Remarks on the higher dimensional Suita conjecture

Abstract

To study the analog of Suita’s conjecture for domains D ⊂ C n D \subset \mathbb {C}^n , n ≥ 2 n \geq 2 , Błocki introduced the invariant F D k ( z ) = K D ( z ) λ ( I D k ( z ) ) F^k_D(z)=K_D(z)\lambda \big (I^k_D(z)\big ) , where K D ( z ) K_D(z) is the Bergman kernel of D D along the diagonal and λ ( I D k ( z ) ) \lambda \big (I^k_D(z)\big ) is the Lebesgue measure of the Kobayashi indicatrix at the point z z . In this note, we study the behaviour of F D k ( z ) F^k_D(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 \mathbb {C}^2 .