On Extensions of Myers' Theorem
On Extensions of Myers' Theorem
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DOI：
10.1112/blms/27.4.392
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发表时间：
1995-07
影响因子：
0.9
通讯作者：
Xue-Mei Li
中科院分区：
文献类型：
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作者：
Xue-Mei Li
设M是紧致黎曼流形，h是M上的光滑函数.设ρh（x）= inf| v| =1（Ricx（v，v）− 2Hess（h）x（v，v））。这里Ricx表示x处的Ricci曲率，Hess（h）是h的Hessian。则M有有限基本群，如果n h − ρh < 0。这里，h =：h + 2L h是Bismut-Witten Laplacian。这导致一个快速证明最近的结果扩展迈尔斯定理的流形大多是积极的曲率。对于非紧流形也有类似的结果。Myers的一个早期结果是，一个Ricci曲率下有界的完备黎曼流形是紧的，并且有有限的基本群。参见例如[9]。从那时起，人们一直在努力得到相同类型的结果，但允许曲率有点负（见Berard和Besson[2]）。Wu [12]证明了Myers定理成立，如果流形允许在一个小直径的集合上有负曲率，而Elworthy和Rosenberg [8]考虑了在一个小体积的集合上有负曲率的流形，随后是Rosenberg和Yang [10]最近的工作。本文利用Bakry [1]的方法得到了一个关于ρ（x）= inf的势核的结果|v| =1 Ricx（v，v），它给出了Myers定理扩展的最近结果的一个快速概率证明。这里Ricx表示x处的Ricci曲率。设M是完备黎曼流形，h是其上的光滑实值函数，假设Ric− 2 Hess（h）是下有界的，其中Hess（h）是由NATO合作研究赠款计划0232/87和SERC赠款GR/H67263部分支持的研究。1991年数学学科分类60 H30，53 C21
Let M be a compact Riemannian manifold and h a smooth function on M . Let ρh(x) = inf |v|=1 (Ricx(v, v)− 2Hess(h)x(v, v)). Here Ricx denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has finite fundamental group if ∆h − ρh < 0. Here ∆h =: ∆ + 2L∇h is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers’ theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds. An early result of Myers says a complete Riemannian manifold with Ricci curvature bounded below by a positive number is compact and has finite fundamental group. See e.g. [9]. Since then efforts have been made to get the same type of result but to allow a little bit of negativity of the curvature (see Berard and Besson[2]). Wu [12] showed that Myers’ theorem holds if the manifold is allowed to have negative curvature on a set of small diameter, while Elworthy and Rosenberg [8] considered manifolds with some negative curvature on a set of small volume, followed by recent work of Rosenberg and Yang [10]. We use a method of Bakry [1] to obtain a result given in terms of the potential kernel related to ρ(x) = inf |v|=1 Ricx(v, v), which gives a quick probabilistic proof of recent results on extensions of Myers’ theorem. Here Ricx denotes the Ricci curvature at x. Let M be a complete Riemannian manifold, and h a smooth real-valued function on it. Assume Ric−2Hess(h) is bounded from below, where Hess(h) ∗Research supported in part by NATO Collaborative Research Grants Programme 0232/87 and by SERC grant GR/H67263. 1991 Mathematical subject classification 60H30,53C21