Infinity-harmonic maps and morphisms

Abstract

We propose a new notion called infinity-harmonic maps between Riemannian 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→∞. Infinity-harmonicity 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 maps. Many examples of infinity-harmonic maps are also given, including some very important and well-known classes of maps between Riemannian manifolds.