On the Suita conjecture for some convex ellipsoids in $\mathbb C^2$

Abstract

It has been recently shown that for a convex domain $\Omega$ in $\mathbb C^n$ and $w\in\Omega$ the function $F_\Omega(w):=\big(K_\Omega(w)\lambda(I_\Omega(w))\big)^{1/n}$, where $K_\Omega$ is the Bergman kernel on the diagonal and $I_\Omega(w)$ the Kobayashi indicatrix, satisfies $1\leq F_\Omega\leq 4$. While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute $F_\Omega$ even numerically and the highest value of it obtained so far is $1.010182\dots$ In this paper we present precise, although rather complicated formulas for the ellipsoids $\Omega=\{|z_1|^{2m}+|z_2|^2<1\}$ (with $m\geq 1/2$) and all $w$, as well as for $\Omega=\{|z_1|+|z_2|<1\}$ and $w$ on the diagonal. The Bergman kernel for those ellipsoids had been known, the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case the function $\lambda(I_\Omega(w))$ is not $C^{3,1}$.