Convergence of Bergman geodesics on CP¹

The space ℋ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ω_t are solutions of a homogeneous complex Monge-Ampère equation in A × X, where A ⊂ ℂ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials φ(t, z) of ω_t may be approximated in a weak C⁰ sense by geodesics φ(t, z) of the finite dimensional symmetric space of Bergman metrics of height N. In this article we prove that φN(t,z) → φ(t, z) in C²([0,1] × X) in the case of toric Kähler metrics on X = CP¹.