In this paper we derive a relative volume comparison of Ricci flow under a certain local curvature condition.It is a refinement of Perelman’s no local collapsing theorem in Perelman(2002).
Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under th...Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.展开更多
In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison th...In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.展开更多
In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|...In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.展开更多
The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.
In this paper,the author considers a class of complete noncompact Riemannian manifoldswhich satisfy certain conditions on Ricci curvature and volume comparison. It is shown thatany harmonic map with finite energy from...In this paper,the author considers a class of complete noncompact Riemannian manifoldswhich satisfy certain conditions on Ricci curvature and volume comparison. It is shown thatany harmonic map with finite energy from such a manifold M into a normal geodesic ball inanother manifold N must be asymptotically constant at the infinity of each large end of M. Arelated existence theorem for harmonic maps is established.展开更多
In this paper,we show that for an Sp(k+1)-invariant metric g on S^(4k+3)(k 1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S^(4k+3),[?])as its conformal infinity is unique up to isom...In this paper,we show that for an Sp(k+1)-invariant metric g on S^(4k+3)(k 1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S^(4k+3),[?])as its conformal infinity is unique up to isometry.Moreover,by the result in Li et al.(2017),g is the Graham-Lee metric(see Graham and Lee(1991))on the unit ball B_(1)■R^(4k+4).We also give an a priori estimate of the Einstein metric g.As a byproduct of the a priori estimates,based on the estimate and Graham-Lee and Lee's seminal perturbation results(see Graham and Lee(1991)and Lee(2006)),we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity(S^(4k+3),[g])when the metric?is Sp(k+1)-invariant.We also generalize the results to the case of conformal infinity(S^(15),[?])with g a Spin(9)-invariant metric in the appendix.展开更多
We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically local...We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic.In this context,our result requires only a partial C2 decay of the metric,namely the full decay of the metric in C1 and the decay of the scalar curvature.In particular,no decay of the Ricci curvature is assumed.In our second result we establish that a local volume ratio condition,when combined with negativity of the scalar curvature at infinity,is sufficient for existence of a solution.Our volume ratio condition appears tight.This paper is based on the DPhil thesis of thefirst author.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11331001 and 11890661)supported by Beijing Natural Science Foundation(Grant No.Z180004)+1 种基金National Natural Science Foundation of China(Grant No.11825105)Ministry of Education of China(Grant No.161001)。
文摘In this paper we derive a relative volume comparison of Ricci flow under a certain local curvature condition.It is a refinement of Perelman’s no local collapsing theorem in Perelman(2002).
基金supported by NSFC (10831008)NKBRPC(2006CB805905)
文摘Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
文摘In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.
基金supported by National Natural Science Foundation of China (Grant No. 11271343)
文摘In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.
基金supported by the National Natural Science Foundation of China (No. 10671066)the Scientific Research Foundation of Qufu Normal University and the Shanghai and Shandong Priority Academic Discipline
文摘The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.
文摘In this paper,the author considers a class of complete noncompact Riemannian manifoldswhich satisfy certain conditions on Ricci curvature and volume comparison. It is shown thatany harmonic map with finite energy from such a manifold M into a normal geodesic ball inanother manifold N must be asymptotically constant at the infinity of each large end of M. Arelated existence theorem for harmonic maps is established.
基金supported by National Natural Science Foundation of China(Grant No.11701326)the Fundamental Research Funds of Shandong University(Grant No.2016HW008)the Young Scholars Program of Shandong University(Grant No.2018WLJH85)。
文摘In this paper,we show that for an Sp(k+1)-invariant metric g on S^(4k+3)(k 1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S^(4k+3),[?])as its conformal infinity is unique up to isometry.Moreover,by the result in Li et al.(2017),g is the Graham-Lee metric(see Graham and Lee(1991))on the unit ball B_(1)■R^(4k+4).We also give an a priori estimate of the Einstein metric g.As a byproduct of the a priori estimates,based on the estimate and Graham-Lee and Lee's seminal perturbation results(see Graham and Lee(1991)and Lee(2006)),we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity(S^(4k+3),[g])when the metric?is Sp(k+1)-invariant.We also generalize the results to the case of conformal infinity(S^(15),[?])with g a Spin(9)-invariant metric in the appendix.
基金supported by the EPSRC Centre for Doctoral Training in Partial Differential Equations(grant number EP/L015811/1).
文摘We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic.In this context,our result requires only a partial C2 decay of the metric,namely the full decay of the metric in C1 and the decay of the scalar curvature.In particular,no decay of the Ricci curvature is assumed.In our second result we establish that a local volume ratio condition,when combined with negativity of the scalar curvature at infinity,is sufficient for existence of a solution.Our volume ratio condition appears tight.This paper is based on the DPhil thesis of thefirst author.
