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

The Calderon problem for two-dimensional manifolds by the BC-method (a corrected version) As was shown by M.Lassas and G.Uhlmann (2001), the smooth two-dimensional compact orientable Riemann manifold with the boundary is uniquely determined by its Dirichlet-to-Neumann 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 identiied 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.