Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature

Abstract

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 λ1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on λ1 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 λ1⩾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 λ1⩾π2d2, 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 λ1 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.