The sphere theorems for manifolds with positive scalar curvature

Juan-Ru Gu Zhejiang University Hong-Wei Xu Zhejiang University

Differential Geometry mathscidoc:1609.10291

Journal of Differential Geometry, 92, (3), 507-545, 2012
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0 > nKmax, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if Mn(n  3) is a compact manifold whose (n . 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric(n.2) min > n(n .2)R0, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.
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@inproceedings{juan-ru2012the,
  title={THE SPHERE THEOREMS FOR MANIFOLDS WITH POSITIVE SCALAR CURVATURE},
  author={Juan-Ru Gu, and Hong-Wei Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914072739140727953},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={3},
  pages={507-545},
  year={2012},
}
Juan-Ru Gu, and Hong-Wei Xu. THE SPHERE THEOREMS FOR MANIFOLDS WITH POSITIVE SCALAR CURVATURE. 2012. Vol. 92. In Journal of Differential Geometry. pp.507-545. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914072739140727953.
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