Flexible curves and Hausdorff dimension

TL;DR

The paper establishes that every log-singular circle welding can be realized by flexible Jordan curves of any prescribed Hausdorff dimension in $[1,2]$ and even by curves of positive area. It develops a refined Bishop-type welding construction using iterated quasiconformal extensions, admissible shapes, and careful control of distortion to produce weldings with targeted geometric properties. The results connect conformal welding, removability, and fractal geometry by showing residuality of flexible curves and by proving dimensional and area-flexible weldings, including the dimension-1 and dimension-$2$ extremes. The work further demonstrates that flexible curves are non-removable and that their weldings can fail to be injective, shedding light on longstanding conjectures about removability and welding injectivity, with potential implications for related areas in complex analysis and geometric function theory.

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.

Figures (21)

• Figure 1: Definition of conformal welding. The conformal maps $f$ and $g$ map the interior and exterior of the unit circle to the two complementary components of a Jordan curve $\gamma$. The map $h\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$ is an orientation-preserving homeomorphism of $\mathbb{S}^{1}$.
• Figure 2: If two non-conformally equivalent curves $\Gamma_{1}, \Gamma_{2}$ yield the same welding $h$, then there exists a homeomorphism $\phi\in\textrm{CH}(\Gamma_{1})$ so that $\phi(\Gamma_{1})=\Gamma_{2}$.
• Figure 3: The curve with welding $h$ is constructed via an iterative procedure, where at each step we quasiconformally extend maps $f_{1}\colon\mathbb{D}\to\Omega_{1}$, $g_{1}\colon\mathbb{C}\setminus\overline{D}\to\Omega_{1}^{*}$ in such a way that the infinity norm between $f_{1}(\xi)$ and $g_{1}(h(\xi))$ is decreased by a fixed ratio.
• Figure 4: Shapes encode conformal embeddings of a quadrilateral into another quadrilateral. The map $E$ from the rectangle $[0,R]\times[0,1]$ into the rectangle $[0,T]\times[0,1]$ is conformal.
• Figure 5: Representation of the quad-vertices, $a$-sides and $b$-sides of a quadrilateral Q, together with its representation via conformal mapping $\varphi$ to a rectangle $[0,M]\times[0,1]$, which maps quad-vertices to vertices. The value $M$ is a conformal invariant.

Theorems & Definitions (72)

• Theorem 1.1
• Corollary 1.2: Chris:Borel Question 11
• Corollary 1.3
• Corollary 1.4
• Conjecture 1.5: MR1274085 Question 2
• Conjecture 1.6
• Corollary 1.7
• Conjecture 1.8
• Corollary 1.9: Chris:Borel Question 14
• Proposition 1.10