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

@inproceedings{Ng2021CaratheodoryBA, title={Carath\'\{e\}odory balls and proper holomorphic maps on multiply-connected planar domains}, author={Tuen Wai Ng and Chiu Chak Tang and Jonathan Tsai}, year={2021} }

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 C for n ≥ 2. This result will follow from an explicit characterization (up to biholomorphisms) of proper holomorphic maps from a non-degenerate finitely-connected…

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