29 may 2017 -- 14:30
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
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 $M^m \rightarrow N^n$ 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 of $\varphi$ and the behaviour at infinity of the density function $\Theta(r)=Vol(M \cap B_r^n)/Vol(B_r^m)$ , where $B_r^n$ and $B_r^m$ are geodesic balls of radius $r$ in $N^n$ and $M^m$, respectively. This is based on joint works with G. Pacelli Bessa, L.P. Jorge, J.F. Montenegro, B.P. Lima, F.B. Vieira.