HIGHER REGULARITY FOR SINGULAR KÄHLER-EINSTEIN METRICS

We study singular Kahler-Einstein metrics that are obtained as noncollapsed limits of polarized Kahler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kahler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein and Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally, we prove a rigidity result for complete @@N -exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon and Hein, which applies to the case of asymptotically conical manifolds.