Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs

TL;DR

This paper uses holomorphic motions to establish quasiconformal rigidity of parabolic germs, providing a conceptual proof of Fatou linearization and extending local germs to near-conformal homeomorphisms.

Contribution

It introduces a novel application of holomorphic motions to prove quasiconformal rigidity and offers a new proof of Fatou linearization for parabolic germs.

Findings

Parabolic germs are quasiconformally rigid.

Holomorphic motions facilitate a conceptual proof of Fatou linearization.

Finite analytic germs can be extended to near-conformal homeomorphisms.

Abstract

By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points.