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PreprintarXiv licencearXiv

Discrete uniformization of polyhedral surfaces

  • Feng Luo
  • Yanwen Luo
  • Zhenghao Rao
  • Xinrong Zhao
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Abstract

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026discrete,
  title = {{Discrete uniformization of polyhedral surfaces}},
  author = {Feng Luo and Yanwen Luo and Zhenghao Rao and Xinrong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.22491},
  eprint = {2608.22491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22491},
}
RIS
TY  - UNPB
TI  - Discrete uniformization of polyhedral surfaces
AU  - Feng Luo
AU  - Yanwen Luo
AU  - Zhenghao Rao
AU  - Xinrong Zhao
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2608.22491
UR  - https://arxiv.org/abs/2608.22491
AB  - The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.
ER  -