Sharp eigenvalue bounds and minimal surfaces in the ball

Abstract

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 Möbius band is achieved by an explicit $S^1$ invariant embedding in $B^4$ as a free boundary surface, the critical Möbius 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 $4\pi$. 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.