Optimal Infinity-Quasiconformal Immersions

Abstract

For a Hamiltonian $K \in C^2(\mathbb{R}^{N \times n})$ and a map $u:\Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,\Omega) \ :=\ \big\|K(Du)\big\|_{L^\infty(\Omega)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} A_\infty u \, :=\, \Big(K_P \otimes K_P + K[K_P]^\bot K_{PP}\Big)(Du):D^2 u \, = \, 0. \tag{2} \] Here $K_P$ is the derivative and $[K_P]^\bot$ is the projection on its nullspace. \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author \cite{K1}-\cite{K6}. Herein we apply our results to Geometric Analysis by choosing as $K$ the dilation function \[ K(P)={|P|^2}{\det(P^\top P)^{-1/n}} \] which measures the deviation of $u$ from being conformal. Our main result is that appropriately defined minimisers of \eqref{1} solve \eqref{2}. Hence, PDE methods can be used to study optimised quasiconformal maps. Nonconvexity of $K$ and appearance of interfaces where $[K_P]^\bot$ is discontinuous cause extra difficulties. When $n=N$, this approach has previously been followed by Capogna-Raich \cite{CR} and relates to Teichm\"uller's theory. In particular, we disprove a conjecture appearing therein.