Reference : Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold
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Physical, chemical, mathematical & earth Sciences : Mathematics
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Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold
Bonsante, Francesco []
Mondello, Gabriele []
Schlenker, Jean-Marc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > >]
[en] Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in M. We prove the uniqueness of smooth minimizing maps from (S,h) to M in a given homotopy class. When (S,h) is fixed, smooth minimizing maps from (S,h) are described by a simple holomorphic data on S: a complex self-adjoint Codazzi tensor of determinant 1. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the holonomy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of F analoguous to the complex length.
Fonds National de la Recherche - FnR

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