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Laplace and Steklov Extremal Metrics via n-Harmonic Maps

Karpukhin, Mikhail and Métras, Antoine (2022) Laplace and Steklov Extremal Metrics via n-Harmonic Maps. Journal of Geometric Analysis, 32 (5). Art. No. 154. ISSN 1050-6926. doi:10.1007/s12220-022-00891-6.

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We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of n-harmonic maps. Our approach extends the well-known results linking extremal metrics for eigenvalues on surfaces with minimal immersions and harmonic maps. In the process, we uncover two previously unknown features of the Steklov eigenvalues. First, we show that in higher dimensions there is a unique normalization involving both the volume of the boundary and of the manifold itself, which leads to meaningful extremal eigenvalue problems. Second, we observe that the critical points of the eigenvalue functionals in a fixed conformal class have a natural geometric interpretation provided one considers the Steklov problem with a density. As an example, we construct a family of free boundary harmonic annuli in the three-dimensional ball and conjecture that they correspond to metrics maximizing the first Steklov eigenvalue in their respective conformal classes.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Karpukhin, Mikhail0000-0003-1935-731X
Métras, Antoine0000-0002-0251-5345
Additional Information:© Mathematica Josephina, Inc. 2022. Received: 12 July 2021 / Accepted: 31 January 2022 / Published: 26 February 2022. The authors would like to thank Daniel Stern for fruitful discussions and Iosif Polterovich for invaluable remarks on the preliminary versions of the manuscript. Antoine Métras’s work is part of a PhD thesis under the supervision of Iosif Polterovich. Mikhail Karpukhin is supported by the US National Science Foundation [DMS-1363432]; and Antoine Métras is supported by the Fonds de recherche du Québec - Nature et Technologies [B2-272578].
Funding AgencyGrant Number
Fonds de recherche du Québec - Nature et technologies (FRQNT)B2-272578
Subject Keywords:Steklov eigenvalues · Laplace eigenvalues · Extremal metric · n-Harmonic maps
Issue or Number:5
Classification Code:Mathematics Subject Classification: 58C40; 58E11; 58E20
Record Number:CaltechAUTHORS:20220301-900033000
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Official Citation:Karpukhin, M., Métras, A. Laplace and Steklov Extremal Metrics via n-Harmonic Maps. J Geom Anal 32, 154 (2022).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113667
Deposited By: George Porter
Deposited On:02 Mar 2022 00:33
Last Modified:10 May 2022 22:01

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