The 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 in the hyperbolic 3-space. Applications of the result to discrete conformal geometry will be discussed. The main tool we use is Schramm's transboundary extremal lengths.