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 ℝ³ with isotropic conductivity function σ > 0 satisfies the equation d(σd^c U)|X = 0, where d^c = i( ∂̄∂), d = ∂+∂ 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 σ on X from the Dirichlet-to-Neumann mapping U|bX →σd^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 ⊂ ℝ². 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.