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【期刊论文】A new characterization of the Clifford torus via scalar curvature pinching
H. W. Xu, Z. Y. Xu
J. Funct. Anal.,2014,267(3):3931-3962
2014年09月26日
We present a new characterization of the Clifford torus via scalar curvature pinching.
Hypersurfaces with constant mean curvature, Rigidity, Scalar curvature, Clifford torus
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【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
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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【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
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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【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
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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【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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