Area and diameter gaps for hyperbolic monotiles
Abstract
For each fixed n, we prove a positive lower bound for the diameter of a compact simple geodesic n-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of n for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer q≥2, a geodesic construction with 2q+3 sides and diameter less than 3/q, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.
Related
- Resonance counting and spectral gaps for hyperbolic surfaces via Fourier decay
- Hyperbolic area of Fatou components and new proofs of no-wandering-domain theorems
- Convex hulls and area minimizing currents in asymptotically locally hyperbolic manifolds
- Escape of mass for finite-area hyperbolic surfaces
- Spectral gaps of double covers of hyperbolic surfaces
BibTeX
@misc{liao2026area,
title = {{Area and diameter gaps for hyperbolic monotiles}},
author = {Yixi Liao and Erxiao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05438},
eprint = {2610.05438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05438},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Area and diameter gaps for hyperbolic monotiles
AU - Yixi Liao
AU - Erxiao Wang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05438
UR - https://arxiv.org/abs/2610.05438
AB - For each fixed n, we prove a positive lower bound for the diameter of a compact simple geodesic n-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of n for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer q≥2, a geodesic construction with 2q+3 sides and diameter less than 3/q, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.
ER -