Infinity-harmonic maps and morphisms

TL;DR

This paper introduces the concept of infinity-harmonic maps and morphisms between Riemannian manifolds, exploring their properties, examples, and the conditions under which they preserve harmonicity, extending classical harmonic map theory.

Contribution

It defines infinity-harmonic maps and morphisms, characterizes those that preserve harmonicity under pullback, and provides numerous examples including important classes of maps.

Findings

Infinity-harmonic maps generalize infinity harmonic functions and are limits of p-harmonic maps as p approaches infinity.

Infinity harmonic morphisms are exactly horizontally homothetic maps.

Examples include metric projections and well-known classes of maps between Riemannian manifolds.

Abstract

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds.