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FB{zA) = {ds\z^)f in the notation of [4]. We consider the boundary behavior of these metrics for fixed £. The notable features are (i) the different limiting behavior in tangential and normal directions (cf. Stein [9]), and (ii) the appearance of the Levi form as the limiting value of a quantity defined inside the domain. THEOREM. Let G be a {bounded) strongly pseudoconvex domain in C with C boundary. Let z0 e dG. Let q> be a C 2 defining function for ÔG such that \Vzq>(z0)\ = 1. Let F(z9£) be either the Carathéodory or the Kobayashi metric on G. Then

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