An extension of a theorem of Paul Yang on negatively pinched curvature

Abstract

We prove that there is no complete Hermitian metric h with bounded torsion on polydisc $$D^{n}$$ such that: the holomorphic bisectional curvature of h is bounded above by a negative constant and the second Ricci curvature of h is bounded below by another negative constant. On the other hand, we also present some interesting examples of domains, which are neither biholomorphic to product nor to homogeneous manifolds, that do not admit such Hermitian metrics.