In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded minimal tori. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory proved by the second author and X. Zhou.
{"title":"Existence of embedded minimal tori in three-spheres with positive Ricci curvature","authors":"Xingzhe Li, Zhichao Wang","doi":"arxiv-2409.10391","DOIUrl":"https://doi.org/arxiv-2409.10391","url":null,"abstract":"In this paper, we prove the strong Morse inequalities for the area functional\u0000in the space of embedded tori and spheres in the three sphere. As a\u0000consequence, we prove that in the three dimensional sphere with positive Ricci\u0000curvature, there exist at least 4 distinct embedded minimal tori. Suppose in\u0000addition that the metric is bumpy, then the three-sphere contains at least 9\u0000distinct embedded minimal tori. The proof relies on a multiplicity one theorem\u0000for the Simon-Smith min-max theory proved by the second author and X. Zhou.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254092","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 homogeneous space $M=Htimes H/Delta K$, where $H/K$ is an irreducible symmetric space and $Delta K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $Htimes H$-invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on $M$, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of $Htimes H$-invariant metrics and study the stability of non-diagonal Einstein metrics on $M$ with respect to the Hilbert action, obtaining that these metrics are unstable with different coindexes for all homogeneous spaces $M$.
{"title":"Stability of non-diagonal Einstein metrics on homogeneous spaces $Htimes H/ ΔK$","authors":"Valeria Gutiérrez","doi":"arxiv-2409.10686","DOIUrl":"https://doi.org/arxiv-2409.10686","url":null,"abstract":"We consider the homogeneous space $M=Htimes H/Delta K$, where $H/K$ is an\u0000irreducible symmetric space and $Delta K$ denotes diagonal embedding.\u0000Recently, Lauret and Will provided a complete classification of $Htimes\u0000H$-invariant Einstein metrics on M. They obtained that there is always at least\u0000one non-diagonal Einstein metric on $M$, and in some cases, diagonal Einstein\u0000metrics also exist. We give a formula for the scalar curvature of a subset of\u0000$Htimes H$-invariant metrics and study the stability of non-diagonal Einstein\u0000metrics on $M$ with respect to the Hilbert action, obtaining that these metrics\u0000are unstable with different coindexes for all homogeneous spaces $M$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254090","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 remove the assumption on the gradient of the Ricci curvature in Hamilton's matrix Harnack estimate for the heat equation on all closed manifolds, answering a question which has been around since the 1990s. New ingredients include a recent sharp Li-Yau estimate, construction of a suitable vector field and various use of integral arguments, iteration and a little tensor algebra.
{"title":"An improved Hamilton matrix estimates for the heat equation","authors":"Lang Qin, Qi S. Zhang","doi":"arxiv-2409.10379","DOIUrl":"https://doi.org/arxiv-2409.10379","url":null,"abstract":"In this paper, we remove the assumption on the gradient of the Ricci\u0000curvature in Hamilton's matrix Harnack estimate for the heat equation on all\u0000closed manifolds, answering a question which has been around since the 1990s.\u0000New ingredients include a recent sharp Li-Yau estimate, construction of a\u0000suitable vector field and various use of integral arguments, iteration and a\u0000little tensor algebra.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254093","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}
For the quaternionic unit ball $mathbb{B}$, let us denote by $mathcal{M}(mathbb{B})$ the set of slice regular M"obius transformations mapping $mathbb{B}$ onto itself. We introduce a smooth manifold structure on $mathcal{M}(mathbb{B})$, for which the evaluation(-action) map of $mathcal{M}(mathbb{B})$ on $mathbb{B}$ is smooth. The manifold structure considered on $mathcal{M}(mathbb{B})$ is obtained by realizing this set as a quotient of the Lie group $mathrm{Sp}(1,1)$, Furthermore, it turns out that $mathbb{B}$ is a quotient as well of both $mathcal{M}(mathbb{B})$ and $mathrm{Sp}(1,1)$. These quotients are in the sense of principal fiber bundles. The manifold $mathcal{M}(mathbb{B})$ is diffeomorphic to $mathbb{R}^4 times S^3$.
