Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
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DOI：
10.4007/annals.2012.176.2.10
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发表时间：
2011-02
影响因子：
4.9
通讯作者：
T. Colding;A. Naber
中科院分区：
文献类型：
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作者：
T. Colding;A. Naber
我们证明了对具有下里奇界的流形的新估计，该估计断言以最小化测地线为中心的球的几何形状最多可以沿测地线以 Holder 连续方式变化。我们给出的例子表明，Holder 指数以及从该估计得出的基本上所有其他结果都是尖锐的。其中一个应用是，正则集对于爱因斯坦度量的任何非折叠极限都是凸的。在流形的潜在崩溃极限仅具有下里奇曲率界的一般情况下，我们证明正则集是弱凸的并且 a.e.凸的。我们还展示了 Cheeger-Colding 的两个猜想。其中之一断言，任何具有均匀里奇曲率下界的流形极限（甚至是折叠的极限）的等距群都是李群。另一个断言任何极限空间的维度在任何地方都是相同的。
We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.