arXiv 1102.24242011-02-11math.DG

具有正数量曲率流形的球面定理

The sphere theorems for manifolds with positive scalar curvature

Juan-Ru Gu, Hong-Wei Xu

备注：: 35 pages

摘要：

利用里奇流和稳定流得到了一些新的可微球面定理。证明了如果$M^n$是一个紧致流形，且其归一化数量曲率和截面曲率满足点态拼挤条件$R_0>σ_{n}K_{\max}$，其中$σ_n\in (\frac{1}{4},1)$是一个显式的正常数，则$M$是一个球面空间形式.这部分地回答了丘关于拼挤定理的猜想。证明了如果$M^n(n\geq3)$是一个紧致流形，且其$(n-2)$阶Ricci曲率和归一化数量曲率满足点态条件$Ric^{(n-2)}_{\min}>τ_n(n-2)R_0,$，其中$τ_n\in (\frac{1}{4},1)$是一个显式正常数，则$M$是球面空间形式的同构.然后，我们将上面的球面定理推广到黎曼流形中的子流形。最后给出了具有弱pinched曲率的子流形的一个分类，改进了Andrews，Baker等人的可微pinching定理。

英文摘要：

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n-2)}_{\min}>τ_n(n-2)R_0,$ where $τ_n\in (\frac{1}{4},1)$ 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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