Minimal Lagrangian Surfaces in CH2 and Representations of Surface Groups into SU(2,1)

Abstract

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus recovering a result of Guichard and Parker-Platis. Our proof involves using the Toda lattice framework to construct an SU(2,1) frame corresponding to a minimal Lagrangian surface. Then the equation of Tzitzeica type is an integrability condition. 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.