As described in the previous week’s talk by Mikhail Karpukhin, there is a rich interplay between isoperimetric problems for Laplace eigenvalues on surfaces and the study of harmonic maps and minimal surfaces in spheres. Over the last 10-15 years, a program initiated by Fraser and Schoen has revealed a similar relationship between isoperimetric problems for the first eigenvalue of the (first-order, nonlocal) Dirichlet-to-Neumann operator on surfaces with boundary, and free boundary minimal surfaces in Euclidean balls. In this talk, I’ll describe recent work with Karpukhin characterizing the asymptotic behavior of these Steklov-maximizing metrics–and associated free boundary minimal surfaces–as the number of boundary components becomes large. A key ingredient is previous work with Karpukhin, Nahon, and Polterovich promoting rigidity results to (quantitative) stability results for metrics maximizing the first Laplace eigenvalue on closed surfaces.