Totally geodesic discs in strongly convex domains

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 $ϕ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_Ω_{n_2}(f(ζ),f(η)) = d^K_{n_1} (ζ,η)$ for all $ζ,η\in Ω_1,$ then $ϕ$ is either holomorphic or anti-holomorphic.

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