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