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PreprintCC BY 4.0arXiv

Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

  • Yuerong Bian
  • Xu Xu
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Abstract

We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{bian2026combinatorial,
  title = {{Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces}},
  author = {Yuerong Bian and Xu Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.25691},
  eprint = {2607.25691},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.25691},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces
AU  - Yuerong Bian
AU  - Xu Xu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2607.25691
UR  - https://arxiv.org/abs/2607.25691
AB  - We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.
ER  -