Intrinsic circle domains (Q2922922)

Koebe's ``Kreisnormierungstheorem'' states that every finitely connected domain \(D\) in the Riemann sphere \(\widehat{\mathbb{C}}\) is conformally equivalent to a circle domain, i.e., a domain \(\Omega\subseteq \widehat{\mathbb{C}}\) whose boundary components are points and circles. The domain \(\Omega\) is unique up to conformal automorphisms of \(\widehat{\mathbb{C}}\).NEWLINENEWLINEThis theorem has been generalized by He and Schramm to the following more general setting:NEWLINENEWLINELet \(R\) be a Riemann surface endowed with its hyperbolic, Euclidean or spherical metric of constant curvature \(-1\), \(0\) and \(+1\), respectively. A \textit{closed geometric disc in \(R\)} is a closed ball of positive radius in \(R\) which is homeomorphic to the closed unit disc in \(\mathbb{C}\), i.e., the radius of the ball is strictly less than the injectivity radius of the metric at the center of the ball. A \textit{circle domain in \(R\)} is a connected open subset of \(R\) whose complementary components are points or closed geometric discs in \(R\). In [Ann. Math. (2) 137, No. 2, 369--406 (1993; Zbl 0777.30002)] \textit{Z.-X. He} and \textit{O. Schramm} showed the following:NEWLINENEWLINELet \(D\) be an open Riemann surface with finite genus and at most countably many ends. Then there exists a closed Riemann surface \(R\) such that \(D\) is conformally homeomorphic to a circle domain \(\Omega\) in \(R.\) Moreover, the pair \((R, \Omega)\) is unique up to conformal homeomorphisms.NEWLINENEWLINEThe author of this article introduces a new type of canonical domains, called \textit{intrinsic circle domains}: A connected open subset \(\Omega\) of a Riemann surface \(R\) is called \textit{intrinsic circle domain in \(R\)} if each component \(L\) of the complement of \(\Omega\) is either a point or a closed geometric disc in the Riemann surface \(\Omega \cup L\). This definition is motivated by certain extremal problems in geometric function theory. The main result of the article, Theorem 2, states the following:NEWLINENEWLINELet \(D\) be a Riemann surface of finite genus \(g\) and finite connectivity. Then there exists a compact Riemann surface \(R\) of genus \(g\) and an intrinsic circle domain \(\Omega\) in \(R\) which is conformally homeomorphic to \(D\). Moreover, the pair \((R, \Omega)\) is unique up to conformal homeomorphisms.NEWLINENEWLINEThe author also discusses the case of countably infinite connectivity and explains how circle packings can be used to find numerical approximations of finitely connected intrinsic circle domains in \(\widehat{\mathbb{C}}\).