Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

TL;DR

This paper proves that on non-compact manifolds with bounded geometry, generic metrics yield distinct, Morse eigenfunctions for the Laplacian, and establishes a spectral equality for coverings under certain conditions.

Contribution

It extends Uhlenbeck's arguments to non-compact manifolds, showing generic metrics produce Morse eigenfunctions and relating the spectrum of coverings to fundamental domains.

Findings

Eigenvalues are distinct and eigenfunctions are Morse for generic metrics.

Spectral properties are preserved under Riemannian coverings with isolated first eigenvalue.

The bottom of the spectrum on coverings equals the supremum over fundamental domains.

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.