The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Abstract

Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $Σ_1$ and $Σ_2$ admit neighborhoods $Ω_1$ and $Ω_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $Ω_1\congΩ_2$, such that $|σ_k(M_1)-σ_k(M_2)|\leq C$ for each $k\in\mathbb{N}$. This follows from a quantitative relationship between the Steklov eigenvalues $σ_k$ of a compact Riemannian manifold $M$ and the eigenvalues $λ_k$ of the Laplacian on its boundary. Our main result states that the difference $|σ_k-\sqrt{λ_k}|$ is bounded above by a constant which depends on the geometry of $M$ only in a neighborhood 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 $Ω_1\congΩ_2$.