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The torus with asymptotically the fewest simple closed geodesics

Contributed by Phuc Thinh Dang, Trong Toan Dao, Nhat Minh Doan

We prove that the modular torus uniquely minimizes the area of the stable-norm unit ball among complete finite-area hyperbolic once-punctured tori, resolving the McShane--Rivin area conjecture. It therefore has fewer simple closed geodesics of length at most $L$ than any fixed nonmodular torus for all sufficiently large $L$ . We also bound the area increase away from the modular torus and determine its leading growth and first correction term as the systole tends to zero, with error bounds independent of the twist.

Primarily human-written textHuman understanding: all partsFarey polygonsHyperbolic surfacesMcShane–Rivin area conjectureMirzakhani functions.modular torussimple closed geodesicsstable norm

Oct 08, 2026PDF ↗View work →

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The torus with asymptotically the fewest simple closed geodesics

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