# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## $\infty$-minimal submanifoldsHTML articles powered by AMS MathViewer

by Nikolaos I. Katzourakis
Proc. Amer. Math. Soc. 142 (2014), 2797-2811 Request permission

## Abstract:

We identify the Variational Principle governing $\infty$-Harmonic maps $u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N$, that is, solutions to the $\infty$-Laplacian $$\Delta _\infty u \ :=\ \Big (Du \otimes Du + |Du|^2 [Du]^\bot \! \otimes I \Big ) : D^2 u\ = \ 0.$$ System \eqref{1} was first derived in the limit of the $p$-Laplacian as $p\rightarrow \infty$ in a 2012 paper of the author and was recently studied further by him. Here we show that \eqref{1} is the βEuler-Lagrange PDEβ of the vector-valued Calculus of Variations in $L^\infty$ for the functional $$\|Du\|_{L^\infty (\Omega )}\ = \ \underset {\Omega }{\textrm {ess} \textrm {sup}} |Du|.$$ We introduce the notion of $\infty$-Minimal Maps, which are Rank-One Absolute Minimals of \eqref{2} with β$\infty$-Minimal Areaβ of the submanifold $u(\Omega ) \subseteq \mathbb {R}^N$, and prove they solve \eqref{1}. The converse is true for immersions. We also establish a maximum principle for $|Du|$ for solutions to \eqref{1}. We further characterize minimal surfaces of $\mathbb {R}^3$ as those locally parameterizable by isothermal immersions with $\infty$-Minimal Area and show that isothermal $\infty$-Harmonic maps are rigid.
References
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• Nikolaos I. Katzourakis
• Affiliation: Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo 14, E-48009, Bilbao, Spain
• Address at time of publication: Department of Mathematics and Statistics, University of Reading, Whiteknights P.Β O. Box 220, Reading RG6 6AX, United Kingdom