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By employing Aronsson's Absolute Minimizers of $L^\infty$ functionals, weprove that Absolutely Minimizing Maps $u:\R^n \larrow \R^N$ solve a"tangential" Aronsson PDE system. By following Sheffield-Smart \cite{SS}, wederive $\De_\infty$ with respect to the dual operator norm and show that suchmaps miss information along a hyperplane when compared to Tight Maps. Werecover the lost term which causes non-uniqueness and derive the completeAronsson system which has \emph{discontinuous coefficients}. In particular, theEuclidean $\infty$-Laplacian is $\De_\infty u = Du \ot Du : D^2u\, +\,|Du|^2[Du]^\bot \De u$ where $[Du]^\bot$ is the projection on the null space of$Du^\top$. We exibit $C^\infty$ solutions having interfaces along which therank of their gradient is discontinuous and propose a modification with $C^0$coefficients which admits \emph{varifold solutions}. Away from the interfaces,Aronsson Maps satisfy a structural property of local splitting to 2 phases, anhorizontal and a vertical; horizontally they possess gradient flows similar tothe scalar case and vertically solve a linear system coupled by a scalarHamilton Jacobi PDE. We also construct singular $\infty$-Harmonic local $C^1$diffeomorphisms and singular Aronsson Maps.

L-Infininity Variational Problems for Maps and the Aronsson PDE System