| Reference : Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold |
| E-prints/Working papers : Already available on another site | |||
| Physical, chemical, mathematical & earth Sciences : Mathematics | |||
| Computational Sciences | |||
| http://hdl.handle.net/10993/40746 | |||
| Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold | |
| English | |
| Bonsante, Francesco [] | |
| Mondello, Gabriele [] | |
Schlenker, Jean-Marc  [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > >] | |
| 2019 | |
| 30 | |
| No | |
| [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 | |
| http://hdl.handle.net/10993/40746 | |
| https://arxiv.org/abs/1910.06557 |
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