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<p>The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar domains. In particular, we show that any conformal self-map of a hyperbolic planar domain with three fixed points is the identity. We also give a new and very simple proof of a theorem by Aumann and Carathéodory that states that the isotropy groups of a hyperbolic planar domain are either finite or the domain is simply-connected.</p>

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Conformal Mappings Through the Lens of Invariant Metrics

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Bharathi Thiruvengadam,
  • Jaikrishnan Janardhanan

摘要

The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar domains. In particular, we show that any conformal self-map of a hyperbolic planar domain with three fixed points is the identity. We also give a new and very simple proof of a theorem by Aumann and Carathéodory that states that the isotropy groups of a hyperbolic planar domain are either finite or the domain is simply-connected.