Curvature characterization of certain bounded domains of holomorphy (Q1318184)

The author studies the relation between the existence of a negatively curved complete hermitian metric on a complex manifold \(M\) and the product structure of (or contained in) \(M\). He introduces the concept of geometric ranks and gives a curvature characterization of the rank one manifolds, which generalizes previous results of P. Yang and N. Mok. It is proved, for example, that the product of two bounded domains of holomorphy does not admit any complete Kähler metric with bisectional curvature bounded between two negative constants. The old techniques of Yau's Schwartz lemma and Cheng-Yau's result on the existence of Kähler- Einstein metrics are used in the proofs.