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 $k\geq0$, 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, $\forall k\geq 0$, $\lambda_{k+1} ^c (C)^{n/2} - \lambda_k ^c (C)^{n/2} \geq 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 $\gamma$.