On the curvature conjecture of Hua Loo-keng

Abstract

It is proved that both the holomorphic sectional and the bisectional curvatures of the conformal Bergman metric

$$ds_1^2 = K^2 (z,\bar z)\frac{{\partial ^2 \log K(z,\bar z)}} {{\partial z^\alpha \partial \bar z^\beta }}dz^\alpha d\bar z^\beta$$

are always negative, where $K(z,\bar z)$ is the Bergman kernel of a bounded domain $\mathcal{D}$ in ℂⁿ. As a subsequent result, the Weyl tensor for a Hermitian manifold is obtained.