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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.20355 |
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Table of 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.