We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.
{"title":"Some recent developments on isometric immersions via compensated compactness and gauge transforms","authors":"Siran Li","doi":"arxiv-2409.08922","DOIUrl":"https://doi.org/arxiv-2409.08922","url":null,"abstract":"We survey recent developments on the analysis of Gauss--Codazzi--Ricci\u0000equations, the first-order PDE system arising from the classical problem of\u0000isometric immersions in differential geometry, especially in the regime of low\u0000Sobolev regularity. Such equations are not purely elliptic, parabolic, or\u0000hyperbolic in general, hence calling for analytical tools for PDEs of mixed\u0000types. We discuss various recent contributions -- in line with the pioneering\u0000works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010);\u0000Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci\u0000equations, the weak stability of isometric immersions, and the fundamental\u0000theorem of submanifold theory with low regularity. Two mixed-type PDE\u0000techniques are emphasised throughout these developments: the method of\u0000compensated compactness and the theory of Coulomb--Uhlenbeck gauges.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254661","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 work we studied the stability of the family of operators $L_a=Delta-aS$, $ainmathbb R$, in a warped product of an infinite interval or real line by one compact manifold, where $Delta$ is the Laplacian and $S$ is the scalar curvature of the resulting manifold.
{"title":"Operator $Δ-aS$ on warped product manifolds","authors":"Ezequiel Barbosa, Mateus Souza, Celso Viana","doi":"arxiv-2409.08818","DOIUrl":"https://doi.org/arxiv-2409.08818","url":null,"abstract":"In this work we studied the stability of the family of operators\u0000$L_a=Delta-aS$, $ainmathbb R$, in a warped product of an infinite interval\u0000or real line by one compact manifold, where $Delta$ is the Laplacian and $S$\u0000is the scalar curvature of the resulting manifold.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254099","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 the Michael-Simon-Sobolev inequality for smooth symmetric uniformly positive definite (0, 2)-tensor fields on compact submanifolds with or without boundary in Riemannian manifolds with nonnegative sectional curvature by the Alexandrov-Bakelman-Pucci (ABP) method. It should be a generalization of S. Brendle in [2].
{"title":"The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields","authors":"Yuting Wu, Chengyang Yi, Yu Zheng","doi":"arxiv-2409.08011","DOIUrl":"https://doi.org/arxiv-2409.08011","url":null,"abstract":"We prove the Michael-Simon-Sobolev inequality for smooth symmetric uniformly\u0000positive definite (0, 2)-tensor fields on compact submanifolds with or without\u0000boundary in Riemannian manifolds with nonnegative sectional curvature by the\u0000Alexandrov-Bakelman-Pucci (ABP) method. It should be a generalization of S.\u0000Brendle in [2].","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198968","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 characterize the set of all conformal Spin(7) forms on an oriented and spin Riemannian eight-manifold $(M,g)$ as solutions to a homogeneous algebraic equation of degree two for the self-dual four-forms of $(M,g)$. When $M$ is compact, we use this result to construct a functional whose self-dual critical set is precisely the set of all Spin(7) structures on $M$. Furthermore, the natural coupling of this potential to the Einstein-Hilbert action gives a functional whose self-dual critical points are conformally Ricci-flat Spin(7) structures. Our proof relies on the computation of the square of an irreducible and chiral real spinor as a section of a bundle of real algebraic varieties sitting inside the K"ahler-Atiyah bundle of $(M,g)$.
{"title":"A functional for Spin(7) forms","authors":"Calin Iuliu Lazaroiu, C. S. Shahbazi","doi":"arxiv-2409.08274","DOIUrl":"https://doi.org/arxiv-2409.08274","url":null,"abstract":"We characterize the set of all conformal Spin(7) forms on an oriented and\u0000spin Riemannian eight-manifold $(M,g)$ as solutions to a homogeneous algebraic\u0000equation of degree two for the self-dual four-forms of $(M,g)$. When $M$ is\u0000compact, we use this result to construct a functional whose self-dual critical\u0000set is precisely the set of all Spin(7) structures on $M$. Furthermore, the\u0000natural coupling of this potential to the Einstein-Hilbert action gives a\u0000functional whose self-dual critical points are conformally Ricci-flat Spin(7)\u0000structures. Our proof relies on the computation of the square of an irreducible\u0000and chiral real spinor as a section of a bundle of real algebraic varieties\u0000sitting inside the K\"ahler-Atiyah bundle of $(M,g)$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198967","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 Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime.
