Completing the extremal theory of uniformization by square domains: the maximizer of Bonk's functional and an independent variational existence proof

Bonk (2016) found an extremal problem whose solution uniformizes a finitely connected plane domain by a square domain: among conformal maps of a finitely connected domain containing the point at infinity, normalized by fixing infinity with derivative one, the functional combining the real part of the mapping coefficient with the deviation of the image's complementary components from square shape has a unique minimizer, and this minimizer is precisely the normalized conformal map onto a domain whose complementary components are axis-parallel squares or points. The proof, however, presupposes the existence of the square-domain map via the Brandt–Harrington uniformization theorem, and the note explicitly records two unresolved questions: it is not clear which conformal map should maximize the functional, and it is open whether the functional — or an associated variational problem for real-valued functions corresponding to the imaginary parts of candidate maps — can yield an existence proof of square-domain uniformization that does not rely on Brandt–Harrington. This problem asks for the completion of that extremal theory: identify the maximizer of the functional (or prove non-attainment), and give a self-contained variational existence proof of square-domain uniformization for finitely connected domains. By a theorem of Schramm, uniformization by square domains is equivalent to Koebe's circle-domain conjecture, so an extremal theory extending beyond finite connectivity is a recorded route toward that famous problem.

Background

Uniformization of planar domains asks for a conformal map — a one-to-one holomorphic bijection — from a given domain onto a domain of some prescribed canonical shape. Here a domain is a connected open subset of the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$, and it is finitely connected if its complement has finitely many connected components. The classical model is uniformization by parallel slit domains, domains whose complementary components are line segments parallel to a fixed direction (or single points): every domain containing $\infty$ is conformally equivalent to a parallel slit domain, and the extremal route to this theorem is the prototype for the present problem. If $f(z)=z+a_{1}/z+\dots$ denotes the expansion near $\infty$ of a conformal map with the hydrodynamic normalization (fixing $\infty$ with derivative $1$), then the functional $f\mapsto\mathrm{Re}(a_{1})$ has a unique minimum, attained exactly by the map onto a slit domain with vertical slits, and a unique maximum, attained by the map onto a slit domain with horizontal slits — both sides of the extremal picture are understood. The natural next target is Koebe's Kreisnormierungsproblem (1908): whether every domain in $\widehat{\mathbb{C}}$ is conformally equivalent to a circle domain, i.e., a domain whose complementary components are points or closed round disks. This remains one of the great open problems of classical function theory. It holds for finitely connected domains (Koebe) and for countably connected domains (He–Schramm, 1993); Schramm's method of transboundary extremal length — a variant of classical extremal length, the conformally invariant quantity measuring how 'wide' a family of curves is (its reciprocal is the conformal modulus), modified so that it behaves well for conformal maps of multiply connected domains — gives an alternative proof in the countable case and shows that the full conjecture is equivalent to uniformization by square domains (this history is recounted in Ntalampekos's 2026 survey). A square domain is a domain in $\widehat{\mathbb{C}}$ containing $\infty$ whose complementary components are single points or closed squares with sides parallel to the real or imaginary axis. That every finitely connected domain maps conformally onto a square domain is essentially classical, going back to Grötzsch (1935), and the Brandt–Harrington uniformization theorem (1980/1982) gives a far stronger statement: for a finitely connected domain one may prescribe each complementary component of the image domain up to a homothety $z\mapsto az+b$, $a>0$. Bonk (2016) added the missing extremal characterization. For a finitely connected $\Omega\subset\widehat{\mathbb{C}}$ with $\infty\in\Omega$, let $\mathcal{F}$ be the family of conformal maps $f\colon\Omega\to D=f(\Omega)\subset\widehat{\mathbb{C}}$ with $f(z)=z+a_{1}/z+\dots$ near $\infty$. For the complementary components $K_{1},\dots,K_{n}$ of $D$, let $A_{j}$ be the Euclidean area of $K_{j}$ and $V_{j}=\max_{w\in K_{j}}\mathrm{Im}(w)-\min_{w\in K_{j}}\mathrm{Im}(w)$ its vertical variation. Bonk's functional is $\mathcal{S}(f)=2\pi\,\mathrm{Re}(a_{1})+\sum_{j=1}^{n}(V_{j}^{2}-A_{j})$. For an axis-parallel square of side $s$ one has $V_{j}^{2}-A_{j}=s^{2}-s^{2}=0$. The summand is not a one-signed deviation from square shape: for an axis-parallel rectangle of width $a$ and height $b$ it equals $b(b-a)$, so it vanishes both on square-shaped components ($a=b$) and on degenerate horizontal components ($b=0$); it is negative for components wider than tall and positive for those taller than wide. Bonk's Theorem 1.1 states that $\mathcal{S}$ has a unique minimizer $f_{0}\in\mathcal{F}$, and that $f_{0}$ is the unique normalized conformal map of $\Omega$ onto a square domain. The proof first obtains a square-domain map from the Brandt–Harrington theorem (in Schramm's formulation) and then proves an inequality $\mathcal{S}(f)\geq 0$ for normalized conformal maps of a square domain, with equality only for the identity, via an asymptotic modulus estimate inspired by Schramm's transboundary extremal length. The extremal characterization is thus not self-contained, and its other side is unknown: the published note records that it is not clear which conformal map should maximize $\mathcal{S}$, and that although a normal-family argument (a normal family of holomorphic maps is one in which every sequence has a locally uniformly convergent subsequence) guarantees a minimizer in the conformal class, there seems to be no simple variational argument forcing the minimizer's image to be a square domain — though a more general variational problem for real-valued functions $v$ (the imaginary parts of candidate conformal maps) might yield the existence directly, in the spirit of classical potential-theoretic methods for uniformization.

