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

Abstract

In this paper, we study the Weyl problem for noncompact convex surfaces in hyperbolic $3$-space $\mathbb{H}^3$. We prove that any planar CBB$(-1)$ surface with countably many ends can be isometrically embedded as the boundary of a closed convex subset $X\subset\mathbb{H}^3$ 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 $\mathbb{H}^3$. Moreover, we show that the assumption of countably many ends can be dropped if Koebe's circle domain conjecture holds.