turkmath.org

We discuss some applications of pluripotential theory for the Bergman kernel and metric. In particular, the pluricomplex Green function can be used to prove Bergman completeness of a large class of domains (much bigger than had been known before). It is also applied to estimate the Bergman kernel from below, and recently very accurate estimates of this kind have been obtained. The main tools are the $\overline{\partial}$-operator and the Monge-Ampere equation. Most of the results are new and nontrivial already in dimension one. For convex domains such a lower bound can be used to simplify Nazarov's complex analytic approach to the Bourgain-Milman inequality in convex analysis.