{"title":"Nonexistence of closed timelike geodesics in Kerr spacetimes","authors":"Giulio Sanzeni","doi":"arxiv-2409.09094","DOIUrl":"https://doi.org/arxiv-2409.09094","url":null,"abstract":"The Kerr-star spacetime is the extension over the horizons and in the\u0000negative radial region of the Kerr spacetime. Despite the presence of closed\u0000timelike curves below the inner horizon, we prove that the timelike geodesics\u0000cannot be closed in the Kerr-star spacetime. Since the existence of closed null\u0000geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)],\u0000this result shows the absence of closed causal geodesics in the Kerr-star\u0000spacetime.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254098","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}
Andreas Arvanitoyeorgos, Nikolaos Panagiotis Souris, Marina Statha
Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds $(M,g)$ whose geodesics are integral curves of Killing vector fields. Equivalently, there exists a Lie group $G$ of isometries of $(M,g)$ such that any geodesic $gamma$ has the simple form $gamma(t)=e^{tX}cdot p$, where $e$ denotes the exponential map on $G$. The classification of g.o. manifolds is a longstanding problem in Riemannian geometry. In this brief survey, we present some recent results and open questions on the subject focusing on the compact case. In addition we study the geodesic orbit condition for the space $SU(5)/s(U(2)times U(2))$.
{"title":"A review of compact geodesic orbit manifolds and the g.o. condition for $SU(5)/s(U(2)times U(2))$","authors":"Andreas Arvanitoyeorgos, Nikolaos Panagiotis Souris, Marina Statha","doi":"arxiv-2409.08247","DOIUrl":"https://doi.org/arxiv-2409.08247","url":null,"abstract":"Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds\u0000$(M,g)$ whose geodesics are integral curves of Killing vector fields.\u0000Equivalently, there exists a Lie group $G$ of isometries of $(M,g)$ such that\u0000any geodesic $gamma$ has the simple form $gamma(t)=e^{tX}cdot p$, where $e$\u0000denotes the exponential map on $G$. The classification of g.o. manifolds is a\u0000longstanding problem in Riemannian geometry. In this brief survey, we present\u0000some recent results and open questions on the subject focusing on the compact\u0000case. In addition we study the geodesic orbit condition for the space\u0000$SU(5)/s(U(2)times U(2))$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198972","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}
Fernando Manfio, João Batista Marques dos Santos, João Paulo dos Santos, Joeri Van der Veken
We classify the hypersurfaces of $mathbb{Q}^3timesmathbb{R}$ with three distinct constant principal curvatures, where $varepsilon in {1,-1}$ and $mathbb{Q}^3$ denotes the unit sphere $mathbb{S}^3$ if $varepsilon = 1$, whereas it denotes the hyperbolic space $mathbb{H}^3$ if $varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $mathbb{Q}^3timesmathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $mathbb{Q}^3timesmathbb{R}$.
