In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is pr...In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.展开更多
In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these fac...In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these facts,we prove that the solution exists for all time and the principal curvatures converge to 1 polynomially fast.展开更多
Assume M is a closed 3-manifold whose universal covering is not S^3.We show that the obstruction to extend the Ricci flow is the boundedness L 3/2-norm of the scalar curvature R(t),i.e.,the Ricci flow can be extended ...Assume M is a closed 3-manifold whose universal covering is not S^3.We show that the obstruction to extend the Ricci flow is the boundedness L 3/2-norm of the scalar curvature R(t),i.e.,the Ricci flow can be extended over finite time T if and only if the||R(t)||L 3/2 is uniformly bounded for 0≤t<T.On the other hand,if the fundamental group of M is finite and the||R(t)||L 3/2 is bounded for all time under the Ricci flow,then M is diffeomorphic to a 3-dimensional spherical space-form.展开更多
基金the National Natural Science Foundation of China(No.10531090).
文摘In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.
基金supported by National Natural Science Foundation of China(Grant No.11831005)a collaboration project funded by National Natural Science Foundation of China and the Research Foundation Flanders(Grant No.11961131001)。
文摘In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these facts,we prove that the solution exists for all time and the principal curvatures converge to 1 polynomially fast.
基金The author would like to express his gratitude to X.Chen who brought this problem to his attention and provided many helpful and stimulating discussions.He is very grateful of V.Apostolov’s detailed suggestions for this paper.He also would like to thank H.Li for discussing and reviewing the paper and R.Haslhofer for useful comments.
文摘Assume M is a closed 3-manifold whose universal covering is not S^3.We show that the obstruction to extend the Ricci flow is the boundedness L 3/2-norm of the scalar curvature R(t),i.e.,the Ricci flow can be extended over finite time T if and only if the||R(t)||L 3/2 is uniformly bounded for 0≤t<T.On the other hand,if the fundamental group of M is finite and the||R(t)||L 3/2 is bounded for all time under the Ricci flow,then M is diffeomorphic to a 3-dimensional spherical space-form.
