Narasimhan–Simha-type metrics on strongly pseudoconvex domains in

Abstract

For a bounded domain D⊂Cn, let KD=KD(z)>0 denote the Bergman kernel on the diagonal and consider the reproducing kernel Hilbert space of holomorphic functions on D that are square integrable with respect to the weight K−dD, where d≥0 is an integer. The corresponding weighted kernel KD,d transforms appropriately under biholomorphisms and hence produces an invariant Kähler metric on D. Thus, there is a hierarchy of such metrics starting with the classical Bergman metric that corresponds to the case d = 0. This note is an attempt to study this class of metrics in much the same way as the Bergman metric has been with a view towards identifying properties that are common to this family. When D is strongly pseudoconvex, the scaling principle is used to obtain the boundary asymptotics of these metrics and several invariants associated with them. It turns out that all these metrics are complete on strongly pseudoconvex domains.