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

Abstract

Given a surface M and a fixed conformal class c one defines Λ_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 Λ_k(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify Λ₁(M,c) and Λ₂(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 Λ₁(M,c), Λ₂(M,c) and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.