Large manifolds with positive Ricci curvature
Large manifolds with positive Ricci curvature
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DOI：
10.1007/s002220050050
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发表时间：
1996-01
影响因子：
3.1
通讯作者：
T. Colding
中科院分区：
文献类型：
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作者：
T. Colding
本文的主要目的是证明一个接近单位n-球面的n维流形，其Ricci曲率大于或等于（n− 1）（在Gromov-Hausdorff拓扑中），其体积接近球面。这证明了[C1]中定理的匡威。也就是说，与[C1]一起，它表明一个Ricci曲率大于或等于（n− 1）的n-流形接近球面当且仅当体积接近球面的体积。特别地，通过[P]，这样的流形同胚于球面。此外，作为这个和[C1]结果的应用，我们证明了一个半径定理，即如果一个Ricci曲率大于或等于（n-1）的n-流形的半径几乎等于;那么它的体积接近于球的体积。为了得到这些结果，我们进一步发展和应用[C1]的估计。虽然[C1]中的主要关注点是大尺度几何形状，但本文的主要关注点是小尺度几何形状。让！n是n-球面Sn的体积，截面曲率为1。
The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n− 1) which is close (in the Gromov–Hausdorff topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n− 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n− 1) has radius almost equal to; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let! n be the volume of the round n-sphere, Sn; with sectional curvature one.