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Metric definition of quasiconformality and exceptional sets

Dimitrios Ntalampekos

TL;DR

The paper generalizes Gehring's metric definition of quasiconformality by introducing eccentric distortion $E_f$ and allowing uncentered, shrinking sets ${A_k}$ with bounded eccentricity; it proves that a uniform bound on $E_f$ across all points is equivalent to quasiconformality. It introduces CNED as a broad class of sets that are negligible for extremal distances and proves that CNED sets are exceptional for this new definition, yielding removability results. A key technical advance is the egg-yolk covering lemma, which enables simultaneous control of domain and range geometry under a homeomorphism and replaces the reliance on Besicovitch-type ball covers. The main theorem shows that, under CNED exceptional sets and a mild measure-theoretic condition, the eccentric-distortion control suffices to deduce quasiconformality, with potential applications to rigidity problems in circle domains and complex dynamics.

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.

Metric definition of quasiconformality and exceptional sets

TL;DR

The paper generalizes Gehring's metric definition of quasiconformality by introducing eccentric distortion and allowing uncentered, shrinking sets with bounded eccentricity; it proves that a uniform bound on across all points is equivalent to quasiconformality. It introduces CNED as a broad class of sets that are negligible for extremal distances and proves that CNED sets are exceptional for this new definition, yielding removability results. A key technical advance is the egg-yolk covering lemma, which enables simultaneous control of domain and range geometry under a homeomorphism and replaces the reliance on Besicovitch-type ball covers. The main theorem shows that, under CNED exceptional sets and a mild measure-theoretic condition, the eccentric-distortion control suffices to deduce quasiconformality, with potential applications to rigidity problems in circle domains and complex dynamics.

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 is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting the set at countably many points. We show as our main theorem that CNED sets are exceptional for the definition of quasiconformality.

Paper Structure

This paper contains 11 sections, 13 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A new definition of quasiconformality
  3. Exceptional sets for the definition of quasiconformality
  4. The egg-yolk covering lemma
  5. Known covering results
  6. Egg-yolk pairs
  7. The egg-yolk covering lemma
  8. Proof of Theorem \ref{['theorem:main']}
  9. Preliminaries
  10. Finite distortion implies absolute continuity
  11. Completing the proof of Theorem \ref{['theorem:main']}

Key Result

Theorem 1.2

Let $\Omega\subset \mathbb R^n$ be an open set and $f\colon \Omega \to \mathbb R^n$ be a topological embedding. Suppose that there exists a constant $H\geq 1$ such that for all $x\in \Omega$ we have Then $f$ is quasiconformal in $\Omega$.

Figures (2)

  • Figure 1: An $M$-egg-yolk pair.
  • Figure 2: Top figure: $B_{i_1}'\cap B_j'\neq \emptyset$, so $\mathop{\mathrm{\mathrm{diam}}}\nolimits(A_{i_1}')\simeq_M \mathop{\mathrm{\mathrm{diam}}}\nolimits (A_j')$. On the other hand, $B_{i_1}$ need not intersect $B_j$ and $\mathop{\mathrm{\mathrm{diam}}}\nolimits(A_j)$ might be much smaller than $\mathop{\mathrm{\mathrm{diam}}}\nolimits(A_{i_1})$. Bottom figure: Formation of $D_{k_m+1}$ by taking the union of $A_{i_1}$ with sets $A_j$ such that $B_{i_1}'\cap B_j'\neq \emptyset$.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1: $5B$-covering lemma, Heinonen:metric*Theorem 1.2, p. 2
  • Theorem 2.2: Besicovitch covering theorem, Mattila:geometry*Theorem 2.7
  • Lemma 2.3: Bojarski:inequality
  • proof : Proof of \ref{['egg:intersect_yolk']}
  • proof : Proof of \ref{['egg:cluster']}
  • Lemma 2.4: Egg-yolk covering lemma
  • ...and 11 more