  We prove existence and regularity of metrics on a surface with boundary which
maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L
the boundary length. We show that such metrics arise as the induced metrics on
free boundary minimal surfaces in the unit ball B^n for some n. In the case of
the annulus we prove that the unique solution to this problem is the induced
metric on the critical catenoid, the unique free boundary surface of revolution
in B^3. We also show that the unique solution on the Mobius band is achieved by
an explicit S^1 invariant embedding in B^4 as a free boundary surface, the
critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many
boundary components we prove the existence of maximizers which are given by
minimal embeddings in B^3. We characterize the limit as the number of boundary
components tends to infinity to give the asymptotically sharp upper bound of
4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and
we give a lower bound on the Morse index for the area functional for free
boundary surfaces in the ball.
