Research Problems in Function Theory — Problem 7.56

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$, and let $\Gamma\subset\widehat{\mathbb C}$ be a Jordan curve through $\infty$. Write $H_+=\{\operatorname{Im}z>0\}$ and $H_-=\{\operatorname{Im}z<0\}$. Let conformal bijections $f_\pm:H_\pm\to\Omega_\pm$ map onto the two complementary domains of $\Gamma$, with boundary extensions satisfying $f_\pm(\infty)=\infty$. Their boundary identification is \[h=(f_-|_{\mathbb R})^{-1}\circ(f_+|_{\mathbb R}):\mathbb R\to\mathbb R. \] It is an increasing homeomorphism. Quasisymmetry means that some $C\ge1$ satisfies $C^{-1}\le(h(x+t)-h(x))/(h(x)-h(x-t))\le C$ for all $x\in\mathbb R$ and $t>0$.

Statement recorded in the supplied source.

Characterize the homeomorphisms $h$ arising by these conformal boundary identifications for arbitrary Jordan curves. In particular, does every increasing homeomorphism of $\mathbb R$ arise? Quasisymmetric identifications correspond to quasicircles. The original update records Oikawa and Huber counterexamples to the every-homeomorphism assertion. [S2]

Short English statement

Characterize the boundary homeomorphisms obtained by conformally welding the two sides of a Jordan curve, including whether every increasing homeomorphism occurs.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307056 (AMR-022-7056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.56, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.