Winding and focussing for geodesics passing a thin cuspidal neck

Abstract

We study geodesics on a family $(M_{\epsilon })$ of manifolds that have a thin neck, which degenerate to a space with an incomplete cuspidal singularity as $\epsilon \to 0$. 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 $\epsilon \to 0$: 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 $\epsilon \to 0$. 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 $\epsilon$, and those much farther away. This multiscale analysis is efficiently expressed in terms of blow-up.

v.2: Corrected minor issues in Section 8. 62 pages, 13 pictures