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arXiv · 2610.35744

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

Xinrong Zhao
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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.

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Xinrong Zhao. 2026-10-26. The Weyl Problem for Planar CBB$(-1)$ Surfaces in Hyperbolic $3$-Space. https://arxiv.org/abs/2610.35744

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