Problem Statement

Let $\Omega\subset\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ be an arbitrary finitely connected domain with $\infty\in\Omega$, and let $\mathcal{F}$ be the class of conformal maps $f\colon\Omega\to D=f(\Omega)\subset\widehat{\mathbb{C}}$ normalized near $\infty$ by $f(z)=z+a_{1}/z+\dots$. For the complementary components $K_{1},\dots,K_{n}$ of $D$, set $A_{j}$ equal to the Euclidean area of $K_{j}$ and $V_{j}=\max_{w\in K_{j}}\mathrm{Im}(w)-\min_{w\in K_{j}}\mathrm{Im}(w)$, and define

$$\mathcal{S}(f)=2\pi\,\mathrm{Re}(a_{1})+\sum_{j=1}^{n}\left(V_{j}^{2}-A_{j}\right).$$

Bonk proved that $\mathcal{S}$ has a unique minimizer in $\mathcal{F}$ and that this minimizer is the unique normalized conformal map of $\Omega$ onto a square domain (a domain whose complementary components are points or axis-parallel squares). The problem is to complete the extremal theory of $\mathcal{S}$, in two linked parts with one common objective — making square-domain uniformization of finitely connected domains follow entirely from the variational principle, as slit-domain uniformization does from $\mathrm{Re}(a_{1})$:

(1) The maximizer. Determine $\sup_{f\in\mathcal{F}}\mathcal{S}(f)$ for a general finitely connected $\Omega$; decide whether the supremum is attained, and if so identify the maximizing map or maps and the geometry of the image domains, in analogy with the slit picture where the minimum and maximum of $\mathrm{Re}(a_{1})$ correspond to the two orthogonal slit directions. If the supremum is not attained, characterize the maximizing sequences and the geometry of the limiting image domains.

(2) Independent existence. Prove, without invoking the Brandt–Harrington uniformization theorem, Grötzsch's construction, or any other prior existence result for square-domain or circle-domain uniformization, that the minimizer of $\mathcal{S}$ over $\mathcal{F}$ — whose existence follows from a normal-family compactness argument — maps $\Omega$ onto a square domain.

