Shortest k-Geodesics on Hyperbolic Surfaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909928455667712 |
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| author | Chen, Changjie |
| author_facet | Chen, Changjie |
| contents | We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of a shortest closed geodesic with exactly $k$ self-intersections in terms of the length $L_\textswab{8}(X)$ of a shortest figure eight curve, improving Basmajian's estimate. We analyze the geometry of a shortest figure eight curve and explicitly build families of words in $π_1(X)$ whose geodesic representatives realize prescribed self-intersection numbers. As a consequence, we improve existing estimates on the maximal self-intersection number $I_k(X)$ of shortest geodesics with at least $k$ self-intersections, reducing the asymptotic upper bound from 512 to 128. This provides a sharper quantitative connection between the geometry and combinatorial complexity of non-simple closed geodesics on hyperbolic surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shortest k-Geodesics on Hyperbolic Surfaces Chen, Changjie Geometric Topology Differential Geometry We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of a shortest closed geodesic with exactly $k$ self-intersections in terms of the length $L_\textswab{8}(X)$ of a shortest figure eight curve, improving Basmajian's estimate. We analyze the geometry of a shortest figure eight curve and explicitly build families of words in $π_1(X)$ whose geodesic representatives realize prescribed self-intersection numbers. As a consequence, we improve existing estimates on the maximal self-intersection number $I_k(X)$ of shortest geodesics with at least $k$ self-intersections, reducing the asymptotic upper bound from 512 to 128. This provides a sharper quantitative connection between the geometry and combinatorial complexity of non-simple closed geodesics on hyperbolic surfaces. |
| title | Shortest k-Geodesics on Hyperbolic Surfaces |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2511.21993 |