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 L2(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 C10 . 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 Ck , the Schwartz kernel is not smooth but Ck-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.
