Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class

Fathi, Albert; Giuliani, Alessandro; Sorrentino, Alfonso

Geodesic

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Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class

Fathi, Albert ¹ ; Giuliani, Alessandro ² ; Sorrentino, Alfonso ^3,⁴

¹ Ecole Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, 46 allée d’Italie, 69364 Lyon Cedex 07, France
² Dipartimento di Matematica, Università degli Studi di Roma Tre, L.go S. Leonardo Murialdo 1, 00146 Roma, Italia
³ Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000, USA
⁴ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wiberforce Road, Cambridge CB3 OWB, UK

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 659-680

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Résumé

Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the cotangent space $T^{*} M$ , strictly convex and superlinear in the momentum variables, we prove uniqueness of certain “ergodic” invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with rotation vector $ρ$ . This result extends generically to the $C^{0}$ -closure of KAM tori.

MR Zbl

Classification : 37J50, 37J40, 53D12

Affiliations des auteurs :

Fathi, Albert ¹ ; Giuliani, Alessandro ² ; Sorrentino, Alfonso ^{3, 4}

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     title = {Uniqueness of invariant {Lagrangian} graphs in a homology or a cohomology class},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {659--680},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {4},
     year = {2009},
     mrnumber = {2647908},
     zbl = {1192.37086},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2009_5_8_4_659_0/}
}

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Fathi, Albert; Giuliani, Alessandro; Sorrentino, Alfonso. Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 659-680. http://geodesic.mathdoc.fr/item/ASNSP_2009_5_8_4_659_0/