$\infty$-minimal submanifolds

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 “Euler-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 “$\infty$-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.