An extremal eigenvalue problem and minimal surfaces in the ball

We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a manifold to the normal derivative of its harmonic extension. We show how the question of finding surfaces with fixed boundary length and largest first eigenvalue is intimately connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary
solutions). We describe recent results on the characterization of optimal surfaces for this problem. This is joint work with Ailana Fraser.