  In the present paper, we study sharp isoperimetric inequalities for the first
Steklov eigenvalue $\sigma_1$ on surfaces with fixed genus and large number $k$
of boundary components. We show that as $k\to \infty$ the free boundary minimal
surfaces in the unit ball arising from the maximization of $\sigma_1$ 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 $\frac{\log k}{k}$. 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.
