigidity theorems for (possibly critical) domains in Gaussian half-space, including an Alexandrov-type soap bubble theorem. Although we provide direct proofs of these latter results, we find it worthwhile to point out that the main tools employed (specifically, certain Pohozhaev and Reilly identities) can be formally understood as limits (when the dimension goes to infinity) of tools previously established by Ciarolo-Vezzoni and Qiu-Xia to handle similar problems in round hemispheres, with the notion of ``convergence'' of weighted manifolds being loosely inspired by the celebrated Poincar'e's limit theorem in the theory of Gaussian random vectors.
{"title":"Critical domains for certain Dirichlet integrals in weighted manifolds","authors":"Levi Lopes de Lima","doi":"arxiv-2409.03554","DOIUrl":"https://doi.org/arxiv-2409.03554","url":null,"abstract":"We start by revisiting the derivation of the variational formulae for the\u0000functional assigning to a bounded regular domain in a Riemannian manifold its\u0000first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains\u0000in certain weighted manifolds. This is further extended to other functionals\u0000defined by certain Dirichlet energy integrals, with a Morse index formula for\u0000the corresponding critical domains being established. We complement these\u0000infinitesimal results by proving a couple of global rigidity theorems for\u0000(possibly critical) domains in Gaussian half-space, including an\u0000Alexandrov-type soap bubble theorem. Although we provide direct proofs of these\u0000latter results, we find it worthwhile to point out that the main tools employed\u0000(specifically, certain Pohozhaev and Reilly identities) can be formally\u0000understood as limits (when the dimension goes to infinity) of tools previously\u0000established by Ciarolo-Vezzoni and Qiu-Xia to handle similar problems in round\u0000hemispheres, with the notion of ``convergence'' of weighted manifolds being\u0000loosely inspired by the celebrated Poincar'e's limit theorem in the theory of\u0000Gaussian random vectors.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199037","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 present OGRePy, the official Python port of the popular Mathematica tensor calculus package OGRe (Object-Oriented General Relativity) - a powerful, yet user-friendly, tool for advanced tensor calculations in mathematics and physics, especially suitable for general relativity. The Python port uses the same robust and performance-oriented algorithms as the original package, and retains its core design principles. However, its truly object-oriented interface, enabled by Python, is more intuitive and flexible than the original Mathematica implementation. It utilizes SymPy for symbolic computations and Jupyter as a notebook interface. OGRePy allows calculating arbitrary tensor formulas using any combination of addition, multiplication by scalar, trace, contraction, partial derivative, covariant derivative, and permutation of indices. Transformations of the tensor components between different index configurations and/or coordinate systems are performed seamlessly behind the scenes as needed, eliminating user error due to combining incompatible representations, and guaranteeing consistent results. In addition, the package provides facilities for easily calculating various curvature tensors and geodesic equations in multiple representations. This paper presents the main features of the package in great detail, including many examples of its use in the context of general relativity research.
{"title":"OGRePy: An Object-Oriented General Relativity Package for Python","authors":"Barak Shoshany","doi":"arxiv-2409.03803","DOIUrl":"https://doi.org/arxiv-2409.03803","url":null,"abstract":"We present OGRePy, the official Python port of the popular Mathematica tensor\u0000calculus package OGRe (Object-Oriented General Relativity) - a powerful, yet\u0000user-friendly, tool for advanced tensor calculations in mathematics and\u0000physics, especially suitable for general relativity. The Python port uses the\u0000same robust and performance-oriented algorithms as the original package, and\u0000retains its core design principles. However, its truly object-oriented\u0000interface, enabled by Python, is more intuitive and flexible than the original\u0000Mathematica implementation. It utilizes SymPy for symbolic computations and\u0000Jupyter as a notebook interface. OGRePy allows calculating arbitrary tensor\u0000formulas using any combination of addition, multiplication by scalar, trace,\u0000contraction, partial derivative, covariant derivative, and permutation of\u0000indices. Transformations of the tensor components between different index\u0000configurations and/or coordinate systems are performed seamlessly behind the\u0000scenes as needed, eliminating user error due to combining incompatible\u0000representations, and guaranteeing consistent results. In addition, the package\u0000provides facilities for easily calculating various curvature tensors and\u0000geodesic equations in multiple representations. This paper presents the main\u0000features of the package in great detail, including many examples of its use in\u0000the context of general relativity research.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199035","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}
Using co-homogeneity one symmetries, we construct a two-parameter family of non-abelian $G_2$-instantons on every member of the asymptotically locally conical $mathbb{B}_7$-family of $G_2$-metrics on $S^3 times mathbb{R}^4 $, and classify the resulting solutions. These solutions can be described as perturbations of a one-parameter family of abelian instantons, arising from the Killing vector-field generating the asymptotic circle fibre. Generically, these perturbations decay exponentially to the model, but we find a one-parameter family of instantons with polynomial decay. Moreover, we relate the two-parameter family to a lift of an explicit two-parameter family of anti-self-dual instantons on Taub-NUT $mathbb{R}^4$, fibred over $S^3$ in an adiabatic limit.
