LARGE EIGENVALUES AND CONCENTRATION

Let M(n) = (M, g) be a compact, connected, Riemannian manifold of dimension n. Let mu be the measure mu=sigma dvol(g), where sigma is an element of C(infinity) (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 sigma on M implies a strong concentration phenomenon for the measure mu. When the density is positive, we show that our result is optimal. Then we investigate the case of a Laplace-type operator D = del*del + T on a vector bundle E over M, and show that the presence of a large gap between the (k+1)-st eigenvalue lambda(k+1) and the k-th eigenvalue lambda(k) implies a concentration phenomenon for the eigensections associated to the eigenvalues lambda(1), . . . . , lambda(k) of the operator D.