10:00 - 11:00, July 12, 2024 (Friday) Room 515, Cosmology Building, NTU
(臺灣大學次震宇宙館 515研討室)
Boundary Rigidity and the Geodesic X-ray Transform in Low Regularity Kevin Lam (University of California, Santa Barbara)

Abstract

We prove that the geodesic X-ray transform is injective on L^{2}(M) when the Riemannian metric is simple but the metric tensor is only finitely differentiable. The number of derivatives needed depends explicitly on dimension, and in dimension 2 we assume g C^{10} . Our proof is based on microlocal analysis of the normal operator: we establish ellipticity and a smoothing property in a suitable sense and then use a recent injectivity result on Lipschitz functions [1].The norma operator is a standard elliptic pseudodifferential operator in the smooth setting [2] where existence of parametrix is a standard result in PDE theory [3]; 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 as the corresponding symbol only satisfies the pseudodifferential operators symbol estimates up to a finite degree.

As an application we use the injectivity of the geodesic X-ray transform to prove that even for metrics at low regularity - the scattering relation on a compact two-dimensional simple manifolds determines the Dirichlet-to-Neumann map - a major component of the proof of boundary rigidity for simple metrics proved in [4] for smooth geometry.