Kai Rajala

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math.CV2026

Uniformization of cofat domains on metric two-spheres

Chengxi Li, Kai Rajala

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$ . Specifically, we show that if $I^c ◯ \subset X$ is a cofat d…

math.CV2025

Exhaustions of circle domains

Dimitrios Ntalampekos, Kai Rajala

Koebe's conjecture asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. We prove that every domain $I^c ◯$ satisfying Koebe's conjecture admit…

math.CV2025

Conformal Uniformization of Domains Bounded by Quasitripods

Behnam Esmayli, Kai Rajala

We prove Koebe's conjecture and a version of Schramm's cofat uniformization theorem for domains $I^c ◯ \subset C$ satisfying conditions involving quasitripods, i.e., quasisymme…

math.CV2024

Rigid circle domains with non-removable boundaries

Kai Rajala

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $I^c ◯$ on the Riemann sphere with conformally non-removable boundary. Her…