Carath\'{e}odory balls and proper holomorphic maps on multiply-connected planar domains

In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carath\'{e}odory metric in some planar domains of finite connectivity greater than $2$, and hence resolve a problem posed by Jarnicki, Pflug and Vigu\'{e} 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, $\Omega$, 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 $\Omega$ 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 $\eta(\cdot,p)$ play an analogous role to the M\"{o}bius transformations that fix $\mathbb{D}$ in finite Blaschke products. These functions $\eta(\cdot,p)$ map $\Omega$ 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 $\Omega$ onto $\mathbb{D}$.