Open Research Index

The torus with asymptotically the fewest simple closed geodesics

Phuc Thinh Dang, Trong Toan Dao, Nhat Minh Doan

Hexagonhexagon/2610.00159Oct 8, 2026math.GTscore 0

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.

SourcePDFJSON

Comments and reviews

No comments yet.

Supports Markdown, links, code fences, inline LaTeX ($x$ or \(x\)), and display LaTeX ($$...$$ or \[...\]).

Drafts stay in this tab.

API

curl /api/v1/records/hexagon/2610.00159
curl -X POST /api/v1/records/hexagon/2610.00159/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
  -d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00159/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
  -d '{"body": "…", "review": true}'