Some new results on eigenvectors via dimension, diameter, and Ricci curvature

被引:103
作者
Bakry, D
Qian, ZM
机构
[1] CNRS, F-31062 Toulouse, France
[2] Univ Toulouse 3, F-31062 Toulouse, France
[3] Univ London Imperial Coll Sci Technol & Med, Dept Math, London SW7 2BZ, England
[4] Shanghai Tie Dao Univ, Shanghai, Peoples R China
关键词
D O I
10.1006/aima.2000.1932
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We generalise for a general symmetric elliptic operator the different notions of dimension. diameter, and Ricci curvature. which coincide with the usual notions in the case of the Laplace-Beltrami operators on Riemannian manifolds. If lambda (1) denotes the spectral gap, that is the: first nonzero eigenvalue, we investigate in this paper the best lower bound on , one can obtain under an upper bound on the dimension. an upper bound on the diameter. and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R > 0, then lambda (1) greater than or equal to nR/(n - 1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong-Yang's estimate asserts that lambda (1) greater than or equal to pi /d, where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for lambda (1), which unifies and sharpens Lichnerowicz and Zhong-Yang's estimates. together with other comparison results concerning the range of the associated eigenfunctions and their derivatives. (C) 2000 Academic Press.
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页码:98 / 153
页数:56
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