The Steklov and Laplacian spectra of Riemannian manifolds with boundary

TL;DR

This paper establishes bounds on the difference between Steklov and Laplacian eigenvalues of Riemannian manifolds with boundary, showing that local boundary geometry controls spectral differences.

Contribution

It provides a quantitative relationship between Steklov and Laplacian spectra, with explicit bounds depending only on boundary neighborhood geometry.

Findings

Bound on eigenvalue differences depending on boundary neighborhood geometry

Explicit constants derived for specific geometric bounds

Relationship between Steklov and Laplacian eigenvalues established

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 $\Omega_1\cong\Omega_2$.