Abstract
We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal mapf:D→Ω can be factored as aK-quasiconformal self-map of the disk (withK independent of Ω) and a mapg:D→Ω with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Ω (with its internal path metric) to the unit disk.
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The author is partially supported by NSF Grant DMS 9800924.
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Bishop, C.J. Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture. Ark. Mat. 40, 1–26 (2002). https://doi.org/10.1007/BF02384499
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DOI: https://doi.org/10.1007/BF02384499