• Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{I}$

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/127/02/0323-0335

• # Keywords

Wu metric; Kobayashi metric; negative holomorphic curvature

• # Abstract

We study the Wu metric on convex egg domains of the form $E_{2m}= \{z \in \mathbb{C}^n : \mid z_1\mid ^{2m} + \mid z_2\mid ^2 +\cdots +\mid z_{n−1}\mid^2 + \mid z_n \mid^2$ < $1\}$, 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$\ddot{a}$hler 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 $E_{2m}$.

• # Author Affiliations

1. Indian Statistical Institute, Chennai 600 113, India.
2. Department of Mathematics, Indian Institute of Technology – Bombay, Mumbai 400 076, India.

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019