Karpukhin, Mikhail; Stern, Daniel; (2023) Min-max harmonic maps and a new characterization of conformal eigenvalues. Journal of the European Mathematical Society 10.4171/jems/1401. (In press).

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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 Λ ₁ ​ (M,c) and Λ 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 Λ ₁ ​ (M,c), Λ ₂ ​ (M,c), and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

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