Jan-Erik Björk: Riemann Surfaces

Abstract: Distributions and more generally currents play a central role in my expository lecture. Among the results may be mentioned the classic result due to Abel, which gives a necessary and sufficient condition for the distribution and position of zeroes and poles of a meromorphic function on a compact Riemann surface. For open Riemann surfaces a major result due to Behnke and Stein asserts that the inhomogeneous d-bar equation is always solvable. I will sketch a proof of these results following a device by Malgrange using currents. Let me mention that the Behnke-Stein theorem implies the uniformisation theorem for open Riemann surfaces.