  In this note we establish an expression for the Steklov spectrum of warped
products in terms of auxiliary Steklov problems for drift Laplacians with
weight induced by the warping factor. As an application, we show that a compact
manifold with connected boundary diffeomorphic to a product admits a family of
Riemannian metrics which coincide on the boundary, have fixed volume and
arbitrarily large first non-zero Steklov eigenvalue. These are the first
examples of Riemannian metrics with these properties on three-dimensional
manifolds.
