Research Problems in Function Theory — Problem 7.43

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 $D\subset\mathbb C$ be a bounded simply connected domain. A conformal self-map means a biholomorphic map $f:D\to D$. Call $D$ conformally rigid if there is $\varepsilon>0$ such that every such $f$ satisfying $|f(z)-z|<\varepsilon$ for every $z\in D$ is the identity. For a prime end represented by a nested chain of crosscuts, its impression is the intersection of the closures of the associated nested end domains. Saying the prime end is a singleton means that this impression is a single boundary point.

Statement recorded in the supplied source.

Must every bounded simply connected domain that is not conformally rigid have singleton impressions for all its prime ends? Equivalently, is non-rigidity sufficient for every prime end to be a singleton, as well as being necessary under the source's stated implication? [S2] The printed premise says singleton prime ends imply non-rigidity; the requested converse is the reverse implication. Update7.43 records Gaier's counterexample to the actual converse. [S2]

Short English statement

Does failure of conformal rigidity force every prime end to have a single-point impression?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307043 (AMR-022-7043). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.43 and its complete update. https://arxiv.org/abs/1809.07200.