Tensor tomography on Asymptotically Hyperbolic surfaces: the reconstruction problem

Abstract: On a Riemannian manifold (M,g), the tensor tomography problem consists of inferring what is reconstructible of a
symmetric m-tensor field from the collection of its "longitudinal" integrals along all geodesics through M,
i.e., its geodesic X-ray transform. On manifolds with boundary, this problem arises in X-ray computerized
tomography, travel-time tomography, and tomography in elastic media, among other settings. On asymptotically
hyperbolic surfaces, it is also related to linearized entanglement entropy functionals in the AdS/CFT
correspondence, out of which one may consider the reconstruction of a bulk metric. In many of these geometric
situations, the (non-trivial) kernel of the X-ray transform over m-tensors is known precisely. From the perspective
of the inverse problem, it remains to address range characterization issues, and reconstruction, including the
design of "good" representatives modulo kernel to be reconstructed from X-ray data.

In this talk, I will discuss an iterated transverse-tracefree decomposition of tensor fields on
asymptotically hyperbolic surfaces, and will explain how to envision its reconstruction from X-ray data, based
on a few key steps which are geometry-dependent. I will then explain how these key steps can be explicitly
achieved on the Poincare disk, where in this case, a full range characterization of the X-ray
transform over tensor fields will also be presented.

Based on joint works with Nikolaos Eptaminitakis (Hannover U) and Yuzhou Zou (Oakland U).