  We identify the Variational Principle governing inifinity-Harmonic maps, that
is solutions to the Infinity-Laplacian. The system was first derived in the
limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in
[K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued
Calculus of Variations in L-inifinity for the L-inifinity norm of the gradient.
We introduce the notion of inifinity-Minimal Maps, whch are Rank-One Absolute
Minimals of with inifinity-Minimal Area" of the range submanifold and prove
they solve the inifinity-laplacian. The converse is true for immersions. We
also establish a maximum principle for |Du| for solutions. We further
characterize minimal surfaces of 3-space as those locally parameterizable by
isothermal immersions with inifinity-minimal area and show that isothermal
inifinity-Harmonic maps are rigid.
