Pub Date : 2021-07-19DOI: 10.1142/s0129167x22500471
M. Cheng, Man-Chun Lee, Luen-Fai Tam
Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)ge sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $leq 2pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^infty$ metrics with isolated point singularities.
{"title":"Singular Metrics with Negative Scalar Curvature","authors":"M. Cheng, Man-Chun Lee, Luen-Fai Tam","doi":"10.1142/s0129167x22500471","DOIUrl":"https://doi.org/10.1142/s0129167x22500471","url":null,"abstract":"Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)ge sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $leq 2pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^infty$ metrics with isolated point singularities.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73268099","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}
Pub Date : 2021-02-20DOI: 10.7546/CRABS.2021.01.02
G. Nakova
The present paper is a continuation of our previous work, where a class of half lightlike submanifolds of almost contact B-metric manifolds was introduced. We study curvature properties of totally and screen totally umbilical such submanifolds as well as of the corresponding semi-Riemannian submanifolds with respect to the associated B-metric.
{"title":"Totally Umbilical Radical Screen Transversal Half Lightlike Submanifolds of Almost Contact B-metric Manifolds","authors":"G. Nakova","doi":"10.7546/CRABS.2021.01.02","DOIUrl":"https://doi.org/10.7546/CRABS.2021.01.02","url":null,"abstract":"The present paper is a continuation of our previous work, where a class of half lightlike submanifolds of almost contact B-metric manifolds was introduced. We study curvature properties of totally and screen totally umbilical such submanifolds as well as of the corresponding semi-Riemannian submanifolds with respect to the associated B-metric.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81095433","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, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.
{"title":"Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition","authors":"Keita Kunikawa, Y. Sakurai","doi":"10.1090/proc/15768","DOIUrl":"https://doi.org/10.1090/proc/15768","url":null,"abstract":"In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74307663","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}
Pub Date : 2020-12-16DOI: 10.1142/9789811273230_0010
Christian Baer, B. Hanke
Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.
{"title":"Boundary Conditions for Scalar Curvature","authors":"Christian Baer, B. Hanke","doi":"10.1142/9789811273230_0010","DOIUrl":"https://doi.org/10.1142/9789811273230_0010","url":null,"abstract":"Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82114301","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}
Pub Date : 2020-12-08DOI: 10.21685/2072-3040-2021-1-5
K. I. Sheina
The basic automorphism group ${A}_B(M,F)$ of a Cartan foliation $(M, F)$ is the quotient group of the automorphism group of $(M, F)$ by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates of the dimension of these groups are obtained. For some class of Cartan foliations with integrable an Ehresmann connection, a method for finding of basic automorphism groups is specified.
{"title":"Basic automorphisms of cartan foliations covered by fibrations","authors":"K. I. Sheina","doi":"10.21685/2072-3040-2021-1-5","DOIUrl":"https://doi.org/10.21685/2072-3040-2021-1-5","url":null,"abstract":"The basic automorphism group ${A}_B(M,F)$ of a Cartan foliation $(M, F)$ is the quotient group of the automorphism group of $(M, F)$ by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates of the dimension of these groups are obtained. For some class of Cartan foliations with integrable an Ehresmann connection, a method for finding of basic automorphism groups is specified.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"219 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74661914","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}
Consider a compact manifold $M$ with smooth boundary $partial M$. Suppose that $g$ and $tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $partial M$ and agrees with $tilde{g}$ in a neighborhood of $partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.
{"title":"Positivity of Curvature On Manifolds With Boundary","authors":"Tsz-Kiu Aaron Chow","doi":"10.1093/IMRN/RNAB071","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB071","url":null,"abstract":"Consider a compact manifold $M$ with smooth boundary $partial M$. Suppose that $g$ and $tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $partial M$ and agrees with $tilde{g}$ in a neighborhood of $partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74052148","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}
Pub Date : 2020-12-01DOI: 10.7546/jgsp-57-2020-45-85
C. Marle
Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold were studied, and their possible applications in Physics and Cosmology were considered, by the French mathematician and physicist Jean-Marie Souriau. They are presented here with detailed proofs of all the stated results. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar'e disk and the Poincar'e half-plane.
{"title":"On Gibbs States of Mechanical Systems with Symmetries","authors":"C. Marle","doi":"10.7546/jgsp-57-2020-45-85","DOIUrl":"https://doi.org/10.7546/jgsp-57-2020-45-85","url":null,"abstract":"Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold were studied, and their possible applications in Physics and Cosmology were considered, by the French mathematician and physicist Jean-Marie Souriau. They are presented here with detailed proofs of all the stated results. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar'e disk and the Poincar'e half-plane.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88466909","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 first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.
{"title":"Perelman-type no breather theorem for noncompact Ricci flows","authors":"Liang Cheng, Yongjia Zhang","doi":"10.1090/TRAN/8436","DOIUrl":"https://doi.org/10.1090/TRAN/8436","url":null,"abstract":"In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80968040","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}
Pub Date : 2020-11-24DOI: 10.1142/s0129167x22500422
U. Semmelmann, Changliang Wang, McKenzie Y. Wang
In this article we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${rm G}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly we prove that nearly parallel $rm G_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space ${rm SO}(5)/{rm SO}(3)_{irr} $ which is a $7$-dimensional homology sphere with a proper nearly parallel ${rm G}_2$ structure.
{"title":"Linear instability of Sasaki Einstein and nearly parallel G2 manifolds","authors":"U. Semmelmann, Changliang Wang, McKenzie Y. Wang","doi":"10.1142/s0129167x22500422","DOIUrl":"https://doi.org/10.1142/s0129167x22500422","url":null,"abstract":"In this article we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${rm G}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly we prove that nearly parallel $rm G_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space ${rm SO}(5)/{rm SO}(3)_{irr} $ which is a $7$-dimensional homology sphere with a proper nearly parallel ${rm G}_2$ structure.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84781165","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 smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin-manifolds of dimension six or at least ten which admit such a metric and are a connected sum of some manifold and $S^n times S^n$ or $S^n times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order which lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.
{"title":"Bundles with Non-multiplicativeÂ-Genus and Spaces of Metrics with Lower Curvature Bounds","authors":"Georg Frenck, Jens Reinhold","doi":"10.1093/IMRN/RNAA361","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA361","url":null,"abstract":"We construct smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin-manifolds of dimension six or at least ten which admit such a metric and are a connected sum of some manifold and $S^n times S^n$ or $S^n times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order which lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86060733","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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