Harmonic Maps And Harmonic Complex Structure

The thesis is separated in two parts.In first part, we study the complex analogousness of Liouville type theorems for harmonic maps. We mainly prove that, under an assumption, a harmonic map of finite (?) energy from Cn (n≥2) to any Kahler manifold must be a holomorphic map.In the second part, the author introduce a new concept so called harmonic complex structure. It is a new structure intermediates between complex structure and Kahler structure. Harmonic complex structure is the natural generalization of Kahler structure, just as minimal submanifold is the generalization of totally geodesic submanifold. We obtain a series of results about harmonic complex structures. Particularly, we prove that S6 with standard metric can not admit any harmonic complex structure. Moreover, we also get some interesting results of the trace of A∈Γ(TM(?)TM*).