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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 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 X⊂ℍ3 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.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026weyl,
  title = {{The Weyl Problem for Planar CBB$(-1)$ Surfaces in Hyperbolic $3$-Space}},
  author = {Xinrong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35744},
  eprint = {2610.35744},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35744},
}
RIS
TY  - UNPB
TI  - The Weyl Problem for Planar CBB(−1) Surfaces in Hyperbolic $3$-Space
AU  - Xinrong Zhao
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35744
UR  - https://arxiv.org/abs/2610.35744
AB  - In this paper, we study the Weyl problem for noncompact convex surfaces in hyperbolic $3$-space 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⊂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 ℍ^3. Moreover, we show that the assumption of countably many ends can be dropped if Koebe's circle domain conjecture holds.
ER  -