Conformal Uniformization of Domains Bounded by Quasitripods

Behnam Esmayli; Kai Rajala

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arXiv · 2401.08485

Conformal Uniformization of Domains Bounded by Quasitripods

Behnam Esmayli·Kai Rajala

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Abstract

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

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BibTeX RIS

Behnam Esmayli, Kai Rajala. 2024-01-16. Conformal Uniformization of Domains Bounded by Quasitripods. https://arxiv.org/abs/2401.08485

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