On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator

Abstract

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace--Beltrami operator has an eigenvalue of a given multiplicity. The starting point of our investigation is the "Strong Arnold Hypothesis" introduced by Colin de Verdière, which posits that Laplace eigenvalues of higher multiplicity split up under perturbation exactly as eigenvalues of symmetric matrices do. There exists a simple geometric characterization, due to Colin de Verdière and Besson, of metrics which satisfy the Strong Arnold Hypothesis. Using Besson's characterization, we prove the Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension, and use this to obtain the precise codimension of the set of metrics admitting an eigenvalue of any given multiplicity. Furthermore, we show that the Strong Arnold Hypothesis is satisfied for all metrics admitting eigenvalues of multiplicity at most six, and discuss several examples of metrics violating it.