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 study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that t...In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.展开更多
Given a projective map F: M→N of a complete Riemannian manifold to a Riemannianmanifold with the sectional curvature bounded above by a negative constant, we prove that f decreases volume up to a constant depending ...Given a projective map F: M→N of a complete Riemannian manifold to a Riemannianmanifold with the sectional curvature bounded above by a negative constant, we prove that f decreases volume up to a constant depending only on the curvatures of M and N. This generalizes theresult due to Har’el.展开更多
It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses...It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses the volume growth of a manifold with asymptotically nonnegative Ricci curvature.展开更多
In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is ...In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau＇s theorem.展开更多
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, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nif...In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.展开更多
In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
In this paper we prove that tile set of Riemannian manifolds with parallel Ricci curvature, lower bounds for sectional curvature and injectivity radius and a upper bound for volume is coo compact in Gromov-Hausdroff t...In this paper we prove that tile set of Riemannian manifolds with parallel Ricci curvature, lower bounds for sectional curvature and injectivity radius and a upper bound for volume is coo compact in Gromov-Hausdroff topology. As an application we also prove a pinching result which states that a Ricci flat manifold is flat under certain conditions.展开更多
The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space.This formula generalizes the volume of classical domains a...The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space.This formula generalizes the volume of classical domains as tubes or domains given as motions along the submanifold.Finally,some stereological considerations regarding this formula are provided.展开更多
A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrins...A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.展开更多
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 study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.
文摘Given a projective map F: M→N of a complete Riemannian manifold to a Riemannianmanifold with the sectional curvature bounded above by a negative constant, we prove that f decreases volume up to a constant depending only on the curvatures of M and N. This generalizes theresult due to Har’el.
文摘It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses the volume growth of a manifold with asymptotically nonnegative Ricci curvature.
基金Supported by NSFC (10401042)Foundation of Department of Education of Zhejiang Province.
文摘In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau＇s theorem.
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Project supported by the National Natural Science Foundation of China(Nos.10971055,11171096)the Research Fund for the Doctoral Program of Higher Education of China(No.20104208110002)the Funds for Disciplines Leaders of Wuhan(No.Z201051730002)
文摘In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.
基金supported by JSPS Grant-in-Aid for Scientific Research(B)(No.16H03937)the fund of Fukuoka University(No.225001)+2 种基金the National Natural Science Foundation of China(No.12171164)the Natural Science Foundation of Guangdong Province(No.2019A1515011451)GDUPS(2018)。
文摘In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
基金Supported by National Natural Science Foundation of China (19971081)
文摘In this paper we prove that tile set of Riemannian manifolds with parallel Ricci curvature, lower bounds for sectional curvature and injectivity radius and a upper bound for volume is coo compact in Gromov-Hausdroff topology. As an application we also prove a pinching result which states that a Ricci flat manifold is flat under certain conditions.
基金Project supported by the Spanish Ministry of Science and Technology Grants MTM2005-O8689-G02-02 and MTM 2004-06015-C02-01.
文摘The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space.This formula generalizes the volume of classical domains as tubes or domains given as motions along the submanifold.Finally,some stereological considerations regarding this formula are provided.
基金This work was supported by grants:VEGA 1/0269/09,APVV-0351-07,APVV-RPEU-0004-07(K.Mikula and M.Balazovjech)and APVV-0247-06(D.Sevcovic).
文摘A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.
