Inverse Scattering from Conformal Infinity for Totally Geodesic Defects in Hyperbolic Space

We study inverse scattering from conformal infinity for impenetrable topological defects whose interaction surfaces are totally geodesic in hyperbolic space \(\mathbb H^n\), \(n \geq 2\). For a fixed spectral parameter \(\lambda_0 > 0\), the prescribed inputs are boundary labels \(\xi \in \partial_\infty \mathbb H^n\). Each label selects an incoming Helgason mode in the hyperbolic interior. Given a defect \(\mathcal P \Subset \mathbb H^n\), this mode generally fails to satisfy the homogeneous trace condition on the interaction surface and hence generates an outgoing correction. The leading coefficient of this correction at conformal infinity defines the measured far-field pattern. The central question is whether \(\mathcal P\) can be recovered from the far-field patterns corresponding to one or finitely many prescribed boundary labels. This gives a formally determined inverse problem at one fixed spectral parameter.
Our first main result establishes the unique determination of totally geodesic defects, an admissible class that includes both bulk components and hypersurface-supported ones. For Dirichlet-type defects, a single boundary label suffices. For Neumann-type defects, \(n+1\) boundary labels are sufficient and in general necessary. These labels are required to satisfy the natural affine-independence condition at conformal infinity.
Our second main result provides quantitative stability estimates within the same framework. The hyperbolic Hausdorff distance between two defects is controlled by the discrepancy of their far-field patterns at conformal infinity. The proof combines continuation from conformal infinity with quantitative geodesic reflection across totally geodesic hypersurfaces. Taken together, these results yield a qualitative and quantitative far-field inverse scattering theory at conformal infinity, based on formally determined data.