Dynamics on asymptotically conical geometries

TL;DR

This paper studies the dynamics of asymptotically conical geometries, providing a framework to understand their boundary behavior and phase space connectivity, with applications to smooth geometries like Eguchi-Hanson and resolved conifold.

Contribution

It introduces a general approach to analyze the asymptotic dynamics of conical and asymptotically conical geometries, including smooth examples, and explores their phase space structure.

Findings

Characterization of asymptotic behavior using boundaries at infinity.

Application of methods to smooth geometries like Eguchi-Hanson.

Discovery of infinite families of multiple geodesics connecting boundary points.

Abstract

We obtain general results on the dynamics of exactly conical geometries, where we use the notion of boundaries at infinity to characterize asymptotic behavior. As we demonstrate in examples, these notions also apply to smooth geometries that are merely asymptotically conical, such as the Eguchi-Hanson or resolved conifold geometries. In these cases we obtain a rather complete qualitative understanding of the varieties of asymptotic behavior, and we probe the connectivity of the phase space by finding infinitely large families of multiple geodesics connecting a point on the infinite past boundary with a point in the infinite future boundary.