Characterization of Asymptotically Smooth Curves

TL;DR

This paper constructs examples and provides a characterization of asymptotically smooth curves, revealing nuanced distinctions in conformality, smoothness, and symmetry properties relevant to complex analysis and geometric function theory.

Contribution

It offers the first explicit example of an asymptotically conformal curve that is not asymptotically smooth and characterizes asymptotically smooth curves through conformality and approximability.

Findings

An explicit example of a non-asymptotically smooth asymptotically conformal curve.

Demonstration that certain function spaces are not contained within others.

Complete characterization of asymptotically smooth curves in terms of conformality and approximability.

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.