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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?

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

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