HARMONIC MAPS FROM COMPACT KAHLER MANIFOLDS WITH POSITIVE SCALAR CURVATURE TO KAHLER MANIFOLDS OF STRONGLY SEMINEGATIVE CURVATURE

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. If one reduces the assumption on the Ricci curvature to one on the scalar curvature, such a vanishing theorem does not hold in general. This raises the question: What information can we obtain from the existence of a non-constant harmonic map? This paper gives an answer to this problem when both manifolds are Kahler; the results obtained are optimal.