Totally geodesic discs in strongly convex domains

Abstract

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1,n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $\varphi :({\Omega }_1,d_{{\Omega }_1}^K)\to ({\Omega }_2,d_{{\Omega }_2}^K)$ is an isometry, i.e. $ d^K_Ω_{n_2}(f(ζ),f(η)) = d^K_{n_1} (ζ,η)$ for all $\zeta ,\eta \in {\Omega }_1,$ then $\varphi$ is either holomorphic or anti-holomorphic.

12 pages