∞-minimal submanifolds

We identify the Variational Principle governing ∞ \infty -Harmonic maps u : Ω ⊆ R n ⟶ R N u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N , that is, solutions to the ∞ \infty -Laplacian Δ ∞ u   :=   ( D u ⊗ D u + | D u | 2 [ D u ] ⊥ ⊗ I ) : D 2 u   =   0. \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 p -Laplacian as p → ∞ 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 ∞ L^\infty for the functional ‖ D u ‖ L ∞ ( Ω )   =   ess sup Ω | D u | . \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 ( Ω ) ⊆ R N 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 | D u | |Du| for solutions to \eqref{1}. We further characterize minimal surfaces of R 3 \mathbb {R}^3 as those locally parameterizable by isothermal immersions with ∞ \infty -Minimal Area and show that isothermal ∞ \infty -Harmonic maps are rigid.