Article (Scientific journals)
Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold
Bonsante, Francesco; Mondello, Gabriele; SCHLENKER, Jean-Marc
2023In American Journal of Mathematics, 145 (4), p. 995-1049
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Abstract :
[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.
Disciplines :
Mathematics
Author, co-author :
Bonsante, Francesco
Mondello, Gabriele
SCHLENKER, Jean-Marc ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC)
External co-authors :
yes
Language :
English
Title :
Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold
Publication date :
2023
Journal title :
American Journal of Mathematics
ISSN :
0002-9327
eISSN :
1080-6377
Publisher :
The Johns Hopkins University Press, United States
Volume :
145
Issue :
4
Pages :
995-1049
Peer reviewed :
Peer Reviewed verified by ORBi
Focus Area :
Computational Sciences
Funders :
FNR - Fonds National de la Recherche [LU]
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