Date: Monday, April 14, 2014
Location: 3096 East Hall (4:00 PM to 5:00 PM)
Title: 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 $0$ whose boundaries undergo a holomorphic motion $\partial U_0 \to \partial U_t$ over the unit disk $\mathbb D$. We investigate when there exists a family of Riemann maps $({\mathbb D},0) \to (U_t,0)$ which depends holomorphically on $t$. We give six equivalent conditions which provide analytic, dynamical and measuretheoretic characterizations for the existence of such family, and explore the consequences. Somewhat curiously, one of these equivalent conditions is the harmonicity of the map $t \mapsto \log \, r_t$, where $r_t$ is the conformal radius of the pointed disk $(U_t,0)$.
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Speaker: Saeed Zakeri
Institution: Queens College/CUNY Graduate Center
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