Conformal fitness and uniformization of holomorphically moving disks

Zakeri, Saeed

doi:10.1090/tran/6362

Conformal fitness and uniformization of holomorphically moving disks
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Trans. Amer. Math. Soc. 368 (2016), 1023-1049 Request permission

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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