Existence of Ahlfors Islands Maps Between Compact Surfaces

Definitions and mathematical statement

Let $X$ and $Y$ be compact Riemann surfaces, with $X$ hyperbolic, meaning that its universal covering surface is conformally equivalent to $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$. Following the original statement, let $W\subseteq Y$ be a domain, that is, a nonempty connected open subset.

The supplied record does not define an Ahlfors islands map. Retain this source terminology by writing \[ \mathcal I(W,X)=\{f:W\to X : f\text{ is an Ahlfors islands map in the source's intended sense}\}. \] This notation does not supply the missing definition: the precise islands condition, its boundary quantifiers, and any exceptional or excluded maps must be checked in the cited source before this becomes a complete formal statement.

The problem is to characterize the triples $(Y,W,X)$ satisfying the preceding assumptions for which \[ \mathcal I(W,X)\ne\varnothing. \] The target-dependence question asks whether, for each fixed admissible pair $(Y,W)$, this existence property is independent of the compact hyperbolic target. Explicitly, determine whether \[ \mathcal I(W,X_1)\ne\varnothing \quad\Longleftrightarrow\quad \mathcal I(W,X_2)\ne\varnothing \] holds for every pair of compact hyperbolic Riemann surfaces $X_1,X_2$, and characterize the dependence if it does not.

Unresolved scope qualification. The record does not explicitly say whether $W$ must be proper. The notation above records the domain assumption without resolving whether $W=Y$ is admissible; that boundary case and the missing definition require source review.

Short English statement

Characterize when a domain in a compact Riemann surface admits an Ahlfors islands map to a compact hyperbolic Riemann surface, and determine whether existence changes with the target. The supplied record leaves the exact islands-map definition and the admissibility of the whole-surface domain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001396$, source problem code catalogue entry, titled "Existence of Ahlfors Islands Maps Between Compact Surfaces." The original statement explicitly calls $W$ a domain. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied source citation: Mini-Workshop: The Escaping Set in Transcendental Dynamics (2010), Oberwolfach Reports, pp. 2927--2964, DOI 10.4171/owr/2009/54. https://doi.org/10.4171/owr/2009/54. This is also the supplied literature-source link labeled "original OWR report." The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature-source label: "Rempe Rippon exotic Ahlfors islands." https://arxiv.org/abs/1008.1724.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports existence on the sphere and torus for arbitrary proper domains, examples on hyperbolic surfaces, and unresolved general existence criteria and target dependence. These are supplied dated curation claims, not independently verified conclusions. The sphere, torus, and hyperbolic-surface wording does not explicitly distinguish source and target roles. The background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This is supplied verification metadata; no independent source consultation is asserted here.