arXiv math.GT digest — 09 Jun 2026 (1 paper)
One primary math.GT new submission from the Monday 8 June 2026 arXiv session: Sabourau proves sharp systole-volume and inradius-volume inequalities for finite-volume hyperbolic 3-manifolds, identifying the figure-eight knot complement, its sister, the Gieseking manifold, and the Weeks–Matveev–Fomenko manifold as the respective extremal cases — and completing Gendulphe's uniqueness theorem for minimal inradius.
Vistazo a la investigación
Today's listing closes with one paper. The Monday 8 June 2026 arXiv session had a single primary math.GT new submission; three other entries appearing in the feed are cross-lists from math.AT, math.CO, and math.DG and are excluded per the channel's scope.
1. Systole, inradius and rigidity of cusped hyperbolic 3-manifolds
Authors: Stephane Sabourau
arXiv: 2606.06777 — Submitted 4 Jun 2026
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Abstract. We establish optimal inequalities relating the systole and the inradius to the volume of finite-volume hyperbolic 3-manifolds. In the cusped orientable case, we refine a theorem of Gendulphe by proving a sharp systole-volume inequality whose unique extremal manifold is the figure-eight knot complement. Excluding the figure-eight knot complement, we obtain a stronger inequality whose extremal manifold is the sister of the figure-eight knot complement. We also establish analogous optimal systole-volume inequalities for closed orientable hyperbolic 3-manifolds, where the extremal manifolds are the Weeks–Matveev–Fomenko manifold, the manifold Vol3, and the Meyerhoff manifold. In the second part of the article, we study the inradius. We prove optimal inradius-volume inequalities for orientable and nonorientable cusped hyperbolic 3-manifolds, identifying respectively the sister of the figure-eight knot complement and the Gieseking manifold as the extremal cases. We also prove that the Gieseking manifold is the unique cusped hyperbolic 3-manifold of minimal inradius, thereby completing a result of Gendulphe, who had previously established the corresponding lower bound.
Key result
The paper establishes sharp, fully characterised inequalities between three geometric invariants of finite-volume hyperbolic 3-manifolds — systole, inradius, and volume — and identifies the extremal manifold in each case:
- Cusped orientable (systole): A sharp systole-volume inequality holds, with the figure-eight knot complement as the unique extremal manifold. Removing that exception, a strictly stronger bound holds with the sister of the figure-eight knot complement as the new extremal.
- Closed orientable (systole): Analogous sharp inequalities hold; the extremal manifolds in a natural hierarchy are the Weeks–Matveev–Fomenko manifold, Vol3, and the Meyerhoff manifold.
- Cusped (inradius, both orientable and nonorientable): Sharp inradius-volume inequalities are proved, with the sister of the figure-eight knot complement (orientable case) and the Gieseking manifold (nonorientable case) as the extremal cases. The paper further resolves the uniqueness question for minimal inradius among all cusped hyperbolic 3-manifolds: the Gieseking manifold is the sole minimiser, completing a theorem of Gendulphe who had obtained the lower bound without characterising equality. 1
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Techniques and mathematical tools
The paper works in the framework of finite-volume hyperbolic 3-manifolds and draws on several tools:
- Thick-thin decomposition and the geometry of Margulis tubes and cusp regions, which control the relationship between injectivity radius, inradius, and cusp geometry.
- Volume estimates for small balls and cusp regions in hyperbolic space, translating local geometric data into global volume bounds.
- Comparison geometry in spaces of constant negative curvature, with precise formulas for volumes of metric balls.
- Census-based extremal identification: the proof that named manifolds are exactly the extremal cases relies on the known computer-certified census of small-volume hyperbolic 3-manifolds (SnapPy/SnapPea, Snap) together with results of Cao–Meyerhoff, Gabai–Meyerhoff–Milley, and Agol on minimum-volume cusped and closed hyperbolic 3-manifolds.
- Earlier systolic geometry methods of Gendulphe and Sabourau himself, including Guth-type bounds and Adams–Reid-type cusp estimates.
Core proof idea
The strategy differs between the systole and inradius halves.
For the systole bounds, Sabourau refines Gendulphe's approach. The central step shows that any closed geodesic of length forces a ball of controlled hyperbolic volume around it, producing an inequality of the form with explicit sharp constants. Sharpness is verified by checking which low-volume manifolds from the census saturate the inequality. The figure-eight knot complement achieves both small volume and short systole simultaneously, making it the global minimiser. Once it is removed from contention, a case analysis on cusp geometry yields a strictly stronger bound for all remaining manifolds, with the sister manifold as the new extremal.
For the inradius bounds, the argument exploits the relationship between the inradius $r(M)$ — the radius of the largest embedded ball — and the cusp region structure. The inradius is bounded below by a function of the volume; the Gieseking manifold, the unique nonorientable cusped hyperbolic 3-manifold of smallest volume, realises the minimum. The uniqueness statement (Gieseking is the sole minimiser) closes Gendulphe's earlier partial result: Gendulphe had proved the inequality but had not characterised equality. Sabourau identifies equality precisely as the Gieseking case, completing the rigidity statement. 1
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Cross-list entries in the Monday 8 June 2026 math.GT feed — Stegemeyer (2606.07260, primary math.AT), Baldridge–McCarty (2606.06643, primary math.CO), and Li–Ma–Zheng (2606.06619, primary math.DG) — are excluded as the channel covers primary math.GT new submissions only.
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