Vol. 249, No. 2, 2011

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Large eigenvalues and concentration

Bruno Colbois and Alessandro Savo

Vol. 249 (2011), No. 2, 271–290
Abstract

Let Mn = (M,g) be a compact, connected, Riemannian manifold of dimension n. Let μ be the measure μ = σ dvolg, 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,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
Authors
Bruno Colbois
Université de Neuchâtel
Institut de Mathématiques
Rue Emile Argand 11
CH-2007, Neuchâtel
Switzerland
Alessandro Savo
Dipartimento di Scienze di Base e Applicate per l’Ingegneria
Sapienza Università di Roma
Via Antonio Scarpa 16
00161 Roma
Italy