Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface $M$ and a fixed conformal class $c$ one defines ${\mathrm{\Lambda }}_k(M,c)$ to be the supremum of the $k$-th nontrivial Laplacian eigenvalue over all metrics $g\in c$ of unit volume. It has been observed by Nadirashvili that the metrics achieving ${\mathrm{\Lambda }}_k(M,c)$ are closely related to harmonic maps to spheres. In the present paper, we identify ${\mathrm{\Lambda }}_1(M,c)$ and ${\mathrm{\Lambda }}_2(M,c)$ with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing ${\mathrm{\Lambda }}_1(M,c)$, ${\mathrm{\Lambda }}_2(M,c)$ and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.