CNED sets: countably negligible for extremal distances

TL;DR

This paper introduces CNED sets, a generalization of NED sets, and demonstrates their significance in quasiconformal removability, unifying many known results and expanding classes of removable sets.

Contribution

The paper establishes that several known removable sets are also CNED, provides a new criterion for (C)NED sets, and shows countable unions of such sets are also (C)NED, broadening the scope of quasiconformal removability.

Findings

CNED sets generalize NED sets and are quasiconformally removable.

Sets of $\sigma$-finite Hausdorff $(n-1)$-measure are CNED.

Countable unions of closed CNED sets are CNED.

Abstract

The author has recently introduced the class of CNED sets in Euclidean space, generalizing the classical notion of NED sets, and shown that they are quasiconformally removable. A set $E$ is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting $E$ at countably many points. We prove that several classes of sets that were known to be removable are also CNED, including sets of $\sigma$-finite Hausdorff $(n-1)$-measure and boundaries of domains with $n$-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and suggests that the CNED condition should also be necessary for removability. We give a new necessary and sufficient criterion for closed sets to be (C)NED. Applying this criterion, we show that countable unions of closed (C)NED sets are (C)NED. Therefore we enlarge significantly the known classes of quasiconformally removable sets.