Asymptotic behavior of geodesics in conformally compact manifolds

TL;DR

This paper investigates the asymptotic behavior of geodesics near the boundary of conformally compact manifolds, revealing regularity properties and how the boundary's smooth structure is determined by the manifold's geometry.

Contribution

It extends understanding of geodesic extensions to the boundary beyond the hyperbolic case, showing $C^{1,eta}$ regularity and smooth dependence on initial conditions in general conformally compact manifolds.

Findings

Non-trapped geodesics extend to the boundary with $C^{1,eta}$ regularity.

Endpoints of geodesics depend smoothly on initial conditions.

The conformal infinity's smooth structure is canonically determined by $(X,g)$.

Abstract

We study the asymptotic behavior of geodesics near the boundary of a conformally compact Riemannian manifold $(X,g)$. In the case where the sectional curvature at infinity is constant (the asymptotically hyperbolic case) it is known that non-trapped geodesics extend to the conformal infinity as smoothly immersed curves in $\overline{X}$ and that the conformal infinity can be locally smoothly parametrized by the initial directions of such non-trapped geodesics starting at a given point. We show that in the general case non-trapped geodesics typically only extend to the conformal infinity with $C^{1,\alpha}$ regularity, due to the presence of a logarithmic singularity sourced by the gradient of the limiting sectional curvature. In spite of this singular behavior, we show that the endpoints of such geodesics depend smoothly on the initial conditions, so that the conformal infinity can still be locally smoothly parametrized using the initial velocities of geodesics. In particular, the smooth structure of the conformal infinity is canonically determined by $(X,g)$. We also show how to specify `initial data' for geodesics at the conformal infinity and obtain an asymptotic expansion that parametrizes the geodesics tending toward a given boundary point.