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

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