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Univalency of harmonic mappings between surfaces.

An important generalization of this result was obtained by Schoen and Yau in [30]. The authors extend the theorem to the case of harmonic mappings between twodimensional Riemannian manifolds with boundary, under the assumption that the Gaussian curvature of the image is nowhere positive. (In addition, they show univalency in the case of compact surfaces without boundary under the same curvature restriction. This result was also obtained by Sampson (cf. [29], Thm. 11).)