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 ${\Sigma }_1$ and ${\Sigma }_2$ admit neighborhoods ${\Omega }_1$ and ${\Omega }_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of ${\Omega }_1\cong {\Omega }_2$, such that $|{\sigma }_k(M_1)-{\sigma }_k(M_2)|\le C$ for each $k\in \mathbb{N}$. This follows from a quantitative relationship between the Steklov eigenvalues ${\sigma }_k$ of a compact Riemannian manifold $M$ and the eigenvalues ${\lambda }_k$ of the Laplacian on its boundary. Our main result states that the difference $|{\sigma }_k-\sqrt{{\lambda }_k}|$ 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 ${\Omega }_1\cong {\Omega }_2$.

31 pages, 1 figure