Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Abstract

We study the boundary rigidity problem for domains in Rⁿ: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ_g(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform I_g. We prove that the normal operator N_g=I^*_gI_g is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.