On Isometries of Intrinsic Metrics in Complex Analysis

We study isometries of the Kobayashi and Carathéodory metrics on strongly pseudoconvex and strongly convex domains in C and prove: (i) Let Ω1 and Ω2 be strongly pseudoconvex domains in C and f : Ω1 → Ω2 an isometry. Suppose that f extends as a C map to Ω̄1. Then f |∂Ω1 : ∂Ω1 → ∂Ω2 is a CR or anti-CR diffeomorphism. Hence it follows that Ω1 and Ω2 must be biholomorphic or anti-biholomorphic. (ii) The isometry group of a strongly convex domain in C is compact unless the domain is biholomorphic to the ball. The main tools for these results are a metric version of the Pinchuk rescaling technique and Lempert’s theory of extremal discs.