Remarks on the Higher Dimensional Suita Conjecture

Abstract

To study the analog of Suita's conjecture for domains D subset of C-n, n >= 2, Blocki introduced the invariant F-D(k) (z) = K-D(z)lambda(I-D(k) (z)), where K-D(z) is the Bergman kernel of D along the diagonal and lambda(I-D(k) (z)) is the Lebesgue measure of the Kobayashi indicatrix at the point z. In this note, we study the behaviour of F-D(k) (z) (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in C-2.