Holomorphic Motions and Related Topics

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.