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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】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】The second pinching theorem for hypersurfaces with constant mean curvature in a sphere
H. W. Xu, Z. Y. Xu
Math. Ann.,2013,356(2):869-883
2013年08月20日
We prove the second pinching theorem for hypersurfaces with constant mean curvature in a sphere.
Hypersurfaces with constant mean curvature, the second pinching theorem, , the second fundamental form, Chern', s conjecture
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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】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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