Conformal Uniformization of Domains Bounded by Quasitripods

TL;DR

This work proves Koebe’s conjecture for a new class of domains by uniformizing domains whose non-point complement components are spread quasitripods satisfying a packing condition. The authors develop and apply Schramm’s transboundary modulus, establishing a uniform modulus bound for a critical path family, then construct convergent approximations by circle domains to obtain a conformal map to a circle domain that respects component types. Key innovations include a detailed decomposition of complement components into large/good/bad sets and a density-based admissibility built to handle detours around quasitripods, with energy bounds controlled by the quasitripod and packing constants. The results extend to cospread domains and are stable under quasi-Möbius transformations, yielding Möbius-invariant uniformization classes and strengthening Koebe-type uniformization beyond cofat or uniformly fat domains.

Abstract

We prove Koebe's conjecture and a version of Schramm's cofat uniformization theorem for domains $Ω\subset \mathbb C$ satisfying conditions involving quasitripods, i.e., quasisymmetric images of the standard tripod. If the non-point complementary components of $Ω$ contain uniform quasitripods with large diameters and satisfy a packing condition, then there exists a conformal map $f\colonΩ\to D$ onto a circle domain $D$. Moreover, $f$ preserves the classes of point-components and non-point components. The packing condition is satisfied if $Ω$ is cospread, i.e., if the complementary components contain uniform quasitripods in all scales.

Figures (6)

• Figure 1: $\Omega_j$ is the complement of the union of the solid quasitripods, conformally mapped by $f_j$ onto a circle domain. More components will be included in $C_N(\Omega_k)$ as $k$ increases.
• Figure 2: The annulus $\mathbb{A}(a,R)$ and some of the complementary components of $\Omega$, as in Theorem \ref{['thm:mainestimate']}. Paths can use complementary components as shortcuts to join the two boundary circles of the annulus.
• Figure 3: The dotted path shows a sample $\alpha \in \Gamma(a_p,b_p,\tau)$. It has two subpaths that each join $\mathbb{S}(b_p,r_p)$ to $\mathbb{D}(a_p,4\tau r_p)$.
• Figure 4: Some complementary components of the domain $\Omega$ constructed in the proof of Proposition \ref{['tripodexample']}.
• Figure 5: Example of a domain whose non-trivial complementary components (the "plus" signs) satisfy the conditions of Theorem \ref{['tripodkoebe']} and their diameters are not $\ell^2$-summable.

Theorems & Definitions (51)

• Theorem 1.1: Sch:95
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Proposition 1.5
• Corollary 1.6
• Lemma 2.1: Sch:95, Lemma 1.1
• Lemma 2.2
• Proposition 2.3
• Theorem 3.1