Quasianalyticity and geometric rigidity in anisotropic Calderón's problem

Abstract

The anisotropic Calderón problem in dimensions $n\ge3$ remains open for general smooth metrics~\cite{Uhlmann2009}. We establish uniqueness results in two complementary regimes. In the first, the identity principle for quasianalytic functions propagates boundary information and yields uniqueness in general geometry, including a partial-boundary consequence; under a prescribed normal geometry, quasianalyticity is needed only in the distinguished direction. In the second, suitable symmetry or one-sided ordering assumptions lead to uniqueness at $C^\infty$ regularity with full or restricted boundary access. Taken together, the results exhibit a tradeoff among regularity, geometric structure, and boundary access: quasianalyticity supplies continuation in general geometry, while symmetry or one-sided order replaces that continuation at $C^\infty$ regularity.