Abstract
Diameter estimates for Kahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for L infinity estimates for the Monge-Ampere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long-standing problem of uniform diameter bounds and Gromov-Hausdorff convergence of the Kahler-Ricci flow, for both finite-time and long-time solutions.