Minimal Lagrangian surfaces in {mathbb {CH}^2}and representations of surface groups into SU(2, 1) |
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Authors: | John Loftin Ian McIntosh |
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Affiliation: | 1. Department of Mathematics and Computer Science, Rutgers-Newark, Newark, NJ, 07102, USA 2. Department of Mathematics, University of York, York, YO10 5DD, UK
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Abstract: | We use an elliptic differential equation of ?i?eica (or Toda) type to construct a minimal Lagrangian surface in ${mathbb {CH}^2}$ 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 ${mathbb {R}}$ -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 ?i?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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