Geometric estimates and comparability of Eisenman volume elements with the Bergman kernel on (C-)convex domains

Abstract

We establish geometric upper and lower estimates for the Carathéodory and Kobayashi-Eisenman volume elements on the class of non-degenerate convex domains, as well as on the more general class of non-degenerate $C$-convex domains. As a consequence, we obtain explicit universal lower bounds for the quotient invariant both on non-degenerate convex and $C$-convex domains. Here the bounds we derive, for the above mentioned classes in $C^n$, only depend on the dimension n for a fixed $n\ge 2$. Finally, it is shown that the Bergman kernel is comparable with these volume elements up to small/large constants depending only on n.