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2019年05月21日

【期刊论文】The topological sphere theorem for complete submanifolds

K. Shiohama， H. W. Xu

Compositio Math.，1997，107（2）：221-232

1997年02月01日

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold M in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of M has a positive lower bound. Making use of the Lawson-Simons formula for the nonexistence of stable k -currents, we eliminate the k-th homology group for all 1<k<n-1. We then observe that the fundamental group of M is trivial. It should be emphasized that our result is optimal.

the second fundamental form,， Ricci curvature,， integral homology,， stable currents

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2019年05月21日

【期刊论文】On closed minimal submanifolds in pinched Riemannian manifolds

H. W. Xu

Trans. Amer. Math. Soc.，1995，347（2）：1743-1751

1995年05月01日

摘要

In this paper, we prove the generalized Simons-Chern-do Carmo-Kobayashi-Lawson theorem for closed minimal submanifolds in pinched Riemannian manifolds.

closed minimal submanifolds， rigidity theorem， pinched Riemannian manifolds

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2019年05月21日

【期刊论文】The extension and convergence of mean curvature flow in higher comdimsion

K. F. Liu， H. W. Xu， F. Ye， E. T. Zhao

Trans. Amer. Math. Soc.，2018，370（3）：2231-2262

2018年03月01日

摘要

In this paper, we investigate the convergence of the mean curvature flow of closed submanifolds in Euclidean space . We show that if the initial submanifold satisfies some suitable integral curvature conditions, then along the mean curvature flow it will shrink to a round point in finite time.

Mean curvature flow,， submanifold,， maximal existence time,， convergence theorem,， integral curvature

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2019年05月20日

【期刊论文】On Chern's conjecture for minimal hypersurfaces and rigidity of self-shrinkers

H. W. Xu， Z. Y. Xu， H. W. Xu， Z. Y. Xu

J. Funct. Anal.，2017，273（3）：3406-3425

2017年09月26日

摘要

In this paper, we first give a refined version of Ding–Xin's rigidity theorem for minimal hypersurfaces in a sphere. We then improve Ding–Xin's rigidity theorem for self-shrinkers in the Euclidean space.

Chern conjecture for minimal hypersurfaces， Rigidity theorem， The second， fundamental form， Self-shrinker

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