The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Abstract

Given two compact Riemannian manifolds M1 and M2 such that their respective boundaries Σ1 and Σ2 admit neighbourhoods Ω1 and Ω2 which are isometric, we prove the existence of a constant C such that |σk(M1)−σk(M2)|≤C for each k∈N. The constant C depends only on the geometry of Ω1≅Ω2. This follows from a quantitative relationship between the Steklov eigenvalues σk of a compact Riemannian manifold M and the eigenvalues λk of the Laplacian on its boundary. Our main result states that the difference |σk−λk| is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω1≅Ω2.