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.
LARGE EIGENVALUES AND CONCENTRATION / Bruno, Colbois; Savo, Alessandro. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - 249:2(2011), pp. 271-290. [10.2140/pjm.2011.249.271]
LARGE EIGENVALUES AND CONCENTRATION
SAVO, Alessandro
2011
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.