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2019年05月17日

【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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2019年05月20日

【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

摘要

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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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

摘要

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20日

【期刊论文】Geometric,topological and differentiable rigidity of submanifolds in space forms

H. W. Xu， J. R. Gu

Geom. Funct. Anal.，2013，23（1）：1684-1703

2013年06月20日

摘要

In this paper, the authors investigate rigidity of geometric, topological and differentiable structures of compact submanifolds in a space form.

Submanifolds,， rigidity and sphere theorems,， Ricci curvature,， Ricci flow,， stable currents

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2019年05月20日

【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

摘要

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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