On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian

TL;DR

The paper addresses inverse scattering on asymptotically hyperbolic manifolds with constant scalar curvature $R_g=-n(n+1)$, proving that the scattering matrix at energy $\frac{n+1}{2}$ determines the boundary jet of the metric up to a diffeomorphism and a conformal factor on a common boundary region. The approach leverages the Guillarmou--Guillopé/Chang--González relation between the generalized eigenvalue problem and the Conformal Laplacian, identifying $S(\frac{n+1}{2})$ with the Conformal Laplacian's Dirichlet-to-Neumann map, and then applying boundary-jet results for the DN map from LLS22 to obtain rigidity of the boundary metric. This yields a dimension-free inverse-scattering result (with a stronger 2D variant) and situates the AH inverse problem within a Calderón-type framework for the Conformal Laplacian. The work extends classical Calderón-type recovery to AH geometries by exploiting conformal invariance at the critical energy and a careful analysis of boundary jets, with potential implications for geometric uniqueness in conformally compact settings.

Abstract

In this short note, we use the relation obtained by Guillarmou--Guillopé and Chang--González between the generalized eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the Conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension $n+1$ with constant scalar curvature $-n(n+1)$, we show that the scattering matrix at energy $\frac{n+1}{2}$ determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 2.4: GG07*Section 4
• Lemma 2.5: LLS22*Lemmas 2.1 & 2.2
• proof : Proof of Theorem \ref{['thm:jet']}
• Remark 2.6
• Remark 2.7