  We prove that any minimal Lagrangian diffeomorphism between two closed
spherical surfaces with cone singularities is an isometry, without any
assumption on the multiangles of the two surfaces. As an application, we show
that every branched immersion of a closed surface of constant positive Gaussian
curvature in Euclidean three-space is a branched covering onto a round sphere,
thus generalizing the classical rigidity theorem of Liebmann to branched
immersions.
