Flexible curves and Hausdorff dimension

TL;DR

This paper constructs flexible curves with prescribed Hausdorff dimension and positive area, demonstrating the diversity of non-conformally removable sets and addressing a specific case of a broader conjecture.

Contribution

It introduces methods to produce flexible curves with any given Hausdorff dimension between 1 and 2 and positive area, advancing understanding of non-removable sets.

Findings

Existence of flexible curves with any Hausdorff dimension in [1,2]

Construction of curves with positive area and given welding homeomorphism

Addresses a case of the conjecture on non-conformally removable sets

Abstract

We show that given a log-singular circle homeomorphism $h$ and given any $s\in[1,2]$, there is a flexible curve of Hausdorff dimension $s$ with welding $h$. We also see that there is another curve with welding $h$ and positive area. In particular, this implies that given a flexible curve $\Gamma$, there is a homeomorphism of the plane $\phi\colon\mathbb{C}\to\mathbb{C}$, conformal off $\Gamma$, so that $\phi(\Gamma)$ has positive area. This answers a particular case of the corresponding conjecture for general non-conformally removable sets, for a class of curves that is residual in the space of all Jordan curves.