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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.07166 |
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| _version_ | 1866909369578291200 |
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| author | Cheng, Herng Yi |
| author_facet | Cheng, Herng Yi |
| contents | For each odd $n \geq 3$, we construct a closed convex hypersurface of $\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even $n$, so in a sense we prove that Synge's theorem is "sharp." We also construct stable figure-eights: that is, for each $n \geq 3$ we embed the figure-eight graph in a closed convex hypersurface of $\mathbb{R}^{n+1}$, such that sufficiently small variations of the embedding either preserve its image or must increase its length. These index-zero geodesics and stable figure-eights are mainly derived by constructing explicit billiard trajectories with "controlled parallel transport" in convex polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_07166 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stable closed geodesics and stable figure-eights in convex hypersurfaces Cheng, Herng Yi Differential Geometry 53C22 (Primary) For each odd $n \geq 3$, we construct a closed convex hypersurface of $\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even $n$, so in a sense we prove that Synge's theorem is "sharp." We also construct stable figure-eights: that is, for each $n \geq 3$ we embed the figure-eight graph in a closed convex hypersurface of $\mathbb{R}^{n+1}$, such that sufficiently small variations of the embedding either preserve its image or must increase its length. These index-zero geodesics and stable figure-eights are mainly derived by constructing explicit billiard trajectories with "controlled parallel transport" in convex polytopes. |
| title | Stable closed geodesics and stable figure-eights in convex hypersurfaces |
| topic | Differential Geometry 53C22 (Primary) |
| url | https://arxiv.org/abs/2203.07166 |