Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in 𝐶ⁿ with smooth boundary

Abstract

The Carathéodory and Kobayashi distance functions on a bounded domain G G in C n {{\mathbf {C}}^n} have related infinitesimal forms. These are the Carathéodory and Kobayashi metrics. They are denoted by F ( z , ξ ) F(z,\xi ) (length of the tangent vector ξ \xi at the point z z ). They are defined in terms of holomorphic mappings, from G G to the unit disk for the Carathéodory metric, and from the unit disk to G G for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in C n {{\mathbf {C}}^n} with C 2 {C^2} boundary. ξ \xi is fixed and z z is allowed to approach a boundary point z 0 {z_0} . The quantity F ( z , ξ ) d ( z , ∂ G ) F(z,\xi )d(z,\partial G) is shown to have a finite limit. In addition, if ξ \xi belongs to the complex tangent space to the boundary at z 0 {z_0} , then this first limit is zero, and ( F ( z , ξ ) ) 2 d ( z , ∂ G ) {(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.