The almost rigidity of manifolds with lower bounds on Ricci curvature
The almost rigidity of manifolds with lower bounds on Ricci curvature
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具有里奇曲率下界的流形的近似刚性
DOI：
10.4310/cag.2000.v8.n1.a6
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
C. Sormani
中科院分区：
文献类型：
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作者：
C. Sormani
我们考虑具有二次衰减的Ricci曲率下界和最小体积增长的完备非紧黎曼流形。我们首先证明了一个刚性结果，证明了具有强最小体积增长的一端与翘曲的乘积流形等距。接下来，我们考虑几乎刚性的情形，其中具有非负且二次衰减的Ricci曲率下界的流形具有最小的体积增长。证明了这种流形中的紧致区域渐近于Gromov-Hausdorff拓扑中的翘曲积。证明了具有非负Ricci曲率和线性体积增长的流形具有渐近于等距积的区域。这些证明包括使用最近发展的Cheeger-Colding几乎刚性理论对这些流形上的Busemann函数进行仔细的分析。此外，我们还证明了这类流形中Busemann函数的水平集的直径是次线性增长的。
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology. Manifolds with nonnegative Ricci curvature and linear volume growth are shown to have regions which are asymptotically close to being isometric products. The proofs involve a careful analysis of the Busemann functions on these manifolds using the recently developed Cheeger-Colding Almost Rigidity Theory. In addition, we show that the diameters of the level sets of Busemann functions in such manifolds grow sublinearly.