Boundary Rigidity and the Geodesic X-Ray Transform in Low Regularity

Abstract

We prove that the geodesic X-ray transform is injective on $L^2$ when the Riemannian metric is simple but only finitely differentiable. This result was a collaborative effort with Joonas Ilmavirta and Antti Kykk ̈anen [12]. The number of derivatives needed depends explicitly on the dimension of the manifold, and in dimension two we assume $g ∈ C^{10}$. Our proof isbased on microlocal analysis of the normal operator; we establish ellipticity and a smoothing property in a suitable sense. We then use a recent injectivity result on Lipschitz functions. When the metric tensor is $C^k$, the Schwartz kernel is not smooth but $C^{k-2}$ off the diagonal, which makes standard smooth microlocal analysis inapplicable. With the result above, we also prove that on a simple surface where the metric is $C^{17}$, the scattering relation determines the Dirichlet to Neumann map (DN map) [15] - a result proved in [31] for the case when the metric is smooth. For metrics with finite differentiability we had to modified each technical result used in the original proof; such as properties of the exit time function and the characterization of Cα space (Theorem 5.1.1 [30]). Moreover, surjectivity of $I^*$ in the original proof required the use of microlocal analysis of the normal operator which is not a standard pseudodifferential operator when the metric only has finite regularity- this was addressed in [12]. Finally, using the injectivity of $I$ on Lipschitz one forms for simple $C^{1,1}$ manifolds by [11] we prove an equivalent characterization of harmonic conjugacy using operators determined by the scattering relation (Theorem 1.6 [31]) to prove the titular result. We also prove that the boundary distance function determines the metricat the boundary (which in turns determines the scattering relation) for a closed disk even when the metric is only $C^{1,1}$ and the exponential map is only Lipschitz and does not preserve tangent vectors or differentials pointwise. We also provide a report on the partial progress for the Calder ́on problem for $C^{1,1}$ metrics.