This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eig...This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eigenvalue of M. Then there exists a constant Cm dependent only on m such that展开更多
Let Mn be an n-dimensional compact minimal submanifolds in Sin（1）× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize...Let Mn be an n-dimensional compact minimal submanifolds in Sin（1）× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.展开更多
In this paper,we study a class of Finsler metrics defined by a vector field on a gradient Ricci soliton.We obtain a necessary and sufficient condition for these Finsler metrics on a compact gradient Ricci soliton to b...In this paper,we study a class of Finsler metrics defined by a vector field on a gradient Ricci soliton.We obtain a necessary and sufficient condition for these Finsler metrics on a compact gradient Ricci soliton to be of isotropic S-curvature by establishing a new integral inequality.Then we determine the Ricci curvature of navigation Finsler metrics of isotropic S-curvature on a gradient Ricci soliton generalizing result only known in the case when such soliton is of Einstein type.As its application,we obtain the Ricci curvature of all navigation Finsler metrics of isotropic S-curvature on Gaussian shrinking soliton.展开更多
We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curv...We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.展开更多
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
基金Partially supported by the National Natural Science Foundation of China
文摘This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eigenvalue of M. Then there exists a constant Cm dependent only on m such that
基金supported by National Natural Science Foundation of China (Grant No.11271214)
文摘Let Mn be an n-dimensional compact minimal submanifolds in Sin（1）× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.
基金Supported by the National Natural Science Foundation of China(11371386)the European Union’s Seventh Framework Programme(FP7/2007-2013)under grant agreement(317721)
基金Supported by the National Natural Science Foundation of China(11771020,12171005).
文摘In this paper,we study a class of Finsler metrics defined by a vector field on a gradient Ricci soliton.We obtain a necessary and sufficient condition for these Finsler metrics on a compact gradient Ricci soliton to be of isotropic S-curvature by establishing a new integral inequality.Then we determine the Ricci curvature of navigation Finsler metrics of isotropic S-curvature on a gradient Ricci soliton generalizing result only known in the case when such soliton is of Einstein type.As its application,we obtain the Ricci curvature of all navigation Finsler metrics of isotropic S-curvature on Gaussian shrinking soliton.
基金the National Natural Science Foundation of China (10471001)
文摘We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.
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.
