On the Reconstruction of Conductivity of a Bordered Two-dimensional Surface in R3 from Electrical Current Measurements on Its Boundary

An electrical potential U on a bordered real surface X in R-3 with isotropic conductivity function sigma > 0 satisfies the equation d(sigma d(c)U)|(X) = 0, where d(c) = i(partial derivative - partial derivative), d = partial derivative + partial derivative are real operators associated with a complex ( conformal) structure on X induced by the Euclidean metric of R3. This paper gives an exact reconstruction of the conductivity function s on X from the Dirichlet-to-Neumann mapping U|(bX) -> sigma d(c)U|(bX). This paper extends to the case of Riemann surfaces the reconstruction schemes of R. Novikov (Funkt. Anal. Prilozh. 22(4):11-22, 1988) and of A. Bukhgeim ( J. Inv. Ill-posed Probl. 16: 19-34, 2008), given for the case X subset of R-2. The paper extends and corrects the statements of Henkin and Michel (J. Geom. Anal. 18: 1033-1052, 2008), where the inverse boundary value problem on the Riemann surfaces was first considered.