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Minimal Lagrangian surfaces in \({\mathbb {CH}^2}\)and representations of surface groups into SU(2, 1)

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Abstract

We use an elliptic differential equation of Ţiţeica (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 Ţiţeica-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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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Rutgers-Newark, Newark, NJ, 07102, USA

    John Loftin

  2. Department of Mathematics, University of York, York, YO10 5DD, UK

    Ian McIntosh

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  1. John Loftin
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  2. Ian McIntosh
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Loftin, J., McIntosh, I. Minimal Lagrangian surfaces in \({\mathbb {CH}^2}\)and representations of surface groups into SU(2, 1). Geom Dedicata 162, 67–93 (2013). https://doi.org/10.1007/s10711-012-9717-1

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