  We show that metrics that maximize the k-th Steklov eigenvalue on surfaces
with boundary arise from free boundary minimal surfaces in the unit ball. We
prove several properties of the volumes of these minimal submanifolds. For free
boundary minimal submanifolds in the ball we show that the boundary volume is
reduced up to second order under conformal transformations of the ball. For
two-dimensional stationary integer multiplicity rectifiable varifolds that are
stationary for deformations that preserve the ball, we prove that the boundary
length is reduced under conformal transformations. We also give an overview of
some of the known results on extremal metrics of the Laplacian on closed
surfaces, and give a survey of our recent results from [FS2] on extremal
metrics for Steklov eigenvalues on compact surfaces with boundary.
