[Paper Review] The Scattering Relation on Asymptotically Hyperbolic Manifolds

Antônio Sá Barreto, Yiran Wang|arXiv (Cornell University)|Oct 24, 2014

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper establishes the smoothness and structure of the scattering relation on non-trapping asymptotically hyperbolic manifolds (AHMs), proving that the sojourn time and geodesic flow extend smoothly to the boundary at infinity. It further derives the asymptotic expansion of the distance function on geodesically convex AHMs, showing that the distance function is smooth and its differential is described by the Lagrangian flow of the geodesic vector field, with the sojourn time capturing the leading-order behavior near infinity.

ABSTRACT

We study the scattering relation and the sojourn times on non-trapping asymptotically hyperbolic manifolds and use it to obtain the asymptotics of the distance function on geodesically convex asymptotically hyperbolic manifolds.

Motivation & Objective

• To define and analyze the scattering relation on non-trapping asymptotically hyperbolic manifolds (AHMs), generalizing the classical scattering framework from Euclidean space.
• To extend the sojourn time functional to geodesics connecting two boundary points, generalizing prior results that considered only one boundary approach.
• To establish the smoothness of the scattering relation and its relation to the Lagrangian flow of the geodesic vector field on the cotangent bundle.
• To derive the asymptotic expansion of the distance function on geodesically convex AHMs using the scattering relation and sojourn time.
• To show that the sojourn time $ s = r(z,z') + \log \rho_L + \log \rho_R $ is a smooth function on $ X \times_0 X \setminus \text{Diag}_0 $, linking the geometry of geodesics to the scattering data.

Proposed method

• Uses the geodesic flow $ \exp(tH_p) $ on the cosphere bundle $ S^*\text{Diag} $ to define the Lagrangian submanifold $ \Lambda $, which parametrizes unit-speed geodesics.
• Applies microlocal analysis and the theory of $ C^\infty $-smooth structures on fibered products $ X \times_0 X $ to extend the scattering relation across the boundary at infinity.
• Introduces the sojourn time $ s = t + \log \rho_L + \log \rho_R $, where $ t $ is the geodesic length and $ \rho_L, \rho_R $ are boundary defining functions, to define a smooth extension of the scattering data.
• Relies on the existence of a smooth defining function $ x $ such that $ g = \frac{dx^2}{x^2} + \frac{h(x)}{x^2} $ with $ h(0) = h_0 $, ensuring the metric is asymptotically hyperbolic.
• Uses the exponential map in normal coordinates to parametrize geodesics globally when the manifold is geodesically convex, enabling the construction of a global Lagrangian structure.
• Applies the result from Sá Barreto and Wunsch (2012) on the existence of the limit $ s_\gamma(z',y) = \lim_{t\to\infty}(t + \log \rho(\gamma(t))) $, generalizing it to both endpoints approaching the boundary.

Experimental results

Research questions

• RQ1 How can the scattering relation be defined and extended smoothly to the boundary at infinity for non-trapping asymptotically hyperbolic manifolds?
• RQ2 What is the asymptotic behavior of the distance function between two interior points as both approach the boundary?
• RQ3 How does the sojourn time $ s = r(z,z') + \log \rho_L + \log \rho_R $ relate to the geometry of geodesics and the scattering data?
• RQ4 Under what conditions is the Lagrangian $ \Lambda $, generated by the geodesic flow, a smooth embedded submanifold?
• RQ5 Can the differential of the distance function be described as the graph of a smooth Lagrangian correspondence in the cotangent bundle?

Key findings

• The scattering relation on non-trapping AHMs extends smoothly to the boundary at infinity, forming a $ C^\infty $ embedded Lagrangian submanifold in $ T^*({\mathbb{R}}_s \times X \times_0 X) $.
• The sojourn time $ s = r(z,z') + \log \rho_L + \log \rho_R $ is a smooth function on $ X \times_0 X \setminus \text{Diag}_0 $, capturing the leading-order asymptotic behavior of geodesic length near infinity.
• On geodesically convex AHMs, the Lagrangian $ \Lambda \setminus S^*\text{Diag} $ is the graph of the differential of the distance function $ r(z,z') $, i.e., $ \zeta = d_z r(z,z'), \zeta' = -d_{z'} r(z,z') $.
• The scattering relation is characterized as the restriction of the flowout $ \exp(tH_p)S^*\text{Diag} $ to the boundary at infinity, with $ t $ replaced by $ s = t + \log \rho_L + \log \rho_R $.
• The extension of the scattering relation to $ \partial({\mathbb{R}}_s \times X \times_0 X) $ is smooth and well-defined, with $ s $ being a smooth function of the base variables only.
• The result confirms that the leading singular term in the scattering matrix's Schwartz kernel is related to $ R^{-2i\lambda}H|_{\{\rho_L=\rho_R=0\}} $, where $ H $ is smooth on $ X \times_0 X $, linking microlocal analysis to scattering theory.