Is there a complete Kahler metric on a bounded domain？

Question

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. By the theorems of Cheng-Yau and Mok-Yau, $\Omega$ admits a complete Kahler-Einstein metric if and only if $\Omega$ is a pseudoconvex domain.

My question: Does every bounded domain admit a complete Kahler metric? If not, what is the condition to ensure the existence of such complete Kahler metric?

https://mathoverflow.net/questions/239677/does-a-k%C3%A4hler-manifold-always-admit-a-complete-k%C3%A4hler-metric — Moishe Kohan, Jun 22 '23 at 14:40

score 2 · Answer 1 · answered Jun 23 '23 at 01:36

2

If the domain is Reinhardt, then it has to be logarithmaically convex. A general result may be found in Math. Ann. 257, 191 (1981).

answered Jun 23 '23 at 01:36

peter

51

Is there a complete Kahler metric on a bounded domain？

1 Answers1