Complex-analyticity of harmonic maps and strong rigidity of compact Kähler manifolds.

A harmonic map f between two compact Kähler manifolds is shown to be either holomorphic or conjugate holomorphic under a suitable negativity condition on the curvature of the image manifold and a condition on the rank of df. As a consequence, a compact Kähler manifold of dimension >/=2 that is of the same homotopy type as a compact Kähler manifold with suitable negative curvature condition or as a compact quotient of an irreducible classical bounded symmetric domain must be either biholomorphic or conjugate biholomorphic to it.

On the structure of complete simply-connected Kähler manifolds with nonpositive curvature.

We prove that a complete simply-connected Kähler manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.