Diameters and Green’s functions in K¨ahler geometry Abstract:

Abstract:

Estimates for the diameter and the Green’s function for K¨ahler 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∞ estimates for the Monge-Amp`ere equation, with a key improvement allowing degeneracies of the volume form along subvarieties 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 K¨ahler-Ricci flow, for both finitetime and long-time solutions. This is joint work with B. Guo, J. Song, and J. Sturm.