Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range

@misc{egorov2026minimal,
  title = {{Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range}},
  author = {Andrey Egorov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.14223},
  eprint = {2608.14223},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14223},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range
AU  - Andrey Egorov
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.14223
UR  - https://arxiv.org/abs/2608.14223
AB  - We study the minimum-volume problem for finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number α. In the non-obtuse case one necessarily has π/3≤ α≤ π/2. We prove that throughout the full range in which the regular hyperbolic tetrahedron with dihedral angle α exists, π/3≤ α<arccos1/3, it is the unique minimum-volume equiangular hyperbolic polyhedron with prescribed angle α. The left endpoint is the known ideal case, while at α=arccos(1/3) the regular tetrahedron degenerates to the Euclidean one. The proof combines Andreev's theorem and the Schläfli formula with Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.
ER  -