{"title":"Geometry of the slice regular Möbius transformations of the quaternionic unit ball","authors":"Raul Quiroga-Barranco","doi":"arxiv-2409.09897","DOIUrl":"https://doi.org/arxiv-2409.09897","url":null,"abstract":"For the quaternionic unit ball $mathbb{B}$, let us denote by\u0000$mathcal{M}(mathbb{B})$ the set of slice regular M\"obius transformations\u0000mapping $mathbb{B}$ onto itself. We introduce a smooth manifold structure on\u0000$mathcal{M}(mathbb{B})$, for which the evaluation(-action) map of\u0000$mathcal{M}(mathbb{B})$ on $mathbb{B}$ is smooth. The manifold structure\u0000considered on $mathcal{M}(mathbb{B})$ is obtained by realizing this set as a\u0000quotient of the Lie group $mathrm{Sp}(1,1)$, Furthermore, it turns out that\u0000$mathbb{B}$ is a quotient as well of both $mathcal{M}(mathbb{B})$ and\u0000$mathrm{Sp}(1,1)$. These quotients are in the sense of principal fiber\u0000bundles. The manifold $mathcal{M}(mathbb{B})$ is diffeomorphic to\u0000$mathbb{R}^4 times S^3$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254097","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}
Jaime Cuadros Valle, Ralph R. Gomez, Joe Lope Vicente
We use the Berglund-H"ubsch transpose rule from classical mirror symmetry in the context of Sasakian geometry and results on relative K-stability in the Sasaki setting developed by Boyer and van Coevering to exhibit examples of Sasaki manifolds of complexity 3 or complexity 4 that do not admit any extremal Sasaki metrics in its whole Sasaki-Reeb cone which is of Gorenstein type. Previously, examples with this feature were produced in by Boyer and van Coevering for Brieskorn-Pham polynomials or their deformations. Our examples are based on the more general framework of invertible polynomials. In particular, we construct families of examples of links with the following property: if the link satisfies the Lichnerowicz obstruction of Gauntlett, Martelli, Sparks and Yau then its Berglund-H"ubsch dual admits a perturbation in its local moduli, a link arising from a Brieskorn-Pham polynomial, which is obstructed to admitting extremal Sasaki metrics in its whole Sasaki-Reeb cone. Most of the examples produced in this work have the homotopy type of a sphere or are rational homology spheres.
{"title":"Non-existence of extremal Sasaki metrics via the Berglund-Hübsch transpose","authors":"Jaime Cuadros Valle, Ralph R. Gomez, Joe Lope Vicente","doi":"arxiv-2409.09720","DOIUrl":"https://doi.org/arxiv-2409.09720","url":null,"abstract":"We use the Berglund-H\"ubsch transpose rule from classical mirror symmetry in\u0000the context of Sasakian geometry and results on relative K-stability in the\u0000Sasaki setting developed by Boyer and van Coevering to exhibit examples of\u0000Sasaki manifolds of complexity 3 or complexity 4 that do not admit any extremal\u0000Sasaki metrics in its whole Sasaki-Reeb cone which is of Gorenstein type.\u0000Previously, examples with this feature were produced in by Boyer and van\u0000Coevering for Brieskorn-Pham polynomials or their deformations. Our examples\u0000are based on the more general framework of invertible polynomials. In\u0000particular, we construct families of examples of links with the following\u0000property: if the link satisfies the Lichnerowicz obstruction of Gauntlett,\u0000Martelli, Sparks and Yau then its Berglund-H\"ubsch dual admits a perturbation\u0000in its local moduli, a link arising from a Brieskorn-Pham polynomial, which is\u0000obstructed to admitting extremal Sasaki metrics in its whole Sasaki-Reeb cone.\u0000Most of the examples produced in this work have the homotopy type of a sphere\u0000or are rational homology spheres.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254094","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}
A Hermitian-symplectic metric is a Hermitian metric whose K"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be K"ahlerian (i.e., admitting a K"ahler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for some special types of compact Hermitian manifolds, including the Chern flat manifolds, non-balanced Bismut torsion parallel manifolds (which contains Vaisman manifolds as a subset), and quotients of Lie groups which are either almost ableian or whose Lie algebra contains a codimension $2$ abelian ideal that is $J$-invariant. The last case presents adequate algebraic complexity which illustrates the subtlety and intricacy of Streets-Tian Conjecture.
