Constant Scalar Curvature Metrics on Toric Surfaces

The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist. Received: May 2008, Revision: December 2008, Accepted: December 2008