Remarks on the higher dimensional Suita conjecture

Abstract

To study the analog of Suita's conjecture for domains $D\subset {\mathbb{C}}^n$, $n\ge 2$, Błocki 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 ${\mathbb{C}}^2$.