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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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【期刊论文】Rigidity of Einstein manifolds with positive scalar curvature
H. W. Xu, J. R. Gu
Math. Ann.,2014,358(2):169-193
2014年06月20日
The purpose of this paper is to prove some new rigidity theorems for Einstein manifolds and submanifolds.
Einstein manifolds, rigidity theorems, positive scalar curvature
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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 sphere theorem for manifolds with positive scalar curvature
J. R. Gu, H. W. Xu
Journal of Differential Geometry,2012,92(3):507-545
2012年05月17日
We prove some new differentiable sphere theorems via the Ricci flow and stable currents. We extend the sphere theorems in Riemannian geometry to submanifolds in a Riemannian manifold. We give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors. We also show that the Yau conjecture is false.
Differentiable sphere theorem, Ricci flow, stable currents, the Yau conjecture
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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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