Isoperimetric control of the Steklov spectrum

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniforml bounded
above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus.
We also establish a relationship between the Steklov eigenvalues of a
domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.