Positive vector bundles and harmonic maps (Q1121573)

Concerning the holomorphicity of a mapping between complex manifolds, the authors have gained the following results: (1) If f: (N,g)\(\to (M,h)\) is a harmonic map between Kähler manifolds where N is compact of dimension n and (a) Ric \(N\geq 0\) and \(=0\) somewhere, (b) \(E:=f^*(F)\) \((F:=TM_{(1,0)}\), T is the tangent functor) n-semipositive, then f is holomorphic. (2) Result (1) is formulated and proved in the case of boundary conditions. (3) If (M,h) is a Kähler manifold with positive bisectional curvature, (N,g) is another complete Kähler manifold and f: (N,g)\(\to (M,h)\) is a stable harmonic map, then under some further delicate analytic conditions f is holomorphic again. (4) Result (3) is applied to the special case \(N={\mathbb{C}}\) for drawing further conclusions. It is also shown that for pluriharmonic f one of the crucial assumptions of (3) is always satisfied.