Both parts are posed for all finitely connected domains $\Omega$, including degenerate (point) complementary components and all finite connectivities $n\geq 0$; in the doubly connected case they agree with the explicit annulus and rectangle maps. The extension of this extremal problem to countably or infinitely connected domains — which, by Schramm's equivalence between Koebe's conjecture and square-domain uniformization, is the square-domain route toward the Kreisnormierungsproblem — is an explicitly recorded open direction of the surrounding program and part of this problem's context, but it is not part of the acceptance boundary stated here.

References

1. Mario Bonk, "Uniformization by square domains", The Journal of Analysis 24 (2016), no. 1, 103–110. DOI: 10.1007/s41478-016-0015-4, https://doi.org/10.1007/s41478-016-0015-4 (preprint: arXiv:1606.00349, https://arxiv.org/abs/1606.00349)
2. Herbert Grötzsch, "Eine Bemerkung zum Koebeschen Kreisnormierungsprinzip", Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Naturwissenschaftliche Klasse 87 (1935), 319–324. Zbl: 0014.16602, https://zbmath.org/0014.16602
3. Manfred Brandt, "Ein Abbildungssatz für endlich-vielfach zusammenhängende Gebiete", Bulletin de la Société des Sciences et des Lettres de Łódź 30 (1980), no. 3. Zbl: 0463.30006, https://zbmath.org/0463.30006
4. A. N. Harrington, "Conformal mappings onto domains with arbitrarily specified boundary shapes", Journal d'Analyse Mathématique 41 (1982), 39–53. DOI: 10.1007/BF02803393, https://doi.org/10.1007/BF02803393
5. Z.-X. He and O. Schramm, "Fixed points, Koebe uniformization and circle packings", Annals of Mathematics (2) 137 (1993), 369–406. DOI: 10.2307/2946541, https://doi.org/10.2307/2946541
6. O. Schramm, "Square tilings with prescribed combinatorics", Israel Journal of Mathematics 84 (1993), 97–118. DOI: 10.1007/BF02761693, https://doi.org/10.1007/BF02761693
7. O. Schramm, "Transboundary extremal length", Journal d'Analyse Mathématique 66 (1995), 307–329. DOI: 10.1007/BF02788827, https://doi.org/10.1007/BF02788827
8. O. Schramm, "Conformal uniformization and packings", Israel Journal of Mathematics 93 (1996), 399–428. DOI: 10.1007/BF02761115, https://doi.org/10.1007/BF02761115
9. Alexander Yu. Solynin and Nadeesha C. Vidanage, "Uniformization by rectangular domains: A path from slits to squares", Journal of Mathematical Analysis and Applications 486 (2020), no. 2, 123927. DOI: 10.1016/j.jmaa.2020.123927, https://doi.org/10.1016/j.jmaa.2020.123927
10. Dimitrios Ntalampekos, "Uniformization problems in the plane: A survey", preprint (2026). arXiv:2603.15098, https://arxiv.org/abs/2603.15098
11. Christina Karafyllia and Dimitrios Ntalampekos, "Uniformization of Gromov hyperbolic domains by circle domains", preprint (2024). arXiv:2405.13782, https://arxiv.org/abs/2405.13782
12. Kai Rajala, "Uniformization of planar domains by exhaustion", American Journal of Mathematics 147 (2025), no. 4, 1025–1051. DOI: 10.1353/ajm.2025.a966291, https://doi.org/10.1353/ajm.2025.a966291 (preprint: arXiv:2111.00845, https://arxiv.org/abs/2111.00845)
13. Dimitrios Ntalampekos and Kai Rajala, "Exhaustions of circle domains", International Mathematics Research Notices 2025 (2025), no. 19, article rnaf296. DOI: 10.1093/imrn/rnaf296, https://doi.org/10.1093/imrn/rnaf296 (preprint: arXiv:2312.06840, https://arxiv.org/abs/2312.06840)
14. Yi Zhong, "Koebe uniformization of nondegenerate domains with bounded gap-ratio", preprint (2024). arXiv:2408.03484, https://arxiv.org/abs/2408.03484
15. Xiaoguang Wang and Yi Zhong, "Koebe uniformization for infinitely connected attracting Fatou domains", preprint (2024). arXiv:2406.13524, https://arxiv.org/abs/2406.13524