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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.20355 |
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| _version_ | 1866908686044102656 |
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| author | Grieser, Daniel Lye, Jørgen Olsen |
| author_facet | Grieser, Daniel Lye, Jørgen Olsen |
| contents | We study geodesics on a family $(M_\varepsilon)$ of manifolds that have a thin neck, which degenerate to a space with an incomplete cuspidal singularity as $\varepsilon\to0$. There are essentially two classes of geodesics passing the waist, i.e. the cross section where the neck is thinnest: 1. Those hitting the waist almost vertically. We find that these exhibit a surprising focussing phenomenon as $\varepsilon\to0$: certain exit directions will be preferred, for a generic limiting singularity. 2. Those hitting the waist obliquely at a uniformly non-vertical angle. They wind around the neck more and more as $\varepsilon\to0$. We give a precise quantitative description of this winding. We illustrate both phenomena by numerical solutions. Our results rest on a detailed analysis at the two relevant scales: the points whose distance to the waist is of order $\varepsilon$, and those much farther away. This multiscale analysis is efficiently expressed in terms of blow-up. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Winding and focussing for geodesics passing a thin cuspidal neck Grieser, Daniel Lye, Jørgen Olsen Differential Geometry 53C22, 37D40, 53D25 We study geodesics on a family $(M_\varepsilon)$ of manifolds that have a thin neck, which degenerate to a space with an incomplete cuspidal singularity as $\varepsilon\to0$. There are essentially two classes of geodesics passing the waist, i.e. the cross section where the neck is thinnest: 1. Those hitting the waist almost vertically. We find that these exhibit a surprising focussing phenomenon as $\varepsilon\to0$: certain exit directions will be preferred, for a generic limiting singularity. 2. Those hitting the waist obliquely at a uniformly non-vertical angle. They wind around the neck more and more as $\varepsilon\to0$. We give a precise quantitative description of this winding. We illustrate both phenomena by numerical solutions. Our results rest on a detailed analysis at the two relevant scales: the points whose distance to the waist is of order $\varepsilon$, and those much farther away. This multiscale analysis is efficiently expressed in terms of blow-up. |
| title | Winding and focussing for geodesics passing a thin cuspidal neck |
| topic | Differential Geometry 53C22, 37D40, 53D25 |
| url | https://arxiv.org/abs/2510.20355 |