  We consider the relationship of the geometry of compact Riemannian manifolds
with boundary to the first nonzero eigenvalue sigma_1 of the
Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus
gamma and k boundary components we obtain the upper bound sigma_1L(\partial
\Sigma) \leq 2(2gamma+k)\pi. We attempt to find the best constant in this
inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric
metrics we show that the best constant is achieved by the induced metric on the
portion of the catenoid centered at the origin which meets a sphere
orthogonally and hence is a solution of the free boundary problem for the area
functional in the ball. For a general class of (not necessarily rotationally
symmetric) metrics on the annulus, which we call supercritical, we prove that
$\sigma_1(\sig)L(\p\Sigma)$ is dominated by that of the critical catenoid with
equality if and only if the annulus is conformally equivalent to the critical
catenoid by a conformal transformation which is an isometry on the boundary. We
prove general upper bounds for conformal metrics on manifolds of any dimension
which can be properly conformally immersed into the unit ball in terms of
certain conformal volume quantities. We show that these bounds are only
achieved when the manifold is minimally immersed by first Steklov
eigenfunctions. We also use these ideas to show that any free boundary solution
in two dimensions has area at least \pi.
