Regularity of asymptotically conical Ricci-flat K\"ahler metrics

Abstract

Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical K\"ahler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat K\"ahler cone admits a Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class. We prove the sharp rate of convergence of the metric to the cone metric. For compact K\"ahler classes this is the same as for the Ricci-flat ALE metrics of P. Kronheimer and D. Joyce.