Research Problems in Function Theory — Problem 7.24

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.

Put $E=\{z\in\mathbb C:|z|<1\}$. Fix distinct $0,a,b\in E$, where $a>0$ and $b=|b|e^{i\beta}$ with $-\pi<\beta\le\pi$. A continuum here is a nonempty compact connected set. Let $\mathcal K$ consist of continua $K$ with $\{a,b\}\subset K\subset E\setminus\{0\}$ and $E\setminus K$ connected. A continuous branch of argument on $K$ gives the branch-independent difference \[V(K)=\arg_K(b)-\arg_K(a)=\beta+2\pi n,\qquad n\in\mathbb Z;\] write $\mathcal K_n$ for the corresponding class. A meromorphic quadratic differential is $q=\sigma(z)\,dz^2$. A horizontal trajectory is a curve $z(t)$ along which $\sigma(z(t))z'(t)^2$ is positive real away from its singular endpoints. Call $K$ natural when it is such a trajectory for a differential having simple poles at $0,a,b$, no other pole in $E$, and real boundary values as a quadratic differential on $\partial E$; explicitly, its pullback $\sigma(e^{it})(ie^{it})^2dt^2$ is real. For a doubly connected domain conformally equivalent to $\{r<|w|<1\}$, use modulus $(2\pi)^{-1}\log(1/r)$.

Statement recorded in the supplied source.

(i) Does every class $\mathcal K_n$ contain a natural continuum? (ii) Determine all natural continua in $\mathcal K$. (iii) For each $n$, determine how the modulus of $E\setminus K$ varies as $K$ ranges over the natural continua in $\mathcal K_n$. [S2] Update7.24(i) reports Hamilton's affirmative answer to the first question; the classification and modulus questions remain separate parts of the recorded scope. [S2]

Short English statement

Determine the existence and classification of quadratic-differential trajectories joining the two points in each winding class, and the conformal moduli of their complements.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307024 (AMR-022-7024). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.24 and its complete update. https://arxiv.org/abs/1809.07200.