Global uniqueness holds for the anisotropic Calderón problem on compact Riemannian manifolds with boundary when the metric is a small perturbation of the Euclidean metric.
The partial data Calder\'on problem in dimension three
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abstract
We consider an inverse boundary value problem for the time-independent Schr\"odinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.
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The Calder\'on problem for near-Euclidean metrics
Global uniqueness holds for the anisotropic Calderón problem on compact Riemannian manifolds with boundary when the metric is a small perturbation of the Euclidean metric.