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

• # Fulltext

Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/127/03/0463-0470

• # Keywords

Wu metric; Kobayashi metric; negative holomorphic curvature

• # Abstract

We study the Wu metric for the non-convex 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 0 < m < 1/2. We give explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$. We verify that the Wu metric is a continuous Hermitian metric on $E_{2m}$, real analytic everywhere except along the complex hypersurface $Z = \{(0, z_{2}, . . . , z_{n}) \in E_{2m}\}$. We also show that the holomorphic curvature of the Wu metric for this noncompactfamily 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 $E_{2m}$.

• # 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
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