*preprint*

**Inserted:** 4 dec 2020

**Year:** 2018

**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
$\Omega_1\cong\Omega_2$.