Totally geodesic discs in strongly convex domains

Abstract

We prove that every isometry from the unit disk Δ in ${\mathbb{C}}$ , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ to a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^m}$ , is either holomorphic or anti-holomorphic.