Index, Intersections, and Multiplicity of Min-Max Geodesics
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912056886689792 |
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| author | Marx-Kuo, Jared Sarnataro, Lorenzo Stryker, Douglas |
| author_facet | Marx-Kuo, Jared Sarnataro, Lorenzo Stryker, Douglas |
| contents | We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines.
We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on $S^2$ for which the $p$-width is only achieved by $p$ copies of a single geodesic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02580 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Index, Intersections, and Multiplicity of Min-Max Geodesics Marx-Kuo, Jared Sarnataro, Lorenzo Stryker, Douglas Differential Geometry We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on $S^2$ for which the $p$-width is only achieved by $p$ copies of a single geodesic. |
| title | Index, Intersections, and Multiplicity of Min-Max Geodesics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.02580 |