Conformal Mappings Through the Lens of Invariant Metrics
Bharathi Thiruvengadam, Jaikrishnan Janardhanan
TL;DR
The paper shows that balls with respect to invariant metrics on hyperbolic planar domains are finitely-connected, and uses this to obtain streamlined proofs of classical conformal-mapping results. By restricting holomorphic maps to Kobayashi and Carathéodory balls, it derives global conclusions from local geometric structure, including a short proof of the Maskit theorem and the Aumann–Carathéodory isotropy theorem. The main technical contributions are the finite-connectivity of Kobayashi balls (and of certain Carathéodory balls) and their use to obtain simple, geometric proofs in conformal mapping theory. The approach highlights the power of invariant metrics in translating local hyperbolic geometry into global rigidity results, with potential extension to complex manifolds covered by hyperbolic convex domains.
Abstract
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 the 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.