Limiting Carleman weights and anisotropic inverse problems

TLDR: In this article, the authors considered the anisotropic Calderon problem and related inverse problems, and characterized those Riemannian manifolds which admit limiting Carleman weights, and gave a complex geometrical optics construction for a class of such manifolds.

Abstract: In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.