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.
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