Nonsimple closed geodesics with given intersection number on hyperbolic surfaces

Wujie Shen

Nonsimple closed geodesics with given intersection number on hyperbolic surfaces

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arxiv 2305.00638 v2 pith:EXECH6FW submitted 2023-05-01 math.GT

Wujie Shen This is my paper · ORCID

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keywords closedhyperbolicnumbercoshfinite-typegeodesicgeodesicsgiven

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abstract

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is independent of it.

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Shortest filling geodesics on hyperbolic surfaces
math.GT 2025-06 conditional novelty 7.0 of 10

A filling multi-geodesic on a genus g hyperbolic surface has length at least half the perimeter of a regular right-angled (8g-4)-gon, and this bound is sharp.

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