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If M had a complete Kahler metric h with positive Ricci curvature then by the Bochner Weitzenbock formula (M ,h) would have no nonzero L^2 harmonic one forms . Now L^2 condition on one forms is a conformally invariant condition. However the disc has many nonzero L^2 harmonic one forms .This leads to a contradiction.In case of Riemann surfaces note that Ricci curvature coincides with Scalar curvature .