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Conformal fitness and uniformization of holomorphically moving disks

Author: Saeed Zakeri
Journal: Trans. Amer. Math. Soc. 368 (2016), 1023-1049
MSC (2010): Primary 37Fxx, 30C85
Published electronically: May 6, 2015
MathSciNet review: 3430357
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Abstract: Let $ \{U_t\}_{t \in \mathbb{D}}$ be a family of topological disks on the Riemann sphere containing the origin 0 whose boundaries undergo a holomorphic motion over the unit disk $ \mathbb{D}$. We study the question of when there exists a family of Riemann maps $ g_t:(\mathbb{D},0) \to (U_t,0)$ which depends holomorphically on the parameter $ t$. We give five equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of the family $ \{ g_t \}_{t \in \mathbb{D}}$, and explore the consequences.

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Additional Information

Saeed Zakeri
Affiliation: Department of Mathematics, Queens College and Graduate Center of CUNY, Queens, New York 11367

Received by editor(s): February 11, 2013
Received by editor(s) in revised form: December 14, 2013
Published electronically: May 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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