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On the graph theorem for Lagrangian minimizing tori

  • * Corresponding author: Rafael O. Ruggiero

    * Corresponding author: Rafael O. Ruggiero
The research project is partially supported by CNPq, FAPERJ (Cientistas do nosso estado), Pronex de Geometria, Pronex de Sistemas Dinómicos (Brazil), CNRS, unité FR2291 FRUMAM.
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  • We study the graph property for Lagrangian minimizing submanifolds of the geodesic flow of a Riemannian metric in the torus $ (T^{n},g) $, $ n>2 $. It is well known that the transitivity of the geodesic flow in a minimizing Lagrangian submanifold implies the graph property. We replace the transitivity by three kind of assumptions: (1) $ r $-density of the set of recurrent orbits for some $ r>0 $ depending on $ g $, (2) $ r $-density of the limit set, (3) every point is nonwandering. Then we show that a Lagrangian, minimizing torus satisfying one of such assumptions is a graph.

    Mathematics Subject Classification: Primary: 53D25; Secondary: 53D12, 37J50.


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  • [1] L. Amorim, Y.-G. Oh and J. O. Dos Santos, Exact Lagrangian submanifolds, Lagrangian Spectral Invariants and Aubry-Mather Theory, arXiv: 1603.06966v4
    [2] M. C. Arnaud, On a Theorem Due to Birkhoff, GAFA, 20 (2010), 1307-1316.  doi: 10.1007/s00039-010-0091-6.
    [3] M. C. Arnaud, When are invariant sumanifolds of symplectic dynamics Lagrangian?, Discrete and Continuos Dynamical Systems-A, 34 (2014), 1811-1827.  doi: 10.3934/dcds.2014.34.1811.
    [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978, Berlin, Heidelberg, New York.
    [5] V. I. Arnold, Sturm theorems and symplectic geometry, Functional Anal. and Its Appl., 19 (1985), 1-10. 
    [6] V. Bangert, Mather sets for twist maps and geodesic on tori, In U. Kirchgraber, H. O. Walther (eds.) Dynamics Reported, Vol. 1, B. G. Teubner, J. Wiley, 1988, 1-56
    [7] V. Bangert, Minimal geodesics, Erg. Th. Dyn. Sys., 10 (1990), 263-286.  doi: 10.1017/S014338570000554X.
    [8] P. Bernard and J. O. dos Santos, A geometric definition of the Mañé-Mather set an a theorem of Marie-Claude Arnaud, Math Proc. Cambridge Philos. Soc., 152 (2012), 167-178.  doi: 10.1017/S0305004111000685.
    [9] M. Bialy, Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two dimensional torus, Communications in Math. Physics, 126 (1989), 13-24.  doi: 10.1007/BF02124329.
    [10] M. Bialy and L. Polterovich, Geodesic flows on the two dimensional torus and phase transitions ''commensurability-noncommensurability", Funk. Anal. Appl., 20 (1986), 223-226. 
    [11] M. Bialy and L. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom, Invent. math., 97 (1989), 291-303.  doi: 10.1007/BF01389043.
    [12] M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627.  doi: 10.1007/BF01444639.
    [13] M. Bialy and L. Polterovich, Invariant tori and symplectic topology, Amer. Math. Soc. Transl., 171 (1996), 23-33.  doi: 10.1090/trans2/171/03.
    [14] A. Candel and L. Conlon, Foliations I. Graduate Studies in Mathematics, vol. 23. Amer. Math. Soc. Providence, Rhode Island, 2000.
    [15] M. J. Dias Carneiro and R. Ruggiero, On Variational and topological properties of C1 invariant Lagrangian tori, Ergodic Theory and Dynamical systems, 24 (2004), 1909-1935.  doi: 10.1017/S0143385704000379.
    [16] M. J. Dias Carneiro and R. Ruggiero, On Birkhoff Theorems for Lagrangian invariant tori with closed orbits, Manuscripta Mathematica, 119 (2006), 411-432.  doi: 10.1007/s00229-005-0619-5.
    [17] C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61.  doi: 10.1090/S0002-9947-1971-0279830-1.
    [18] G. ContrerasJ. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits Ⅱ, Bol. Soc. Bras. Mat., 28 (1997), 155-196.  doi: 10.1007/BF01233390.
    [19] G. ContrerasJ. M. GambaudoR. Iturriaga and G. Paternain, The asymptotic Maslov index and its applications, Ergodic Theory Dynam. Systems, 23 (2003), 1415-1443.  doi: 10.1017/S0143385703000063.
    [20] G. ContrerasR. IturriagaG. Paternain and M. Paternain, Lagrangian graphs, minimizing measures, and Mañé's critical values,, Geom. Funct. Anal., 8 (1998), 788-809.  doi: 10.1007/s000390050074.
    [21] G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Th. Dyn. Sys., 19 (1999), 901-952.  doi: 10.1017/S014338579913387X.
    [22] G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 1999.
    [23] P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, J. Differential Geometry, 8 (1973), 437-463.  doi: 10.4310/jdg/1214431801.
    [24] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Lyon (notes), 2000.
    [25] L. W. Green, Surfaces without conjugate points, Transactions of the American Mathematical Society, 76 (1954), 529-546.  doi: 10.1090/S0002-9947-1954-0063097-3.
    [26] L. W. Green, A theorem of E. Hopf, Michigan Math. J., 5 (1958), 31-34.  doi: 10.1307/mmj/1028998009.
    [27] M. Herman, Inégalités a priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 70 (1989), 47-101. 
    [28] E. Hopf, Closed surfaces without conjugate points, Proceedings of the National Academy of Sciences, 34 (1948), 47-51.  doi: 10.1073/pnas.34.2.47.
    [29] O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math., 250 (1971), 124-129.  doi: 10.1515/crll.1971.250.124.
    [30] R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.  doi: 10.1088/0951-7715/5/3/001.
    [31] R. Mañé, Global variational methods in conservative dynamics, IMPA, Rio de Janeiro, 1993.
    [32] J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.
    [33] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Claredon Press, Oxford, 1995.
    [34] J. M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., 26 (1924), 25-60.  doi: 10.1090/S0002-9947-1924-1501263-9.
    [35] J. M. Morse, Calculus of variations in the large, AMS Colloquium Publications XVIII, Providence, RI, 1996.
    [36] G. Paternain, Geodesic Flows, Progress in Mathematics vol. 180, Birkhauser, Boston, Mass. 1999. doi: 10.1007/978-1-4612-1600-1.
    [37] J. Pesin, Geodesic flows on closed Riemmanian manifolds without focal points, Math. USSR Izvestija, 11 (1977), 1195-1228. 
    [38] L. Polterovich, The second Birkhoff theorem for optical Hamiltonian systems, Proceedings of the AMS, 113 (1991), 513-516.  doi: 10.1090/S0002-9939-1991-1043418-3.
    [39] R. Ruggiero, Manifolds admitting continuous foliations by geodesics, Geometriae Dedicata, 78 (1999), 161-170.  doi: 10.1023/A:1005228901975.
    [40] C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math., 100 (1990), 301-320.  doi: 10.1007/BF01231188.
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