Large eigenvalues and concentration

Bruno Colbois and Alessandro Savo

Vol. 249 (2011), No. 2, 271–290

DOI: 10.2140/pjm.2011.249.271

Abstract

Let Mⁿ = (M,g) be a compact, connected, Riemannian manifold of dimension n. Let μ be the measure μ = σ dvol_g, 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 λ_k+1 and the k-th eigenvalue λ_k implies a concentration phenomenon for the eigensections associated to the eigenvalues λ₁,…,λ_k of the operator D.

Keywords

eigenvalues, upper bounds, Laplace-type operators, concentration

Mathematical Subject Classification 2000

Primary: 58J50

Secondary: 35P15

Milestones

Received: 14 December 2009

Accepted: 2 April 2010

Published: 1 February 2011