Conformal Uniformization of Domains Bounded by Quasitripods

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 $f:\Omega \to D$ 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.

43 pages, 6 figures