The Kobayashi Metric of a Complex Ellipsoid in ℂ^{2}

Brian E. Blank Dashan Fan David Klein Steven G. Krantz Daowei Ma Myung-Yull Pang

Abstract

The infinitesimal Kobayashi metric of an ellipsoid of the form

E_{m} = {(z_{1}, z_{2}) ∈ ℂ^{2} : |z_{1}|^{2} + |z_{2}|^{2m} < 1}

is calculated explicitly, modulo a parameter that is determined by solving a transcendental equation. Using this result, we show that the metric is C^{1} on the tangent bundle away from the zero section. We also describe software that will calculate, using a Monte Carlo method, the infinitesimal Kobayashi metric on a domain of the form

Ω_{ρ} = {(z_{1}, z_{2}) ∈ ℂ^{2} : ρ(z_{1}, z_{2}) < 0},

where ρ is a real-valued polynomial. We compare results of computer calculations with those obtained from the explicit formula for the Kobayashi metric.