The Weyl Problem for Planar CBB$(-1)$ Surfaces in Hyperbolic $3$-Space

In this paper, we study the Weyl problem for noncompact convex surfaces in hyperbolic $3$-space H³. We prove that any planar CBB$(-1)$ surface with countably many ends can be isometrically embedded as the boundary of a closed convex subset X³ such that each connected component of the ideal boundary of X is either a round disk or a single point. By known rigidity results, such an embedding is unique up to isometries of H³. Moreover, we show that the assumption of countably many ends can be dropped if Koebe's circle domain conjecture holds.