An Optimal Differentiable Sphere Theorem for Complete Manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2010
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| Acceso en línea: | |
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| _version_ | 1866913652235304960 |
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| author | Xu, Hong-Wei Gu, Juan-Ru |
| author_facet | Xu, Hong-Wei Gu, Juan-Ru |
| contents | A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1005_2557 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | An Optimal Differentiable Sphere Theorem for Complete Manifolds Xu, Hong-Wei Gu, Juan-Ru Differential Geometry A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$. |
| title | An Optimal Differentiable Sphere Theorem for Complete Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1005.2557 |