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arXiv · 1302.2704

Conformal Fitness and Uniformization of Holomorphically Moving Disks

Saeed Zakeri
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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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Saeed Zakeri. 2013-02-12. Conformal Fitness and Uniformization of Holomorphically Moving Disks. https://doi.org/10.1090/tran/6362

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