Arkiv för Matematik

, Volume 40, Issue 1, pp 1–26 | Cite as

Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

  • Christopher J. Bishop
Article

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.

Keywords

Convex Hull Unit Disk Sharp Constant Internal Path Lipschitz Homeomorphism 

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Copyright information

© Institut Mittag-Leffler 2002

Authors and Affiliations

  • Christopher J. Bishop
    • 1
  1. 1.Mathematics DepartmentState University of New York at Stony BrookStony BrookUSA

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