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

Given a surface \(M\) and a fixed conformal class \(c\) one defines \(\Lambda_k(M,c)\) 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 \(\Lambda_k(M,c)\) are closely related to harmonic maps to spheres. In the present paper, we identify \(\Lambda_1(M,c)\) and \(\Lambda_2(M,c)\) 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 \(\Lambda_1(M,c)\), \(\Lambda_2(M,c)\) and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.