{"title":"Streets-Tian Conjecture on several special types of Hermitian manifolds","authors":"Yuqin Guo, Fangyang Zheng","doi":"arxiv-2409.09425","DOIUrl":"https://doi.org/arxiv-2409.09425","url":null,"abstract":"A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is\u0000given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states\u0000that a compact complex manifold admitting a Hermitian-symplectic metric must be\u0000K\"ahlerian (i.e., admitting a K\"ahler metric). The conjecture is known to be\u0000true in dimension $2$ but is still open in dimensions $3$ or higher. In this\u0000article, we confirm the conjecture for some special types of compact Hermitian\u0000manifolds, including the Chern flat manifolds, non-balanced Bismut torsion\u0000parallel manifolds (which contains Vaisman manifolds as a subset), and\u0000quotients of Lie groups which are either almost ableian or whose Lie algebra\u0000contains a codimension $2$ abelian ideal that is $J$-invariant. The last case\u0000presents adequate algebraic complexity which illustrates the subtlety and\u0000intricacy of Streets-Tian Conjecture.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254095","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 hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $triangle vec{H}=lambda vec{H}$ ($lambda$ a constant) in the pseudo-Euclidean space $mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such hypersurface in $mathbb{E}_{s}^{5}$ with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in $mathbb{E}_{s}^{5}$ with diagonal shape operator must be minimal.
{"title":"Hypersurfaces satisfying $triangle vec {H}=λvec {H}$ in $mathbb{E}_{lowercase{s}}^{5}$","authors":"Ram Shankar Gupta, Andreas Arvanitoyeorgos","doi":"arxiv-2409.08630","DOIUrl":"https://doi.org/arxiv-2409.08630","url":null,"abstract":"In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$\u0000satisfying $triangle vec{H}=lambda vec{H}$ ($lambda$ a constant) in the\u0000pseudo-Euclidean space $mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain\u0000that every such hypersurface in $mathbb{E}_{s}^{5}$ with diagonal shape\u0000operator has constant mean curvature, constant norm of second fundamental form\u0000and constant scalar curvature. Also, we prove that every biharmonic hypersurface in $mathbb{E}_{s}^{5}$\u0000with diagonal shape operator must be minimal.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254101","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 biconservative hypersurfaces $M$ in space forms $overline M^{n+1}(c)$ with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean curvature and constant scalar curvature.
{"title":"Biconservative hypersurfaces in space forms $overline{M}^{lowercase{n+1}}(lowercase{c})$","authors":"Ram Shankar Gupta, Andreas Arvanitoyeorgos","doi":"arxiv-2409.08593","DOIUrl":"https://doi.org/arxiv-2409.08593","url":null,"abstract":"In this paper we study biconservative hypersurfaces $M$ in space forms\u0000$overline M^{n+1}(c)$ with four distinct principal curvatures whose second\u0000fundamental form has constant norm. We prove that every such hypersurface has\u0000constant mean curvature and constant scalar curvature.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254660","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 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":"17 1","pages":""},"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}
We obtain a correspondence between irreducible real differential spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions to an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the K"ahler-Atiyah bundle of $(M,g)$. In particular, we obtain an intrinsic algebraic characterization of $mathrm{G}_2^*$-structures and we provide the first explicit characterization of isotropic irreducible spinors in signature $(4,3)$ parallel under a general connection on the spinor bundle, which we apply to the spinorial lift of metric connections with torsion.
{"title":"Differential spinors for $mathrm{G}_2^*$ and isotropic structures","authors":"C. S. Shahbazi, Alejandro Gil-García","doi":"arxiv-2409.08553","DOIUrl":"https://doi.org/arxiv-2409.08553","url":null,"abstract":"We obtain a correspondence between irreducible real differential spinors on\u0000pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions to an\u0000associated differential system for three-forms that satisfy a homogeneous\u0000algebraic equation of order two in the K\"ahler-Atiyah bundle of $(M,g)$. In\u0000particular, we obtain an intrinsic algebraic characterization of\u0000$mathrm{G}_2^*$-structures and we provide the first explicit characterization\u0000of isotropic irreducible spinors in signature $(4,3)$ parallel under a general\u0000connection on the spinor bundle, which we apply to the spinorial lift of metric\u0000connections with torsion.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254663","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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