Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"
Abstract
Description
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 $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs over $C$. First, we give a sharp universal lower bound for $λ_k ^c (C)$ extending to all $k$ a result obtained by Friedlander and Nadirashvili for $k=1$. Then, we show that the sequence $ \{λ_k ^c (C) \}$, that we call "conformal spectrum", is strictly increasing and satisfies, $\forall k\geq 0$, $λ_{k+1} ^c (C)^{n/2} - λ_k ^c (C)^{n/2} \geq n^{n/2} ω_n $, where $ω_n $ is the volume of the $n$-dimensional standard sphere. When $M$ is an orientable surface of genus $γ$, we also consider the supremum $λ_k ^{top} (γ)$ of $λ_k(g)$ over the set of all the area one Riemannian metrics on $M$, and study the behavior of $λ_k ^{top} (γ)$ in terms of $γ$.