Conformal fitness and uniformization of holomorphically moving disks
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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.References
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Additional Information
- Saeed Zakeri
- Affiliation: Department of Mathematics, Queens College and Graduate Center of CUNY, Queens, New York 11367
- Email: saeed.zakeri@qc.cuny.edu
- Received by editor(s): February 11, 2013
- Received by editor(s) in revised form: December 14, 2013
- Published electronically: May 6, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 1023-1049
- MSC (2010): Primary 37Fxx, 30C85
- DOI: https://doi.org/10.1090/tran/6362
- MathSciNet review: 3430357