Research Problems in Function Theory — Problem 6.59

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 $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ be the Riemann sphere, and let $D\subset\widehat{\mathbb C}$ be a domain containing $\infty$. Its complement $K$ is compact. A continuous angle assignment to its connected components means a continuous real-valued function $a:K\to\mathbb R$ constant on each component; equivalently, it is continuous on the quotient space of components. The value $a(K_j)$ specifies an unoriented line direction modulo $\pi$. A rectilinear slit domain has complement consisting of pairwise disjoint closed straight segments, allowing degenerate segments consisting of one point. Boundary components correspond under a conformal equivalence.

Statement recorded in the supplied source.

Does every such $(D,a)$ admit a conformal equivalence $\phi:D\to\widehat{\mathbb C}\setminus\bigcup_j I_j$ for which the boundary component $K_j$ corresponds to a segment $I_j$ parallel to $e^{ia(K_j)}$? The prescribed directions are allowed to have infinitely many distinct values. [S2] Update 6.59 records FitzGerald and Weening's existence result when the angle assignment has finite range. [S2]

Short English statement

Can an arbitrary continuously prescribed direction for each complementary component be realized by a conformal straight-slit representation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306059 (AMR-022-6059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.59 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.