Holomorphic Motions and Related Topics
Frederick Gardiner, Yunping Jiang, and Zhe Wang

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 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.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Holomorphic and Operator Theory
