Analyzing the Wu metric on a class of eggs in ${\mathbb{C}}^n$ -- II

Abstract

We study the Wu metric for the non-convex domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $0<m<1/2$. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$ are obtained. The Wu metric is then verified to be a continuous Hermitian metric on $E_{2m}$ which is real analytic everywhere except along the complex hypersurface $Z=\{(0,z_2,\dots ,z_n)\in E_{2m}\}$. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of $m$. This verifies a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.