Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

Details Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

2010-01-14

arxiv.org/abs/1001.2506v1

Mathematics

Differential Geometry

33 pages, 1 figure

Scientific paper

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.

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