Winding and focussing for geodesics passing a thin cuspidal neck

TL;DR

This paper investigates the behavior of geodesics passing through a thin neck in a family of manifolds degenerating to a cuspidal singularity, revealing a focusing phenomenon and winding behavior with detailed multiscale analysis.

Contribution

It provides a novel multiscale analysis of geodesics near a degenerating neck, describing focusing and winding phenomena as the neck becomes thinner.

Findings

Vertical hitting geodesics exhibit a focusing phenomenon.

Oblique geodesics wind increasingly around the neck as it thins.

Numerical solutions illustrate the theoretical phenomena.

Abstract

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.