Stability of spectra of Hodge-de Rham laplacians

Abstract

In this paper we investigate the stability of eigenspaces of the Laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact, oriented, Riemannian manifold with boundary (this includes, in particular, both Neumann and Dirichlet conditions for the Laplace-Beltrami operator on functions). More precisely, our main result is that the gap between corresponding eigenspaces (precise definition will be recalled below) measured using the L∞ norm, converges to zero when smooth metrics g converge to g0 in the C 1 topology. It is quite well known (cf. [3] or [14]) that the eigenvalues of the Laplacian vary continuously under C 0-continuous perturbations of the metric. It is perhaps less well known, but implicit in the work of Cheeger [3], that eigenspaces vary continuously as subspaces of L2 when the metric is perturbed C 0-continuously. We reprove this C 0 L2 stability in Sect. 4 for completeness and in order to be able to use certain notation, conventions and partial results in the proof of C 1 L∞ stability in Sect. 5. The second section of the paper contains a review of the Hodge theory for the Laplace operator with absolute and relative boundary conditions. We also state here the Sobolev embedding theorems and the basic a priori estimates for the square root d + δ of the Laplacian ∆. We need to work with d + δ rather than ∆ = (d + δ)2 since the coefficients of ∆ depend on the second derivatives of the metric tensor and we allow only C 1-continuous perturbations of the metric and do not assume any bounds on the second derivatives. In the third section we review following Kato [12] and Osborn [16] general results from functional analysis concerning perturbation theory for compact operators on Banach spaces, that reduce proving convergence of eigenvalues and eigenspaces