Distortion and topology

TL;DR

This paper investigates conditions under which maps with finite distortion extend to boundary homeomorphisms and explores the relationship between distortion, topology, and geometric invariants in complex analysis.

Contribution

It introduces a separation condition on high-distortion regions that ensures boundary extension and applies these results to Riemann surfaces and boundary germs of conformal and quasisymmetric maps.

Findings

Separation condition guarantees homeomorphic boundary extension.

Explicit distortion estimates in terms of the separation data.

Links between high distortion support and surface topology.

Abstract

For a self mapping $f:\mathbb{D}\to \mathbb{D}$ of the unit disk in $\mathbb{C}$ which has finite distortion, we give a separation condition on the components of the set where the distortion is large - say greater than a given constant - which implies that $f$ extends homeomorphically and quasisymetrically to the boundary $\mathbb{S}$ and thus $f$ shares its boundary values with a quasiconformal mapping whose distortion can be explicitly estimated in terms of the data. This result holds more generally. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate densely on the boundary but does not allow a component to run out to the boundary. The lift of a Jordan domain in a Riemann surface to its universal cover $\mathbb{D}$ is always uniformly separated in modulus and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map and topology of the surface - again with explicit and good estimates. As part of our investigations we study mappings $\varphi:\mathbb{S}\to\mathbb{S}$ which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of $\varphi$. We extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.