Research Problems in Function Theory — Problem 6.63

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 $H=\{z:\operatorname{Re}z>0,\ 0\le\operatorname{Im}z\le1\}$. For an increasing homeomorphism $\alpha:(0,\infty)\to(a,\infty)$ with $a\ge0$, identify $x$ on the lower edge with $\alpha(x)+i$ on the upper edge. A conformal sewing is such an identification whose quotient carries a Riemann-surface structure extending the complex structure of the strip interior. Its end at infinity is called hyperbolic if a sufficiently far-out annular neighbourhood of that end has finite conformal modulus, with the end represented by a boundary circle rather than a puncture.

Statement recorded in the supplied source.

If the sewing defined by $\alpha$ has a hyperbolic end at infinity, does there exist a continuous function $\varepsilon:(0,\infty)\to(0,\infty)$ such that every conformal sewing defined by a homeomorphism $\beta$ of the same type, satisfying $|\beta(x)-\alpha(x)|<\varepsilon(x)$ for every $x>0$, also has a hyperbolic end at infinity? [S2]

Short English statement

Is the hyperbolic type of a sewn half-strip stable under a sufficiently small pointwise perturbation of its sewing map?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306063 (AMR-022-6063). 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 6.63 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.