On the growth of the Bergman metric near a point of infinite type

Abstract

We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints -- some of them necessary for a non-trivial Bergman space. However, these are mild constraints: unlike most earlier works on this subject, we are able to make estimates for non-convex pseudoconvex models as well. In fact, the domains we can analyse range from being mildly infinite-type to very flat at infinite-type boundary points.