Every Gromov elliptic domain is meromorphically pseudoconvex

Abstract

We prove that every Gromov elliptic domain $\Omega$ in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood $V$ such that $\Omega\cap V$ becomes Stein after removing a set which is, locally in $\Omega\cap V$, the union of the polar sets of finitely many meromorphic functions on $\Omega\cap V$. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain $\Omega=\operatorname{int}\overline\Omega$ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete K\"ahler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds $\mathbb C^n\setminus\overline{\mathbb B^n}$ and $\mathbb C^n\setminus\mathbb R^n$, $n\geq2$, are not elliptic, and that no proper domain with Lipschitz boundary in $\mathbb P^n$, $n\geq2$, is elliptic.