Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow
Abstract
We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.
Related
- Fractal Hyper-Trees: Combinatorial Enumeration, Symmetry Properties, and Ultrametric Structures
- Geometric ideal triangulations of hyperbolic 3-manifolds
- Non-geometric ideal triangulations with all edge valences at least six
- Hyper-derivative Algebraic Geometry Codes via Local Expansions
- Improved criteria for geometric properties of confluent hypergeometric functions
BibTeX
@misc{uemura2026local,
title = {{Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow}},
author = {Soichiro Uemura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34108},
eprint = {2609.34108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34108},
}
RIS
TY - UNPB
TI - Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow
AU - Soichiro Uemura
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34108
UR - https://arxiv.org/abs/2609.34108
AB - We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.
ER -