Carathéodory balls and proper holomorphic maps on multiply-connected planar domains

Abstract

In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carathéodory metric in some planar domains of finite connectivity greater than $2$, and hence resolve a problem posed by Jarnicki, Pflug and Vigué in 1992. We also establish a corresponding result for some pseudoconvex domains in $\mathbb{C}^n$ for $n \ge 2$. This result will follow from an explicit characterization (up to biholomorphisms) of proper holomorphic maps from a non-degenerate finitely-connected planar domain, $Ω$, onto the standard unit disk $\mathbb{D}$ which answers a question posed by Schmieder in 2005. Similar to Bell and Kaleem's characterization of proper holomorphic maps in terms of Grunsky maps (2008), our characterization of proper holomorphic maps from $Ω$ onto $\mathbb{D}$ is an analogous result to Fatou's famous result that proper holomorphic maps of the unit disk onto itself are finite Blashcke products. Our approach uses a harmonic measure condition of Wang and Yin (2017) on the existence of a proper holomorphic map with prescribed zeros. We will see that certain functions $η(\cdot,p)$ play an analogous role to the Möbius transformations that fix $\mathbb{D}$ in finite Blaschke products. These functions $η(\cdot,p)$ map $Ω$ conformally onto the unit disk with circular arcs (centered at 0) removed and map $p$ to $0$ and can be given in terms of the Schottky-Klein prime function. We also extend a result of Grunsky (1941) and hence introduce a parameter space for proper holomorphic maps from $Ω$ onto $\mathbb{D}$.