Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary

Abstract:

The Carathéodory and Kobayashi distance functions on a bounded domain $G$ in ${{\mathbf {C}}^n}$ have related infinitesimal forms. These are the Carathéodory and Kobayashi metrics. They are denoted by $F(z,\xi )$ (length of the tangent vector $\xi$ at the point $z$). They are defined in terms of holomorphic mappings, from $G$ to the unit disk for the Carathéodory metric, and from the unit disk to $G$ for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in ${{\mathbf {C}}^n}$ with ${C^2}$ boundary. $\xi$ is fixed and $z$ is allowed to approach a boundary point ${z_0}$. The quantity $F(z,\xi )d(z,\partial G)$ is shown to have a finite limit. In addition, if $\xi$ belongs to the complex tangent space to the boundary at ${z_0}$, then this first limit is zero, and ${(F(z,\xi ))^2}d(z,\partial G)$ has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathéodory metric.