An extension theorem of holomorphic functions on hyperconvex domains

Abstract

Let $n\ge 3$ and $\Omega$ be a bounded domain in ${\mathbb{C}}^n$ with a smooth negative plurisubharmonic exhaustion function $\phi$. As a generalization of Y. Tiba's result, we prove that any holomorphic function on a connected open neighborhood of the support of $(i\partial \bar{\partial }\phi {)}^{n-2}$ in $\Omega$ can be extended to the whole domain $\Omega$. To prove it, we combine an $L^2$ version of Serre duality and Donnelly-Fefferman type estimates on $(n,n-1)$- and $(n,n)$- forms.

7 pages, a mistake in the Corollary 1.3 and some typos are corrected. It is accepted to appear in Proceedings of the American Mathematical Society