Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space

Abstract

In this paper, we study the rigidity of noncompact convex sets in hyperbolic 3-space. We prove that any intrinsic isometry between the boundaries of two closed, noncompact convex subsets of hyperbolic 3-space of dimension at least two extends to a global isometry of the ambient space, provided that their ideal boundaries are circle-type closed sets with countably many connected components. Moreover, the same conclusion holds if the ideal boundaries consist of finitely many mutually disjoint disks together with a set of one-dimensional Hausdorff measure zero. This result generalizes a recent rigidity theorem of Luo, Luo, and Rao by allowing the ideal boundaries to contain disk components. As a direct consequence, we establish a uniqueness result concerning the Weyl problem for convex surfaces in hyperbolic 3-space, as proposed by Luo and Wu. In particular, our approach provides an alternative proof of the discrete Schwarz lemma. The proof uses Pogorelov's rigidity theorem for compact convex bodies in ℝ³, the Pogorelov map, and properties of locally convex surfaces in ℝ³.

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BibTeX RIS

BibTeX

@misc{zhao2026rigidity,
  title = {{Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space}},
  author = {Xinrong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15402},
  eprint = {2606.15402},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15402},
}

RIS

TY  - UNPB
TI  - Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
AU  - Xinrong Zhao
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2606.15402
UR  - https://arxiv.org/abs/2606.15402
AB  - In this paper, we study the rigidity of noncompact convex sets in hyperbolic 3-space. We prove that any intrinsic isometry between the boundaries of two closed, noncompact convex subsets of hyperbolic 3-space of dimension at least two extends to a global isometry of the ambient space, provided that their ideal boundaries are circle-type closed sets with countably many connected components. Moreover, the same conclusion holds if the ideal boundaries consist of finitely many mutually disjoint disks together with a set of one-dimensional Hausdorff measure zero. This result generalizes a recent rigidity theorem of Luo, Luo, and Rao by allowing the ideal boundaries to contain disk components. As a direct consequence, we establish a uniqueness result concerning the Weyl problem for convex surfaces in hyperbolic 3-space, as proposed by Luo and Wu. In particular, our approach provides an alternative proof of the discrete Schwarz lemma. The proof uses Pogorelov's rigidity theorem for compact convex bodies in ℝ^3, the Pogorelov map, and properties of locally convex surfaces in ℝ^3.
ER  -

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