Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"

Abstract

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\ge 0$, we consider the conformal invariant ${\lambda }_k^c(C)$ defined as the supremum of the $k$-th eigenvalue ${\lambda }_k(g)$ of the Laplace-Beltrami operator ${\Delta }_g$, where $g$ runs over $C$. First, we give a sharp universal lower bound for ${\lambda }_k^c(C)$ extending to all $k$ a result obtained by Friedlander and Nadirashvili for $k=1$. Then, we show that the sequence $\{{\lambda }_k^c(C)\}$, that we call "conformal spectrum", is strictly increasing and satisfies, $\forall k\ge 0$, ${\lambda }_{k+1}^c(C{)}^{n/2}-{\lambda }_k^c(C{)}^{n/2}\ge n^{n/2}{\omega }_n$, where ${\omega }_n$ is the volume of the $n$-dimensional standard sphere. When $M$ is an orientable surface of genus $\gamma$, we also consider the supremum ${\lambda }_k^{top}(\gamma )$ of ${\lambda }_k(g)$ over the set of all the area one Riemannian metrics on $M$, and study the behavior of ${\lambda }_k^{top}(\gamma )$ in terms of $\gamma$.