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Characterization of Asymptotically Smooth Curves

Katsuhiko Matsuzaki, Fei Tao

TL;DR

This work develops a deep link between the geometry of boundary curves and analytic function spaces by showing there exists an explicit asymptotically conformal chord-arc curve that is not asymptotically smooth, which implies strict inclusions ${\rm VMOA}(\mathbb{D}) \subset {\rm BMOA}(\mathbb{D}) \cap B_0(\mathbb{D})$ and ${\rm SS}(\mathbb{S}) \subset {\rm SQS}(\mathbb{S}) \cap {\rm S}(\mathbb{S})$. It establishes a sharp geometric characterization of asymptotic smoothness via uniform approximability together with asymptotic conformality, bridging local and global regularity. The paper also develops Teichmüller-space machinery, introducing an intermediate space $T_{B_0}$ between $T_V$ and $T_B$ and analyzing its properties, including strict inclusions and several open questions. These results clarify the correspondence between analytic space memberships (VMOA/BMOA, Bloch, Carleson measures) and the geometry of quasicircles, with implications for the structure of Teichmüller spaces and boundary regularity phenomena in conformal mapping theory.

Abstract

We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and BMOA does not necessarily lie in VMOA, and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.

Characterization of Asymptotically Smooth Curves

TL;DR

This work develops a deep link between the geometry of boundary curves and analytic function spaces by showing there exists an explicit asymptotically conformal chord-arc curve that is not asymptotically smooth, which implies strict inclusions and . It establishes a sharp geometric characterization of asymptotic smoothness via uniform approximability together with asymptotic conformality, bridging local and global regularity. The paper also develops Teichmüller-space machinery, introducing an intermediate space between and and analyzing its properties, including strict inclusions and several open questions. These results clarify the correspondence between analytic space memberships (VMOA/BMOA, Bloch, Carleson measures) and the geometry of quasicircles, with implications for the structure of Teichmüller spaces and boundary regularity phenomena in conformal mapping theory.

Abstract

We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and BMOA does not necessarily lie in VMOA, and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.

Paper Structure

This paper contains 10 sections, 14 theorems, 165 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Construction of the Example
  3. Verification of the Example Curve
  4. Lemmas
  5. Verification
  6. Characterization of Asymptotically Smooth Curves
  7. Teichmüller Spaces
  8. The Little Universal Teichmüller Space
  9. The BMO and VMO Teichmüller Space
  10. An Intermediate Teichmüller Space Between BMO and VMO

Key Result

Theorem 1.1

There exists an asymptotically conformal chord-arc curve that is not asymptotically smooth.

Figures (7)

  • Figure 1: The Frenet frame field along $\Gamma$
  • Figure 2: Illustration of $f_h$ embedded into the local Frenet frame of $\Gamma$
  • Figure 3: Graphs of $f_{\frac{1}{n\cdot 2^{n}}}\left(2^{n}\left(t-\frac{1}{2^{n}}\right)\right)$
  • Figure 4: Illustration of how $\gamma_n^{(2)}$ is obtained from $\gamma_n^{(1)}$
  • Figure 5: Construction of the curve $\gamma$
  • ...and 2 more figures

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • ...and 23 more