From Steklov to Laplace: free boundary minimal surfaces with many boundary components

Abstract

In the present paper, we study sharp isoperimetric inequalities for the first Steklov eigenvalue ${\mathit{\sigma }}_1$ on surfaces with fixed genus and large number k of boundary components. We show that as $k\to \mathrm{\infty }$ the free boundary minimal surfaces in the unit ball arising from the maximization of ${\mathit{\sigma }}_1$ converge to a closed minimal surface in the boundary sphere arising from the maximization of the first Laplace eigenvalue on the corresponding closed surface. For some genera, we prove that the corresponding areas converge at the optimal rate $\frac{logk}{k}$. This result appears to provide the first examples of free boundary minimal surfaces in a compact domain converging to closed minimal surfaces in the boundary, suggesting new directions in the study of free boundary minimal surfaces, with many open questions proposed in the present paper. A similar phenomenon is observed for free boundary harmonic maps associated to conformally constrained shape optimization problems.