Quasisymmetric Koebe uniformization of metric surfaces with uniformly relatively separated boundary

Abstract

We study the problem of determining when metric surfaces can be mapped quasisymmetrically onto a circle domain with uniformly relatively separated boundary components. Mario Bonk¹ completely characterized this for domains in the sphere. He proved that if the boundary components of a domain in the sphere are uniformly relatively separated uniform quasicircles then the domain is quasisymmetric to a circle domain. However, Merenkov and Wildrick² showed the existence of a metric surface whose boundary components are uniformly relatively separated uniform quasicircles which fails to be quasisymmetric to a circle domain. They offered an alternative characterization for metric surfaces using properties which are not invariant under quasisymmetries, and they expressed interest in replacing these with properties which are. In this dissertation, we introduce what we call the 2-transboundary Loewner property. This first appeared in Bonk's work¹. It is an analog of the Loewner property of Heinonen and Koskela³ in terms of Schramm's⁴ transboundary modulus. Using recent quasiconformal uniformization results of Rajala⁵ and Ikonen⁶, we prove that under some mild assumptions, a metric surface is quasisymmetric to a circle domain with uniformly relatively separated boundary components if and only if it is 2-transboundary Loewner. Since the 2-transboundary Loewner property is invariant under quasisymmetries, this answers the question posed by Merenkov and Wildrick. It is also a natural generalization of Bonk's result to metric surfaces, as it is equivalent to his theorem for domains in the sphere. Applying our results, we give new examples of metric surfaces which we show are quasisymmetric to circle domains.