The Kobayashi metric for non-convex complex ellipsoids

Abstract

In the paper we give some necessary conditions for a mapping to be a $\kappa$-geodesic in non-convex complex ellipsoids. Using these results we calculate explicitly the Kobayashi metric in the ellipsoids $\{|z_1{|}^2+|z_2{|}^{2m}<1\}\subset {\mathbf{C}}^2$, where $m<\frac{1}{2}$.