Uniformly perfect domains and convex hulls

TL;DR

This paper establishes an explicit upper bound on the quasiconformal dilatation for uniformly perfect hyperbolic domains, linking geometric properties to conformal mappings.

Contribution

It provides a new explicit bound on quasiconformal dilatation depending solely on the injectivity radius, extending previous results on uniform perfectness.

Findings

Bound on quasiconformal dilatation depends only on injectivity radius

Explicit construction of conformally natural quasiconformal maps

Extension of Marden and Markovic's results to explicit bounds

Abstract

Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect, then there exists a conformally natural quasiconformal map which admits a bounded homotopy to the nearest point retraction. We obtain an explicit upper bound on the quasiconformal dilatation which depends only on the injectivity radius of the domain.