The Steklov and Laplacian spectra of Riemannian manifolds with boundary - Colbois, Bruno; Girouard, Alexandre; Hassannezhad, Asma (My University, 2020)
Given two compact Riemannian manifolds $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighbourhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$ such that $

\sigma_k(M_1)-\sigma_k(M_2)

\leq C$ for each $k\in\N$. The constant $C$ depends only on the geometry of $\Omega_1\cong\Omega_2$. 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 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 $\Omega_1\cong\Omega_2$.