A variational proof for the existence of a conformal metric with preassigned negative Gaussian curvature for compact Riemann surfaces of genus > 1

Given a smooth functionK < 0 we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genusg > 1. We do so by minimizing an appropriate functional using elementary analysis. In particular forK a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genusg > 1. An erratum to this article is available at .