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

Karpukhin, Mikhail and Stern, Daniel (2020) Min-max harmonic maps and a new characterization of conformal eigenvalues. . (Unpublished)

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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.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:The authors would like to thank Iosif Polterovich and Jean Lagacé for remarks on the preliminary version of the manuscript. This project originated during the CRG workshop on Geometric Analysis held at the University of British Columbia in May 2019. The hospitality of the University of British Columbia is gratefully acknowledged
Record Number:CaltechAUTHORS:20201123-143024911
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106795
Deposited By: Tony Diaz
Deposited On:23 Nov 2020 22:37
Last Modified:23 Nov 2020 22:37

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