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

TL;DR

This paper derives explicit formulas for the Wu and Kobayashi metrics on certain non-convex, pseudo-egg domains in complex space, showing the Wu metric's regularity and negative curvature bounds, confirming a conjecture for these domains.

Contribution

It provides explicit metric expressions and curvature bounds for non-convex pseudo-egg domains, verifying a conjecture on Wu metric curvature.

Findings

Explicit formulas for Kobayashi and Wu metrics on $E_{2m}$.

Wu metric is continuous and real analytic except on a hypersurface.

Holomorphic sectional curvature is bounded above by a negative constant.

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, \ldots, 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 $E_{2m}$.