A new proof of a theorem of Petersen  

作　　者：YANG YiHu ZHANG Yi 

机构地区：[1]Department of Mathematics, Shanghai Jiao Tong University [2]Department of Mathematics, Tongji University

出　　处：《Science China Mathematics》2016年第5期935-944,共10页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171253)

摘　　要：Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S^n) → 0; 2) the volume of M Vol(M) → Vol(S^n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications.Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding（1996） proved that the following three conditions are equivalent： 1）dGH（M, S^n） → 0; 2） the volume of M Vol（M） → Vol（S^n）; 3） the radius of M rad（M） →π. By developing a different technique, Petersen（1999） gave the 4th equivalent condition, namely he proved that the n ＋ 1-th eigenvalue of M, λ_（n＋1）（M） → n, is also equivalent to the radius of M, rad（M） →π, and hence the other two.In this paper, we use Colding＇s techniques to give a new proof of Petersen＇s theorem. We expect our estimates will have further applications.

关 键 词：radius eigenvalues Gromov-Hausdorff distance 

分 类 号：O186.12[理学—数学]