Geometry Seminar



Hyperbolic convexity and

conformal reflections


Dr. Edward Crane

University of Bristol, UK & IMR Junior Fellow, HKU





Any connected open subset of the complex plane that omits at least two points carries a natural conformal metric called the hyperbolic metric. A classical result of Jorgensen from the 1950s says that a Euclidean disc is a hyperbolically convex subset of any simply-connected hyperbolic plane domain that contains it. This result has been generalized in various directions by Minda and Solynin.  Here we give an application of these ideas to the problem of finding the best constant in the Hayman-Wu theorem.

Jorgensen's theorem prompted us to look for a conformally invariant characterization of a Euclidean disc in a hyperbolic plane domain. We give a criterion in terms of the existence of a conformal reflection. A related criterion for hyperbolic convexity follows. There is also a quasiconformal analogue of our result.



June 19, 2008 (Thursday)


3:00 – 4:00pm


Room 517, Meng Wah Complex, HKU




All are welcome