Remarks on the higher dimensional Suita conjecture

Abstract

For a domain D ⊂ C n , n ≥ 2, let F k D (z) = KD(z)λ I k D (z) , where KD(z) is the Bergman kernel of D along the diagonal and λ I k D (z) is the Lebesgue measure of the Kobayashi indicatrix at the point z. This biholomorphic invariant was introduced by B locki and in this note, we study the boundary behaviour of F k D (z) near a finite type boundary point where the boundary is smooth, pseudoconvex with the corank of its Levi form being at most 1. We also compute its limiting behaviour near the boundary of certain other basic classes of domains.1991 Mathematics Subject Classification. 32F45, 32A07, 32A25.

Generalizations of the Higher Dimensional Suita Conjecture and Its Relation with a Problem of Wiegerinck

“…Proposition 16. (see [1]) Let D be a strongly pseudoconvex domain in C n . Then lim z→∂D F D (z) = 1.…”

Bergman representative coordinate and constant holomorphic curvature

“…Notice that neither boundary regularity nor completeness of the metric is assumed in the above Suita type problems. Moreover, Błocki and Zwonek [15] obtained a multidimensional version of the Suita conjecture (see also [2,36] for related comparison results).…”