$\infty$-minimal submanifolds

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

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

(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

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

(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.