Min-max harmonic maps and extremal metrics for Laplacian eigenvalues

I'll describe recent work with Mikhail Karpukhin, in which we relate the problem of maximizing Laplacian eigenvalues over unit-area metrics on a given Riemann surface to natural variational constructions of harmonic maps to high-dimensional spheres. Our results give a new proof of the existence of metrics maximizing the first and second Laplacian eigenvalues, and have several applications to shape optimization problems in spectral geometry, for instance providing sharp upper bounds for the first two Steklov eigenvalues on surfaces with fixed genus and any number of boundary components.