Holomorphic Motions and Related Topics

TL;DR

This paper provides an expository overview of holomorphic motions, their continuity properties, and applications to complex dynamics, including proving the Fatou linearization theorem using holomorphic motions.

Contribution

It offers a comprehensive exposition of the holomorphic motion theorem, analyzes tangent vector continuity, and applies these concepts to establish metric equivalences and dynamics results.

Findings

Holomorphic motions have $| ext{epsilon} \, ext{log} \, ext{epsilon}|$ modulus of continuity.

Holomorphic motions induce H"older continuity on mappings.

Holomorphic motions show that Kobayashi's and Teichm"uller metrics coincide on Teichm"uller space.

Abstract

In this article we give an expository account of the holomorphic motion theorem based on work of M\`a\~n\'e-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have $|\epsilon \log \epsilon|$ moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce H\"older continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichm\"uller's metrics on the Teichm\"uller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.