Some classifications of \infty-Harmonic maps between Riemannian manifolds

$\mathrm{\infty }$-Harmonic maps are a generalization of $\mathrm{\infty }$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\mathrm{\infty }$-harmonic maps from and into a sphere, quadratic $\mathrm{\infty }$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\mathrm{\infty }$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\mathrm{\infty }$-harmonic maps between complex Euclidean spaces.