Séance Séminaire

Séminaire Gaston Darboux

A survey on the spectrum of Riemannian manifolds and their minimal submanifolds

In this talk, I will describe some classical and recent results about the spectrum of the Laplace-Beltrami operator on Riemannian manifolds, focusing on noncompact ones. After an overview of the interplay between curvature and spectrum in the intrinsic case, I will then consider minimal immersions Mm -> Nn in Euclidean or hyperbolic space, and show some new criteria to ensure that the Laplace-Beltrami operator of M has purely discrete (respectively, purely essential) spectrum, addressing a question posed by S.T. Yau. The geometric conditions involve the Hausdorff dimension of the limit set and the behaviour at infinity of the density function theta(r) = vol(M \cap Bnr)/vol(Bmr), where Bnr, Bmr are geodesic balls of radius r in Nn and Nm, respectively. This is based on joint works with G.Pacelli Bessa, L.P. Jorge, J.F. Montenegro, B.P. Lima, F.B. Vieira.