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$ \infty$-minimal submanifolds


Author: Nikolaos I. Katzourakis
Journal: Proc. Amer. Math. Soc. 142 (2014), 2797-2811
MSC (2010): Primary 35J47, 35J62, 53C24; Secondary 49J99
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

$\displaystyle \Delta _\infty u \ :=\ \Big (Du \otimes Du + \vert Du\vert^2 [Du]^\bot \! \otimes I \Big ) : D^2 u\ = \ 0.$ (1)

System (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 (1) is the ``Euler-Lagrange PDE'' of the vector-valued Calculus of Variations in $ L^\infty $ for the functional

$\displaystyle \Vert Du\Vert _{L^\infty (\Omega )}\ = \ \underset {\Omega }{\textrm {ess}\,\textrm {sup}} \,\vert Du\vert.$ (2)

We introduce the notion of $ \infty $-Minimal Maps, which are Rank-One Absolute Minimals of (2) with ``$ \infty $-Minimal Area'' of the submanifold $ u(\Omega ) \subseteq \mathbb{R}^N$, and prove they solve (1). The converse is true for immersions. We also establish a maximum principle for $ \vert Du\vert$ for solutions to (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. O. Box 220, Reading RG6 6AX, United Kingdom
Email: n.katzourakis@reading.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12039-9
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.