Riemannian 3-spheres that are hard to sweep out by short curves
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916571998322688 |
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| author | Alshawa, Omar Cheng, Herng Yi |
| author_facet | Alshawa, Omar Cheng, Herng Yi |
| contents | We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each $L > 0$ we construct a Riemannian 3-sphere $M$ with diameter and volume less than 1, so that every 2-parameter family of closed curves in $M$ that satisfies certain topological conditions must contain a curve that is longer than $L$. This obstructs certain min-max approaches to bound the length of the shortest closed geodesic in Riemannian 3-spheres.
We also find obstructions to min-max estimates of the lengths of orthogonal geodesic chords, which are geodesics in a manifold that meet a given submanifold orthogonally at their endpoints. Specifically, for each $L > 0$, we construct Riemannian 3-spheres with diameter and volume less than 1 such that certain orthogonal geodesic chords that arise from min-max methods must have length greater than $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riemannian 3-spheres that are hard to sweep out by short curves Alshawa, Omar Cheng, Herng Yi Differential Geometry 53C23 We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each $L > 0$ we construct a Riemannian 3-sphere $M$ with diameter and volume less than 1, so that every 2-parameter family of closed curves in $M$ that satisfies certain topological conditions must contain a curve that is longer than $L$. This obstructs certain min-max approaches to bound the length of the shortest closed geodesic in Riemannian 3-spheres. We also find obstructions to min-max estimates of the lengths of orthogonal geodesic chords, which are geodesics in a manifold that meet a given submanifold orthogonally at their endpoints. Specifically, for each $L > 0$, we construct Riemannian 3-spheres with diameter and volume less than 1 such that certain orthogonal geodesic chords that arise from min-max methods must have length greater than $L$. |
| title | Riemannian 3-spheres that are hard to sweep out by short curves |
| topic | Differential Geometry 53C23 |
| url | https://arxiv.org/abs/2501.10904 |