Large eigenvalues and concentration

Abstract

Let Mn = (M, g) be a compact, connected, Riemannian manifold of dimension n. Let μ be the measure μ=σdvolg, where σ∈C∞(M) is a nonnegative density. We first show that, under some mild metric conditions that do not involve the curvature, the presence of a large eigenvalue (or more precisely of a large gap in the spectrum) for the Laplacian associated to the density σ on M implies a strong concentration phenomenon for the measure μ. When the density is positive, we show that our result is optimal. Then we investigate the case of a Laplace-type operator D = ▶ ▶+T on a vector bundle E over M, and show that the presence of a large gap between the (k+1)-st eigenvalue λk+1 and the k-th eigenvalue λk implies a concentration phenomenon for the eigensections associated to the eigenvalues λ1, . . ., λk of the operator D. © 2011 by Pacific Journal of Mathematics.