Conformal Uniformization of Domains Bounded by Quasitripods

TL;DR

This paper proves a version of Koebe's conjecture and Schramm's uniformization theorem for domains with boundaries involving quasitripods, establishing conditions under which they can be conformally mapped onto circle domains.

Contribution

It extends uniformization results to domains bounded by quasitripods, providing new conditions for conformal equivalence to circle domains.

Findings

Proves Koebe's conjecture for domains with quasitripod boundaries.

Establishes conditions for conformal maps preserving boundary classes.

Shows cospread domains admit uniform quasitripods at all scales.

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.