The Calderon Problem for Two-Dimensional Manifolds by the BC-Method

Abstract

As was shown by Lassas and Uhlmann [Ann. Sci. École Norm. Sup. (4), 34 (2001), pp. 771--787], the smooth two-dimensional compact orientable Riemann manifold with the boundary is uniquely determined by its Dirichlet-to-Neumann map (DN-map) up to conformal equivalence. We give a new proof of this fact based on relations between the Calderon problem and function algebras: the manifold is identified with the spectrum of the algebra of holomorphic functions determined by the DN-map up to isometry; as such, the manifold is recovered from the DN-map by the use of the Gelfand transform. A simple formula linking the DN-map to the Euler characteristic of the manifold is derived.