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1 n+1 dr is convex and solves the real Monge-Ampère equation detThe proof of this proposition is the same as our proof of Baues-Cortés's result (which we label as Theorem 1 in [2]).Proposition 1 then reduces the problem of finding a semi-flat Calabi-Yau structure on a neighborhood of the "Y" vertex of a graph in R 3 to finding a hyperbolic affine sphere structure on S 2 minus 3 singular points. The next proposition, which follows from [3], provides many such examples.

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We use an elliptic differential equation of T ¸it ¸eica (or Toda) type to construct a minimal Lagrangian surface in CH 2 from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU (2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the R-Fuchsian representations in the SU (2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU (2, 1) frame for the minimal Lagrangian immersion: the Maurer-Cartan equation for this frame is the T ¸it ¸eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

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