Radó–Kneser–Choquet theorem for harmonic mappings between surfaces

Abstract

We simplify and improve a recent result of Martin (Trans AMS 368:647–658, 2016). Then we prove that if f is an orientation preserving harmonic mapping of the unit disk onto a $$C^{3,\alpha }$$ surface $$\Sigma $$ bounded by a Jordan curve $$\gamma \in C^{3,\alpha }$$ , that belongs to the boundary of a convex domain in $$\mathbf {R}^3$$ , then f is a diffeomorphism.