Min-max harmonic maps and a new characterization of conformal eigenvalues
Abstract
Given a surface M and a fixed conformal class c one defines k(M,c) to be the supremum of the k-th nontrivial Laplacian eigenvalue over all metrics g∈ c of unit volume. It has been observed by Nadirashvili that the metrics achieving k(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify 1(M,c) and 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 1(M,c), 2(M,c) and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.