On the Suita Conjecture for Some Convex Ellipsoids in

Abstract

It was recently shown that for a convex domain Ω in and w ∈ Ω, the function , where is the Bergman kernel on the diagonal and the Kobayashi indicatrix, satisfies . While the lower bound is optimal, not much more is known about the upper bound. In general, it is quite difficult to compute even numerically, and the largest value of it obtained so far is 1.010182… . In this article, we present precise, although rather complicated, formulas for the ellipsoids Ω = {|z₁|^2m + |z₂|² < 1} (with m ≥ 1/2) and all w, as well as for Ω = {|z₁| + |z₂| < 1} and w on the diagonal. The Bergman kernel for those ellipsoids was already known; the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case, the function is not C^{3, 1}.

2000 AMS Subject Classification::

• 32F45
• 32A07
• 32A25

Keywords:

• Suita conjecture
• complex ellipsoid
• Bergman kernel
• Kobayashi indicatrix