Analysing the Wu metric on a class of eggs in ℂⁿ – I

Abstract

We study the Wu metric on convex egg domains of the form

$$E_{2m} = \big\{ z \in \mathbb{C}^{n}: \vert{z_{1}} \vert^{2m} + \vert {z_{2}} \vert^{2} + {\cdots} + \vert {z_{n-1}} \vert^{2} + \vert {z_{n}} \vert^{2} <1 \big\}, $$

where m ≥ 1/2,m ≠ 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-Kähler 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 .