The Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space (Q6639709)

A circle domain is a domain in the Riemann sphere \(S^2\) whose boundary components are circles or points. A classical conjecture of Koebe asserts that every planar domain is conformal to a circle domain. In the present paper, it is proved that Koebe's conjecture is equivalent to another conjecture in hyperbolic geometry. \N\NWe proceed to describe this conjecture. Let \(H^3\) be the ball model of hyperbolic \(3\)-space whose boundary is \(S^2\). For a closed set \(Y\) in \(H^3\cup S^2\), let \(C(Y)\) be the hyperbolic convex hull of \(Y\). A closed set \(Y\subset S^2\) is said to be of circle type if its complement is a circle domain. By the work of Thurston, if \(Y\subset S^2\) contains at least two points, then the boundary \(\partial C(Y)\) of \(C(Y)\) in \(H^3\) is a genus-zero complete hyperbolic surface. It was conjectured in [the authors and \textit{T. Wu}, Geom. Topol. 26, No. 3, 937--987 (2022; Zbl 1502.30137)] that, conversely, every genus-zero complete hyperbolic surface is isometric to \(\partial C(Y)\) for some circle type closed set \(Y\subset S^2\).\N\NThe main tool in the proof that this conjecture is equivalent to Koebe's is Schramm's transboundary extremal length. The authors make special effort to present the details of the long proof. In particular, they prove results on the Hausdorff and Alexandrov convergence and related convergence and Arzela-Ascoli type theorems in hyperbolic geometry.