The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

Abstract

In the planar setting the Radó-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 Radó-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.