{"title":"Hypersurfaces of $mathbb{S}^3 times mathbb{R}$ and $mathbb{H}^3 times mathbb{R}$ with constant principal curvatures","authors":"Fernando Manfio, João Batista Marques dos Santos, João Paulo dos Santos, Joeri Van der Veken","doi":"arxiv-2409.07978","DOIUrl":"https://doi.org/arxiv-2409.07978","url":null,"abstract":"We classify the hypersurfaces of $mathbb{Q}^3timesmathbb{R}$ with three\u0000distinct constant principal curvatures, where $varepsilon in {1,-1}$ and\u0000$mathbb{Q}^3$ denotes the unit sphere $mathbb{S}^3$ if $varepsilon = 1$,\u0000whereas it denotes the hyperbolic space $mathbb{H}^3$ if $varepsilon = -1$.\u0000We show that they are cylinders over isoparametric surfaces in $mathbb{Q}^3$,\u0000filling an intriguing gap in the existing literature. We also prove that the\u0000hypersurfaces with constant principal curvatures of\u0000$mathbb{Q}^3timesmathbb{R}$ are isoparametric. Furthermore, we provide the\u0000complete classification of the extrinsically homogeneous hypersurfaces in\u0000$mathbb{Q}^3timesmathbb{R}$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198970","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 develop a min-max theory for hypersurfaces of prescribed mean curvature in noncompact manifolds, applicable to prescription functions that do not change sign outside a compact set. We use this theory to prove new existence results for closed prescribed mean curvature hypersurfaces in Euclidean space and complete finite area constant mean curvature hypersurfaces in finite volume manifolds.
{"title":"Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds","authors":"Douglas Stryker","doi":"arxiv-2409.07330","DOIUrl":"https://doi.org/arxiv-2409.07330","url":null,"abstract":"We develop a min-max theory for hypersurfaces of prescribed mean curvature in\u0000noncompact manifolds, applicable to prescription functions that do not change\u0000sign outside a compact set. We use this theory to prove new existence results\u0000for closed prescribed mean curvature hypersurfaces in Euclidean space and\u0000complete finite area constant mean curvature hypersurfaces in finite volume\u0000manifolds.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198971","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 a compact Riemannian manifold $M$ does not admit any non-trivial $m$-modified homothetic vector fields. In the corresponding case of an $m$-modified conformal vector field $V$, we establish an inequality that implies the triviality of $V$. Further, we demonstrate that an affine Killing $m$-modified conformal vector field on a non-compact Riemannian manifold $M$ must be trivial. Finally, we show that an $m$-modified gradient conformal vector field is trivial under the assumptions of polynomial volume growth and convergence to zero at infinity.
{"title":"On The Triviality Of $m$-Modified Conformal Vector Fields","authors":"Rahul Poddar, Ramesh Sharma","doi":"arxiv-2409.07607","DOIUrl":"https://doi.org/arxiv-2409.07607","url":null,"abstract":"We prove that a compact Riemannian manifold $M$ does not admit any\u0000non-trivial $m$-modified homothetic vector fields. In the corresponding case of\u0000an $m$-modified conformal vector field $V$, we establish an inequality that\u0000implies the triviality of $V$. Further, we demonstrate that an affine Killing\u0000$m$-modified conformal vector field on a non-compact Riemannian manifold $M$\u0000must be trivial. Finally, we show that an $m$-modified gradient conformal\u0000vector field is trivial under the assumptions of polynomial volume growth and\u0000convergence to zero at infinity.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198969","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 renormalized Yang-Mills energy on six dimensional Poincar'e-Einstein spaces can be expressed as the bulk integral of a local, pointwise conformally invariant integrand. We show that the latter agrees with the corresponding anomaly boundary integrand in the seven dimensional renormalized Yang-Mills energy. Our methods rely on a generalization of the Chang-Qing-Yang method for computing renormalized volumes of Poincar'e-Einstein manifolds, as well as known scattering theory results for Schr"odinger operators with short range potentials.
{"title":"Renormalized Yang-Mills Energy on Poincaré-Einstein Manifolds","authors":"A. R. Gover, E. Latini, A. Waldron, Y. Zhang","doi":"arxiv-2409.06995","DOIUrl":"https://doi.org/arxiv-2409.06995","url":null,"abstract":"We prove that the renormalized Yang-Mills energy on six dimensional\u0000Poincar'e-Einstein spaces can be expressed as the bulk integral of a local,\u0000pointwise conformally invariant integrand. We show that the latter agrees with\u0000the corresponding anomaly boundary integrand in the seven dimensional\u0000renormalized Yang-Mills energy. Our methods rely on a generalization of the\u0000Chang-Qing-Yang method for computing renormalized volumes of\u0000Poincar'e-Einstein manifolds, as well as known scattering theory results for\u0000Schr\"odinger operators with short range potentials.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199034","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：