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

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

2010-09-13

arxiv.org/abs/1009.2435v3

Mathematics

Differential Geometry

31 pp, introduction reworked, structure equations for Lagrangian surfaces in CH2 included

Scientific paper

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.

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