An Optimal Differentiable Sphere Theorem for Complete Manifolds
An Optimal Differentiable Sphere Theorem for Complete Manifolds
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DOI：
10.4310/mrl.2010.v17.n6.a10
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发表时间：
2010-05
期刊：
影响因子：
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通讯作者：
Hong Gu
中科院分区：
文献类型：
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作者：
Hong Gu
从子流形几何的角度得到了一个新的可微球定理。一个重要的标量是由具有$c0$的空间形式$F^{n+p}(C)$中的定向完备子流形$M^n$的标量曲率和平均曲率定义的。利用Ricci流的Hamilton-Brendle-Schoen收敛结果和不存在稳定流的Lawson-Simons-Xin公式，证明了如果这个标量的下确界为正，则$M$微分同胚于$S^n$.然后，我们通过标量引进了定向完备黎曼流形$M$的内在不变量$I(M)$，并证明了如果$I(M)和GT；0$，则$M$微分同胚于$S^n$。需要强调的是，我们的可微球定理对于任意的$n(2)$是最优的。
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 $n(\ge2)$.