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The torus with asymptotically the fewest simple closed geodesics

Primarily human-written textHuman understanding: all partsmath.GT — Geometric Topologymath.NT — Number Theory

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

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

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 LL than any fixed nonmodular torus for all sufficiently large LL. 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.

Provenance statement

The ideas of subdividing the reduced region for different comparisons and estimating Taylor coefficients along two lines through the modular point were developed in part from discussions with Ser Peow Tan during the last-named author's visit to the National University of Singapore. ChatGPT (versions 5.4, 5.5, 5.6, and 6.0) was used extensively by the last-named author to assist with technical details, check calculations and arguments, and prepare TikZ figures. The authors developed the proof strategy, directed this work through questions and critical feedback, contributed ideas along the way to improve the results, carefully checked the resulting arguments and computations, and revised the exposition for clarity, simplicity, and accessibility to a broader readership. They take full responsibility for the content.

Tools used

OpenAI
ChatGPTVersion 5.4, 5.5, 5.6, 6.0

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Version history

  1. v1Submitted by Nhat Minh DoanInitial depositCurrentOct 08, 2026