n this dissertation, we show that there is a bijective map ;The proof is motivated by the fact that the Guass map of a constant mean curvature spacelike surface in Minkowski 3-space M;The main tools in the proof are the generalized maximum principle and the method of sub- and super-solutions on complete manifolds. The method of sub- and super-solutions is of course used to prove the existence of solution of the partial differential equation. However, the generalized maximum principle is important in both the uniqueness and existence of the solution.;As an application of harmonic maps, it was proved that the Teichmuller space of a compact Riemann surface of genus greater than one can be parametrized by holomorphic quadratic differentials on the surface. So, it is interesting to study the analogue for general Teichmuller spaces T(G) of any Fuchsian group G, especially the universal Teichmuller space, that is, G =
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