Limiting Carleman weights and anisotropic inverse problems

TL;DR

This paper characterizes Riemannian manifolds admitting limiting Carleman weights and uses this to establish uniqueness in anisotropic inverse problems, extending previous results beyond real-analytic metrics.

Contribution

It provides a geometric characterization of manifolds with limiting Carleman weights and develops a complex geometrical optics approach for these manifolds.

Findings

Characterization of manifolds with limiting Carleman weights

Construction of complex geometrical optics solutions

Uniqueness results for anisotropic inverse problems

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-Sjoestrand-Uhlmann (Ann. of Math. 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 X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.