Winding and focussing for geodesics passing a thin cuspidal neck

TL;DR

This work analyzes geodesics on a family of manifolds with a thin neck that degenerates to a cuspidal singularity, uncovering two distinct behaviors: obliquely incident geodesics wind around the neck with a precise ε^{-(k-1)} scaling, while almost vertical geodesics exhibit focussing toward distinguished transverse geodesics in the limit. The authors develop a rigorous blow-up (scaling) framework to separate neck and bulk regimes, formulate the geodesic flow as a Hamiltonian system, and perform a detailed front-face analysis using rescaled coordinates to reveal the torques and perturbations driving the focussing phenomenon. The Winding Theorem quantifies winding counts and uniform asymptotics across angles, while the Focussing Theorem shows that, under Morse conditions on S^+, generic limiting behavior concentrates near the minimal transverse geodesic, with stable/unstable manifolds organizing the dynamics. Together, these results connect smooth metric degenerations to limiting cuspidal behavior, providing precise multiscale descriptions and linking geometric degeneration to dynamical systems on the front face. The methods and conclusions illuminate how thin-neck geometries influence geodesic transport and suggest broad applicability to similar metric degenerations and their geodesic flows.

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.

Figures (10)

• Figure 1: Examples of surface \ref{['eqn:example']} with $k=2$ and $\varepsilon=1$ and $\varepsilon=0.2$ respectively. Here $\delta=0.8$.
• Figure 2: Winding: The surface \ref{['eqn:example']} with $k=2$, $\delta=0.8$, and $\varepsilon=1$ and $\varepsilon=0.3$ respectively are shown on the left. Each picture shows a single unit speed geodesic passing the waist $z=0$ at an angle $\phi=\arccos 0.95$. The right hand pictures show the geodesics up close by stretching $(u,v)$ to lie on the unit circle. Numerical solutions.
• Figure 3: Focussing: The surface \ref{['eqn:example']} with $k=2$, eccentricity $\delta=0.7$, and $\varepsilon=1$, $\varepsilon=0.2$, $\varepsilon=0.1$ respectively are shown. Each has ten unit speed geodesics passing the waist $z=0$ vertically, $\varphi =\pi/2$. The points of intersection with the waist are uniformly distributed. The lower right hand picture shows the $\varepsilon=0.1$ plot up close by stretching $(u,v)$ to lie on the unit circle. The dashed red lines are the lines $u=0$ and $v=0$, corresponding to the critical points of $S_0$. The points $u=0$ are maxima, whereas $v=0$ are minima. Numerical solutions.
• Figure 4: Geodesics passing the waist $z=0$ on $M_{\varepsilon}$ for $k=2$ with $\delta=0$ and $\varepsilon=0.1$. Here ten geodesics hitting the waist $z=0$ vertically, $\cos\varphi=0$, are shown. The points of intersection with the waist are uniformly distributed.
• Figure 5: The blow-up $[\mathbb{R}_+\times \mathbb{R};(0,0)]$ is shown on the left. The blown-down space $\mathbb{R}_+\times \mathbb{R}$ is shown on the right. The dotted lines represent $\phi={\operatorname{const}}.$ lines.

Theorems & Definitions (68)

• Remark 1: Motivation for our setup
• Remark 2: Regimes and boundary hypersurfaces
• Theorem 3
• proof
• Proposition 4
• proof
• Remark 5
• Lemma 6
• proof
• Remark 7