ASYMPTOTIC EXPANSIONS OF INVARIANT METRICS OF STRICTLY PSEUDOCONVEX DOMAINS

In this paper we obtain the asymptotic expansions of the Caratheodory and Kobayashi metrics of strictly pseudoconvex domains with C(infinity) smooth boundaries in C(n). The main result of this paper can be stated as following: Main Theorem. Let OMEGA be a strictly pseudoconvex domain with C(infinity) smooth boundary. Let F(OMEGA)(z, X) be either the Caratheodory or the Kobayashi metric of OMEGA. Let delta(z) be the signed distance from z to partial derivativeOMEGA with delta(z) < 0 for z is-an-element-of OMEGA and delta(z) greater-than-or-equal-to 0 for z is-not-an-element-of OMEGA. Then there exist a neighborhood U of partial derivativeOMEGA, a constant C > 0, and a continuous function C(z,X):(U and OMEGA) x C(n) --> R such that [GRAPHICS] and \C(z,X)\ less-than-or-equal-to C\X\ for z is-an-element-of U and OMEGA and X is-an-element-of C(n).