Some classifications of \infty-Harmonic maps between Riemannian manifolds

TL;DR

This paper classifies various types of -harmonic maps, including linear, quadratic, and holomorphic maps, between different Riemannian manifolds, extending understanding of these generalized harmonic maps.

Contribution

It provides complete classifications of linear and quadratic -harmonic maps between spheres, Euclidean spaces, Nil, and Sol geometries, and studies holomorphic -harmonic maps.

Findings

Classified all linear and quadratic -harmonic maps between spheres and Euclidean spaces.

Described all quadratic -harmonic maps between Nil and Euclidean spaces.

Analyzed holomorphic -harmonic maps between complex Euclidean spaces.

Abstract

$\infty$-Harmonic maps are a generalization of $\infty$-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 $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between complex Euclidean spaces.