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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Ian Graham
Journal: Trans. Amer. Math. Soc. 207 (1975), 219-240
MSC: Primary 32F99; Secondary 32F15, 32H15
DOI: https://doi.org/10.1090/S0002-9947-1975-0372252-8
MathSciNet review: 0372252
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0372252-8
Article copyright: © Copyright 1975 American Mathematical Society