## $\infty$-minimal submanifolds

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- by Nikolaos I. Katzourakis
- Proc. Amer. Math. Soc.
**142**(2014), 2797-2811 - DOI: https://doi.org/10.1090/S0002-9939-2014-12039-9
- Published electronically: May 5, 2014
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## 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 \begin{equation}\Delta _\infty u \ :=\ \Big (Du \otimes Du + |Du|^2 [Du]^\bot \! \otimes I \Big ) : D^2 u\ = \ 0.\end{equation} 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 βEuler-Lagrange PDEβ of the vector-valued Calculus of Variations in $L^\infty$ for the functional \begin{equation} \|Du\|_{L^\infty (\Omega )}\ = \ \underset {\Omega }{\textrm {ess} \textrm {sup}} |Du|.\end{equation} 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.

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## Bibliographic Information

**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
- Email: n.katzourakis@reading.ac.uk
- Received by editor(s): June 4, 2012
- Received by editor(s) in revised form: September 7, 2012
- Published electronically: May 5, 2014
- Communicated by: Chuu-Lian Terng
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**142**(2014), 2797-2811 - MSC (2010): Primary 35J47, 35J62, 53C24; Secondary 49J99
- DOI: https://doi.org/10.1090/S0002-9939-2014-12039-9
- MathSciNet review: 3209334