Conformal Metrics of Prescribed Gauss Curvature

Given a Riemann surface (M,g_0), viewed as a two-dimensional Riemannian manifold with background metric g_0, a classical problem in differential geometry is to determine what smooth functions f on M arise as the Gauss curvature of a conformal metric on M. When M = S^2 this is the famous Nirenberg problem. In fact, even when $(M,g_0)$ is closed and has genus greater than 1, this question so far has not been completely settled. In my talk I will present some new results for this problem.