CALDERON INVERSE PROBLEM WITH PARTIAL DATA ON RIEMANN SURFACES

On a fixed smooth compact Riemann surface with boundary (M-0, g), we show that, for the Schrodinger operator Delta(g) + V with potential V epsilon C-1,C-alpha(M-0) for some alpha > 0, the Dirichlet-to-Neumann map N vertical bar(Gamma) measured on an open set Gamma subset of partial derivative M-0 determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.