CNED sets: countably negligible for extremal distances

TL;DR

The paper introduces and develops the class of CNED sets (countably negligible for extremal distances), situating them between classical $NED$ sets and their weak CNED variants. It proves that several removable classes—such as sets with $ heta^{n-1}$-sigma-finite Hausdorff measure and boundaries of domains with $n$-integrable quasihyperbolic distance—are CNED, and establishes a new necessary-and-sufficient criterion for closed CNED sets, along with a CNED stability result under countable unions. The work further connects CNED to Sobolev removability, showing that closed CNED sets are removable for continuous $W^{1,n}$ functions, and provides concrete examples (including a non-measurable CNED set) and applications to circle-domain rigidity. Overall, it unifies several removability results under the CNED framework and raises open questions about the full extent and limitations of CNED in quasiconformal geometry. The development relies on perturbation families (P-families), modulus invariances, and capacity-modulus relations to derive structural results for both NED and CNED sets.

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 $σ$-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.

Figures (4)

• Figure 1:
• Figure 2: The construction of $\widetilde{\gamma}_k$ from $\widetilde{\gamma}_{k-1}$.
• Figure 3: Construction of $\widetilde{\gamma}_{m+1}$ from $\widetilde{\gamma}_m$. Top figure: the case of $\mathcal{W}_0$. Bottom figure: the case of $\mathcal{W}_{\sigma}$. The red curve is $\widetilde{\alpha}_w$ and the green points denote the set $(|\widetilde{\gamma}_{m+1}|\cap E_{m+1})\cap U_w$, which is countable. In fact, a large part of $\widetilde{\alpha}_w$ should be shared with $\alpha_w$ by \ref{["uc':ii"]} but we do not indicate this to simplify the figure.
• Figure 4: The path $\gamma$ (blue) and the paths $\gamma_i$ given by Lemma \ref{['quasi:subpaths']}. There are three constant paths $\gamma_i$. The path $\widetilde{\gamma}$ arises by replacing the arcs of $\gamma$ between the endpoints of $\gamma_i$ with $\gamma_i$.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: Ntalampekos:rigidity_cned
• Theorem 1.9
• Corollary 1.10