Metric upper bounds for Steklov and Laplace eigenvalues

We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. Its proof is based on metric-measure space techniques. The second bound is in terms of the extrinsic diameter of the boundary and its injectivity radius. It is obtained from a concentration inequality, akin to Gromov–Milman concentration for closed manifolds. By applying these bounds to cylinders over closed manifolds, we obtain bounds for eigenvalues of the Laplace operator, in the spirit of Berger–Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius.