A cyclic extension  of the earthquake flow II

Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc

doi:10.24033/asens.2259

Geodesic

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A cyclic extension of the earthquake flow II
[Une extension cyclique du flot des tremblements de terre II]

Bonsante, Francesco ; Mondello, Gabriele ; Schlenker, Jean-Marc

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 811-859.

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Abstract (VO)
Résumé

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichmüller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichmüller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.

Le flot des glissements de terrain, introduit dans [5], est un analogue régulier du flot des tremblements de terre sur l'espace de Teichmüller, qui partage certaines de ses principales propriétés. Nous montrons ici que d'autres propriétés des tremblements de terre s'appliquent aux glissements de terrain. Le flot des glissements de terrain est le flot hamiltonien d'une fonction convexe. L'application de greffage régulière $sgr$ , à valeur dans l'espace de Teichmüller, qui est aux glissements de terrain ce que le greffage est aux tremblements de terre, est propre et surjective par rapport à chacune de ses variables. L'application de greffage régulière $SGr$ , à valeur dans l'espace des structures projectives complexes, est symplectique (à un facteur multiplicatif près). La composition de deux glissements de terrain a un point fixe dans l'espace de Teichmüller. En conséquence, nous obtenons des résultats nouveaux sur les surfaces à courbure de Gauss constantes dans des variétés de dimension 3 hyperboliques ou AdS. Nous montrons aussi que le flot des glissements de terrain a une extension satisfaisante au bord de l'espace de Teichmüller.

Publié le : 2015-05-27

DOI : 10.24033/asens.2259

Classification : 30F60, 53C42, 49Q10, 53C43.
Keywords: Espace de Teichmüller, surfaces hyperboliques, applications minimales lagrangiennes, métrique de Weil-Petersson, tenseurs de Codazzi.
Mots-clés : Teichmüller space, hyperbolic surfaces, minimal Lagrangian maps, Weil-Petersson metric, Codazzi tensors.

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     title = {A cyclic extension  of the earthquake {flow~II}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {811--859},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc. A cyclic extension  of the earthquake flow II. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 811-859. doi : 10.24033/asens.2259. http://geodesic.mathdoc.fr/articles/10.24033/asens.2259/

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