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

  • Mikhail Karpukhin

    University College London, UK
  • Daniel Stern

    Cornell University, Ithaca, USA
Min-max harmonic maps and a new characterization of conformal eigenvalues cover
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Abstract

Given a surface and a fixed conformal class one defines to be the supremum of the -th nontrivial Laplacian eigenvalue over all metrics of unit volume. It has been observed by Nadirashvili that the metrics achieving are closely related to harmonic maps to spheres. In the present paper, we identify and 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 , , and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

Cite this article

Mikhail Karpukhin, Daniel Stern, Min-max harmonic maps and a new characterization of conformal eigenvalues. J. Eur. Math. Soc. 26 (2024), no. 11, pp. 4071–4129

DOI 10.4171/JEMS/1401