Boundary behavior of the Carathéodory and Kobayashi-Eisenman volume elements and the Kobayashi–Fuks metric

Abstract

We will compute the boundary asymptotics of the Carathéodory and Kobayashi-Eisenman vol- ume elements on convex finite type domains and Levi corank one domains in C n using the standard scaling techniques. We will show that their ratio, the so-called C/K ratio or the quotient invariant, can be used to detect strong pseudoconvexity. Some properties of a Kähler metric called the Kobayashi–Fuks metric will also be observed on planar domains as well as on strongly pseudoconvex domains in C^n . We study the localization of this metric near holomor- phic peak points and show that this metric shares several properties with the classical Bergman metric on strongly pseudoconvex domains.