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

TL;DR

This paper investigates the properties of the Wu metric on convex egg-shaped domains in complex space, revealing its smoothness, continuity, and curvature characteristics, and confirming a conjecture for these specific domains.

Contribution

It provides a detailed analysis of the Wu metric on a class of convex egg domains, establishing smoothness properties and confirming a conjecture about its curvature.

Findings

Wu metric is real analytic except on a lower dimensional subvariety.

The metric is continuous for m=1/2 and C^1-smooth for m>1/2.

The Wu metric has strongly negative holomorphic curvature in the sense of currents.

Abstract

We study the Wu metric on convex egg 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 $m \geq 1/2, m \neq 1$. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the Wu metric is shown to be continuous when $m=1/2$ and even $C^1$-smooth for each $m>1/2$, and in all cases, a non-K\"ahler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.