Starting with an immersion (resp., a diffeomorphism) of the boundary at of the Poincaré model of the hyperbolic plane, the author finds a harmonic extension (resp., a harmonic diffeomorphism) . Moreover, is holomorphic (resp., conformal) iff is conformal. The proof proceeds by construction of a suitable barrier map at associated to each – and to obtain from it an a priori growth estimate. He also constructs entire spacelike constant mean curvature surfaces in Minkowski 3-space whose Gauss maps are harmonic diffeomorphisms .
Related results for have been obtained by P. Li and L.-F. Tam [Invent. Math. 105, No. 1, 1-46 (1991; Zbl 0748.58006)].