The Kobayashi metric of a complex ellipsoid in {${\bf C}\sp 2$}

Abstract

The infinitesimal Kobayashi metric of an ellipsoid of the form $$ E_m=\{(z_1,z_2)\in \C^2:|z_1|^2+|z_2|^{2m}<1\} $$ 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 $$ \Omega_\rho=\{(z_1,z_2)\in\C^2:\rho(z_1,z_2)<0\}, $$ where $\rho$ is a real-valued polynomial. We compare results of computer calculations with those obtained from the explicit formula for the Kobayashi metric.