Hyperbolic
convexity and
conformal
reflections
Dr. Edward Crane
Abstract
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.
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Date: |
June 19, 2008 (Thursday) |
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Time: |
3:00 – 4:00pm |
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Place: |
Room 517, Meng Wah Complex, HKU |
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All are
welcome |
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