Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3space
Abstract
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of HeSchramm on the Koebe conjecture, one establishes that every simply connected noncompact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramm's transboundary extremal lengths.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.08001
 Bibcode:
 2019arXiv191008001L
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Complex Variables
 EPrint:
 43 pages, 9 figures