Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space

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 He-Schramm on the Koebe conjecture, one establishes that every genus zero complete hyperbolic surface with countably many topological ends is isometric to the boundary of the convex hull of a closed set whose components are round disks and points in $\partial \mathbb H^3$. The main tool we use is Schramm's transboundary extremal lengths.