Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

Abstract

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the metric is generic for the $\Cl C^{k+2}$-strong topology, then the eigenvalues are distinct and their associated eigenfunctions are Morse. This generalizes to non-compact manifolds some arguments developped by K. Uhlenbeck. We deduce from this result that if $M^n$ has bounded geometry at order $k\geq\frac{n}{2}$ and has an isolated first eigenvalue for its Laplacian, then for any riemannian covering $p : M'\ra M$, we have $\lambda_0(M) = \sup_D \lambda_0(D)$, where $D\subset M'$ runs over all connected fundamental domains for $p$, and $\lambda_0(D)$ is the bottom of the spectrum of $D$ with Neumann boundary conditions.