利用共偶性一对称性，我们在$S^3 times mathbb{R}^4 $上的$G_2$-metrics的渐近局部共轭$mathbb{B}_7$-family的每一个成员上构建了非阿贝尔$G_2$-瞬子的双参数族，并对由此产生的解进行了分类。这些解可以描述为产生渐近圆纤维的基林向量场对无性瞬子单参数族的扰动。一般来说，这些扰动对模型呈指数衰减，但我们发现了一个具有多项式衰减的单参数瞬子族。此外，我们把这个二参数族与在无绝热极限下，Taub-NUT $/mathbb{R}^4$上的反自双瞬子的一个显式二参数族的提升联系起来。
{"title":"$G_2$-instantons on the ALC members of the $mathbb{B}_7$ family","authors":"Jakob Stein, Matt Turner","doi":"arxiv-2409.03886","DOIUrl":"https://doi.org/arxiv-2409.03886","url":null,"abstract":"Using co-homogeneity one symmetries, we construct a two-parameter family of\u0000non-abelian $G_2$-instantons on every member of the asymptotically locally\u0000conical $mathbb{B}_7$-family of $G_2$-metrics on $S^3 times mathbb{R}^4 $,\u0000and classify the resulting solutions. These solutions can be described as\u0000perturbations of a one-parameter family of abelian instantons, arising from the\u0000Killing vector-field generating the asymptotic circle fibre. Generically, these\u0000perturbations decay exponentially to the model, but we find a one-parameter\u0000family of instantons with polynomial decay. Moreover, we relate the\u0000two-parameter family to a lift of an explicit two-parameter family of\u0000anti-self-dual instantons on Taub-NUT $mathbb{R}^4$, fibred over $S^3$ in an\u0000adiabatic limit.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199032","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 investigate the old problem of determining the exact bulk moduli of generic $mathrm{SU}(3)$-structure flux backgrounds of type II string theory. Using techniques from generalised geometry, we show that the infinitesimal deformations are counted by a spectral sequence in which the vertical maps are either de Rham or Dolbeault differentials (depending on the type of the exceptional complex structure (ECS)) and the horizontal maps are linear maps constructed from the flux and intrinsic torsion. Our calculation is exact, covering all possible supergravity $mathrm{SU}(3)$-structure flux backgrounds including those which are not conformally Calabi--Yau, and goes beyond the usual linear approximation in three important ways: (i) we allow for finite flux; (ii) we consider perturbative higher-derivative corrections to the supergravity action; and (iii) we consider obstructions arising from higher-order deformations. Despite these extensions we find that the spectral sequence reproduces the na"ive expectations that come from considering the effective superpotential in the small-flux limit. In particular, by writing the moduli in a form that is independent of the K"ahler potential on the space of ECSs, and arguing the superpotential does not receive higher-derivative corrections, we show that the spectral sequence is perturbatively exact. Further, preliminary results show that a Tian--Todorov-like lemma implies that all the obstructions vanish. This has implications for the tadpole conjecture, showing that such perturbative, higher-order effects do not provide a way of circumventing the bound.
{"title":"All-orders moduli for type II flux backgrounds","authors":"George R. Smith, David Tennyson, Daniel Waldram","doi":"arxiv-2409.03847","DOIUrl":"https://doi.org/arxiv-2409.03847","url":null,"abstract":"We investigate the old problem of determining the exact bulk moduli of\u0000generic $mathrm{SU}(3)$-structure flux backgrounds of type II string theory.\u0000Using techniques from generalised geometry, we show that the infinitesimal\u0000deformations are counted by a spectral sequence in which the vertical maps are\u0000either de Rham or Dolbeault differentials (depending on the type of the\u0000exceptional complex structure (ECS)) and the horizontal maps are linear maps\u0000constructed from the flux and intrinsic torsion. Our calculation is exact,\u0000covering all possible supergravity $mathrm{SU}(3)$-structure flux backgrounds\u0000including those which are not conformally Calabi--Yau, and goes beyond the\u0000usual linear approximation in three important ways: (i) we allow for finite\u0000flux; (ii) we consider perturbative higher-derivative corrections to the\u0000supergravity action; and (iii) we consider obstructions arising from\u0000higher-order deformations. Despite these extensions we find that the spectral\u0000sequence reproduces the na\"ive expectations that come from considering the\u0000effective superpotential in the small-flux limit. In particular, by writing the\u0000moduli in a form that is independent of the K\"ahler potential on the space of\u0000ECSs, and arguing the superpotential does not receive higher-derivative\u0000corrections, we show that the spectral sequence is perturbatively exact.\u0000Further, preliminary results show that a Tian--Todorov-like lemma implies that\u0000all the obstructions vanish. This has implications for the tadpole conjecture,\u0000showing that such perturbative, higher-order effects do not provide a way of\u0000circumventing the bound.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199038","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：