Seminar Event Detail

Complex Analysis, Dynamics and Geometry

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 measure-theoretic 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 $(U_t,0)$.


Speaker:  Saeed Zakeri
Institution:  Queens College/CUNY Graduate Center

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