A quasiconformal variant of the union problem

Abstract

The Union Problem, which has its genesis in the classical Levi problem, asks for a classification of complex manifolds $M$ that can be exhausted by an increasing union of submanifolds $M_j \subset M$ which are all biholomorphic to a fixed domain in $\mathbb C^n$. We explore a quasiconformal variant of this question and seek to classify $n$-Riemannian manifolds $M$ such that each $M_j$ is quasiconformally equivalent to a bounded domain in $\mathbb R^n$. It turns out that this is possible when these quasiconformal equivalences have uniformly bounded dilatations. Using Kiernan's quasiconformal Schwarz lemma when $n=2$ and Ferrand's conformal capacity when $n \geq 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $Ω\setminus A$, where $\sup K_j < \infty$ and $Ω\subset \mathbb R^n$ is a $C^2$-smoothly bounded domain and $A \subset Ω$ is at most finite. As a consequence, we obtain that Gehring's example of a bounded domain in $\mathbb R^n$ which has $C^1$-smooth boundary everywhere except at a point and is known to be quasiconformally inequivalent to the unit ball in $\mathbb R^n$, possesses the additional property that it cannot even be exhausted by quasiconformal images of the unit ball with uniformly bounded dilatations.