Classical estimates for diameters and Green’s functions in Riemannian geometry had required bounds on the Ricci curvature, and limited the applicability of convergence theorems for manifolds. We show how this problem can be overcome in K\”ahler geometry, using new PDE methods which had been instrumental in obtaining L^\infty bounds for fully non-linear equations in complex geometry.

This is joint work with B. Guo and F. Tong on the new PDE methods, and with B. Guo, J. Song, and J. Sturm on bounds for diameters and Green’s functions.