Abstract
Let { U t } t ∈ D \{U_t\}_{t \in \mathbb {D}} be a family of topological disks on the Riemann sphere containing the origin 0 0 whose boundaries undergo a holomorphic motion over the unit disk D \mathbb {D} . We study the question of when there exists a family of Riemann maps g t : ( D , 0 ) → ( U t , 0 ) g_t:(\mathbb {D},0) \to (U_t,0) which depends holomorphically on the parameter t t . We give five equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of the family { g t } t ∈ D \{ g_t \}_{t \in \mathbb {D}} , and explore the consequences.
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