Geodesic X-Ray Transform on Asymptotically Hyperbolic Manifolds

Abstract

This dissertation contains work of the author and joint work with C. Robin Graham concerning the geodesic X-ray transform in the setting of asymptotically hyperbolic manifolds. It is divided into three self contained chapters, each addressing a different question. The topic of the first chapter is the local injectivity of the X-ray transform, extending a result proved by Uhlmann and Vasy on compact manifolds with boundary. Assuming knowledge of the X-ray transform for geodesics contained in a small neighborhood of a boundary point we show local injectivity for asymptotically hyperbolic metrics even modulo O(\rho^5) in dimension 3 and higher. In the second chapter we construct examples of asymptotically hyperbolic metrics demonstrating that in the asymptotically hyperbolic setting absence of conjugate points does not suffice to exclude boundary conjugate points. The construction uses techniques developed by Gulliver and clarifies the definition of a simple asymptotically hyperbolic manifold, formulated by Graham, Guillarmou, Stefanov and Uhlmann. In the third chapter we show a stability estimate for the X-ray transform on simple asymptotically hyperbolic manifolds, extending to this setting the work of Stefanov and Uhlmann on simple compact manifolds with boundary.