The Rad\'o-Kneser-Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

TL;DR

This paper extends the Radó-Kneser-Choquet theorem to $p$-harmonic mappings between Riemannian surfaces, establishing injectivity criteria using approximation by auxiliary maps and homotopy arguments.

Contribution

It proves the injectivity criterion for $p$-harmonic maps between Riemannian surfaces, generalizing the classical theorem beyond harmonic maps.

Findings

Auxiliary mappings solve uniformly elliptic systems.

Each auxiliary map has a positive Jacobian.

Injectivity is preserved through a homotopy using a minimum principle.

Abstract

In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.