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arXiv · 2601.14125

Flexible curves and Hausdorff dimension

Alex Rodriguez
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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 $Γ$, there is a homeomorphism of the plane $ϕ\colon\mathbb{C}\to\mathbb{C}$, conformal off $Γ$, so that $ϕ(Γ)$ 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.

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Alex Rodriguez. 2026-01-20. Flexible curves and Hausdorff dimension. https://arxiv.org/abs/2601.14125

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