Conformal Fitness and Uniformization of Holomorphically Moving Disks

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.

32 pages, 4 figures

Cited by in corpus (2)

• Continuity of capacity of a holomorphic motion
• Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains