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 $n\ge 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $\Omega \setminus A$, where $\sup K_j<\infty$ and $\Omega \subset {\mathbb{R}}^n$ is a $C^2$-smoothly bounded domain and $A\subset \Omega$ 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.

17 pages