From Steklov to Laplace: free boundary minimal surfaces with many boundary components (Q6566411)

The authors establish a connection between free boundary minimal surfaces in the unit ball and closed minimal surfaces in the boundary sphere under some additional hypothesis. One of the key ingredients is the gap condition. There are infinitely many such minimal surfaces as observed in [\textit{R. Petrides}, Geom. Funct. Anal. 24, No. 4, 1336--1376 (2014; Zbl 1310.58022)]. Moreover, the authors provide the optimal rate of convergence of the corresponding areas for some genera. This study presents initial instances of free boundary minimal surfaces within a compact domain converging to closed minimal surfaces on the boundary. This discovery opens new avenues for research into free boundary minimal surfaces, as outlined by the numerous open questions posed in this paper. A parallel observation is made for free boundary harmonic maps related to conformally constrained shape optimization problems.