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Proceedings of the American Mathematical Society

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

 
 

 

$\infty$-minimal submanifolds


Author: Nikolaos I. Katzourakis
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
Published electronically: May 5, 2014
MathSciNet review: 3209334
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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 β€œEuler-Lagrange PDE” 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€œ$\infty$-Minimal Area” 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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Additional 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

Keywords: $\infty$-Harmonic maps, vector-valued calculus of variations in $L^\infty$, vector-valued optimal Lipschitz extensions, quasi-conformal maps, Aronsson PDE, rigidity.
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.