A complete simply connected surface with Gaussian curvature bounded between a negative constant and 0 is called hyperbolic Cartan-Hadamard surface (or simply hyperbolic CH-surface). In this paper the authors study the problem when a given holomorphic quadratic differential form can be realized as the Hopf-differential of a harmonic diffeomorphism. They get the following existence theorem and uniqueness theorem:
Existence-theorem: Let be a hyperbolic CH-surface with Gaussian curvature satisfying for some constant and . Then given any holomorphic quadratic differential on , or , there is a harmonic map from to with Hopf differential given by . Moreover, if or is not a constant, then can be chosen to be a harmonic diffeomorphism into . Furthermore, if and , then can be chosen to be a quasi-conformal harmonic diffeomorphism onto .
Uniqueness-theorem: Let be the Poincaré disk and let be a hyperbolic CH-surface with Gaussian curvature . Let be a holomorphic quadratic differential in BDQ(D). Let and be two orientation preserving harmonic diffeomorphisms from into with the same Hopf differential . Suppose that is complete on for , where , and suppose for all . Then there is an isometry such that .