Metric definition of quasiconformality and exceptional sets

TL;DR

This paper introduces a new metric-based characterization of quasiconformality in Euclidean spaces using sequences of open sets with bounded eccentricity, and explores the role of CNED sets as exceptional sets in this context.

Contribution

It generalizes Gehring's metric definition of quasiconformality with a new approach involving uncentered open sets and introduces CNED sets as a new class of exceptional sets.

Findings

Quasiconformality characterized by sequences of open sets with bounded eccentricity.

CNED sets are shown to be the exceptional sets for this metric quasiconformality definition.

The work links extremal distance negligible sets with the behavior of quasiconformal maps.

Abstract

We show that a homeomorphism of Euclidean space is quasiconformal if and only if at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point whose images also have bounded eccentricity. This generalizes the metric definition of quasiconformality of Gehring that uses balls instead. We also study exceptional sets for this definition, in connection with sets that are negligible for extremal distances. We introduce the class of CNED sets, generalizing the classical notion of NED sets studied by Ahlfors--Beurling. A set $A$ is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting the set $A$ at countably many points. We show as our main theorem that CNED sets are exceptional for the definition of quasiconformality.