Connectivity Detection by Planar-Domain Functions

Definitions and mathematical statement

Question. For the monotone domain functions constructed from Green and kernel functions of planar domains, which expressions require the compared domains to be homotopic, and can these functions detect connectivity? A planar domain is a connected open subset $\Omega\subset{\mathbb{C}}$. Thus connectivity here may mean its order of connectivity, conventionally the number $\kappa(\Omega)$ of connected components of $\widehat{\mathbb{C}}\setminus\Omega$, rather than merely connectedness of $\Omega$; this intended meaning is not explicitly fixed in the source. The order may be finite or infinite unless the domain class is restricted.

To make the two questions precise, the omitted domain functional $F_\Omega$, its Green-function and kernel constructions, their normalizations, and admissible comparison maps must first be specified. The first question asks which claimed expressions or monotonicity comparisons remain valid without the source's homotopy requirement and which require it. The direction of monotonicity and the meaning of homotopic domains are not supplied: homotopy equivalence, a deformation through domains, and a relation between marked comparison maps are different possible conditions. No one of them is silently chosen here.

Once a data space and an equality convention for $F_\Omega$ have been fixed, a possible precise connectivity-detection question is whether\[F_{\Omega_1}=F_{\Omega_2}\quad\Longrightarrow\quad\kappa(\Omega_1)=\kappa(\Omega_2)\] for every pair of admissible domains, or equivalently whether $\kappa(\Omega)$ can be recovered as a function of those data. This conditional notation does not assume the functional is a scalar, fix a kernel, impose finite connectivity, or claim recovery of the entire conformal type. Without the expressions and data-identification convention, the intended question is not yet a determined mathematical assertion.

The supplied triage dated 2026-08-21 labels the item open, explicitly observes that the data-to-connectivity question is not formally specified, and notes related potential-theoretic and conformal-mapping tools without locating a direct theorem deciding the extracted question. That report is a dated search assessment rather than a proof that no such theorem exists.

Short English statement

Identify which Green-function or kernel-based domain comparisons need a homotopy assumption, and whether the resulting data determine the domain's number of complementary components. The functional, comparison relation, and meaning of connectivity are omitted, so the recovery statement is conditional.

Sources

• Local UnsolvedMath registry, permanent ID 30000029, code catalogue entry; Connectivity Detection by Planar-Domain Functions, Oberwolfach Reports Open Problems.
• Funktionentheorie (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/07, pp. 349--406. https://doi.org/10.4171/owr/2004/07. Proposed year 2004; no proposer named.
• Supplied literature label Oberwolfach report: https://ems.press/content/serial-article-files/45883.
• Supplied literature label Conformal-mapping paper: https://epubs.siam.org/doi/abs/10.1137/S0036144598349277. No author list or precise result is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. Verify the dated assessment independently.