Metric surfaces and conformally removable sets in the plane

TL;DR

This paper characterizes conformally removable sets in the plane using metric surface theory, linking removability to quasiconformal maps onto metric surfaces and measure zero sets.

Contribution

It establishes a new characterization of conformally removable sets via quasiconformal maps onto metric surfaces, extending previous understanding.

Findings

A compact set is $S$-removable iff it maps to a measure zero set under a quasiconformal map onto a metric surface.

The characterization fails if maps are into the plane instead of metric surfaces.

A set is $S$-removable or $CH$-removable iff certain quasiconformal homeomorphisms are globally quasiconformal.

Abstract

We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is $S$-removable if and only if there exists a quasiconformal map from the plane onto a metric surface that maps the given set to a set of linear measure zero. The statement fails if we consider maps into the plane rather than metric surfaces. Moreover, we prove that a set is $S$-removable (resp. $CH$-removable) if and only if every homeomorphism from the plane onto a metric surface (resp. reciprocal metric surface) that is quasiconformal in the complement of the given set is quasiconformal everywhere.