Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scattering rigidity for analytic metrics

Yannick Guedes Bonthonneau, Colin Guillarmou, Malo Jézéquel

Abstract

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the metric. More generally, our result holds in the analytic category under the no conjugate point and hyperbolic trapped sets assumptions.

Scattering rigidity for analytic metrics

Abstract

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the metric. More generally, our result holds in the analytic category under the no conjugate point and hyperbolic trapped sets assumptions.
Paper Structure (19 sections, 35 theorems, 145 equations, 1 figure)

This paper contains 19 sections, 35 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scattering map and the normal operator
  3. Dynamical preliminaries
  4. Determination of the metric near the boundary
  5. The normal operator
  6. Proof of Theorem \ref{['Th2']}
  7. Analyticity of the kernel of the Normal Operator away from the diagonal.
  8. The FBI transform
  9. Basic properties
  10. Pseudo-locality properties
  11. Some properties of the analytic wavefront set
  12. General facts about analytic wave front set
  13. Lagrangian deformation and analytic regularity
  14. Analytic radial estimates
  15. Construction of an escape function
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two real analytic negatively curved compact connected Riemannian manifolds (of any dimension) with non-empty analytic strictly convex boundary. Assume that $(\partial M_1,h_1)=(\partial M_2,h_2)$ where $h_i:=g_i|_{T\partial M_i}$ is the metric on the boundary, and

Figures (1)

  • Figure 1: The point $(x,v;S_g(x,v))\in \mathcal{Sc}(M,g)$ corresponds to the geodesic $\gamma\in \mathcal{G}(M,g)$. The map $(x,v)\mapsto S_g(x,v)$ is called the scattering map.

Theorems & Definitions (66)

  • Theorem 1
  • Definition 1.1
  • Theorem 2
  • Conjecture
  • Proposition 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 56 more