The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Research output: Contribution to journalArticle

Original language English J Funct Anal 27 Nov 2019 https://doi.org/10.1016/j.jfa.2019.108409 Accepted/In press - 21 Nov 2019 E-pub ahead of print (current) - 27 Nov 2019

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 $|\sigma_k(M_1)-\sigma_k(M_2)|\leq C$ 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 $\Omega_1\cong\Omega_2$.

Research areas

• math.SP, math.DG, 35P15 (primary), 58C40, 35P20 (secondary)