Tubes and Steklov eigenvalues in negatively curved manifolds

We consider the Steklov eigenvalue problem on a compact pinched negatively curved manifold $M$ of dimension at least three with totally geodesic boundaries. We obtain a geometric lower bound for the first nonzero Steklov eigenvalue in terms of the total volume of $M$ and the volume of its boundary. We provide examples illustrating the necessity of these geometric quantities in the lower bound. Our result can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalue on closed pinched negatively curved manifolds of dimension at least three proved by Schoen in 1982. The proof is composed of certain key elements. We provide a uniform lower bound for the first eigenvalue of the Steklov-Dirichlet problem on a neighborhood of the boundary of $M$ and show that it provides an obstruction to having a small first nonzero Steklov eigenvalue. As another key element of the proof, we give a tubular neighborhood theorem for totally geodesic hypersurfaces in a pinched negatively curved manifold. We give an explicit dependence for the width function in terms of the volume of the boundary and the pinching constant.