Construction of labyrinths in pseudoconvex domains

Abstract

We build in a given pseudoconvex (Runge) domain $D$ of ${\mathbb{C}}^N$ a $\mathcal{O}(D)$ convex set $\Gamma$, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $\gamma :[0,1)\to D$ with ${\lim }_{r\to 1}\gamma (r)\in \partial D$ and omitting $\Gamma$ has infinite length. This solves a problem left open in a recent paper by Alarcón and Forstnerič.