On Eigenvalue Generic Properties of the Laplace-Neumann Operator

TL;DR

This paper proves that for most Riemannian metrics on a compact manifold with boundary, the eigenvalues of the Laplace-Neumann operator are simple, using a new analytic approach to study eigenvalue variation.

Contribution

It introduces a novel method for analyzing eigenvalue variations of the Laplace-Neumann operator, establishing generic simplicity of eigenvalues for a broad class of metrics.

Findings

Eigenvalues vary analytically with metric deformations.

Most metrics have simple Laplace-Neumann eigenvalues.

The set of metrics preserving double eigenvalues is characterized.

Abstract

We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the curve of eigenvalues, which is used as a device to prove that the eigenvalues of the Laplace-Neumann operator are generically simple in the space $\mathcal{M}^k$ of all $C^k$ Riemannian metrics on $M$. This implies the existence of a residual set of metrics in $\mathcal{M}^k$, which make the spectrum of the Laplace-Neumann operator simple. We also give a precise information about the complementary of this residual set, as well as about the structure of the set of the deformation of a Riemannian metric which preserves double eigenvalues.