December  2018, 38(12): 6029-6045. doi: 10.3934/dcds.2018260

On the graph theorem for Lagrangian minimizing tori

1. 

Dep. de Matemática - ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil

2. 

Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22453-900, Brazil

* Corresponding author: Rafael O. Ruggiero

Received  September 2017 Revised  April 2018 Published  September 2018

Fund Project: 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.

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.

Citation: Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
References:
[1]

L. Amorim, Y.-G. Oh and J. O. Dos Santos, Exact Lagrangian submanifolds, Lagrangian Spectral Invariants and Aubry-Mather Theory, arXiv: 1603.06966v4 Google Scholar

[2]

M. C. Arnaud, On a Theorem Due to Birkhoff, GAFA, 20 (2010), 1307-1316.  doi: 10.1007/s00039-010-0091-6.  Google Scholar

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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.  Google Scholar

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978, Berlin, Heidelberg, New York.  Google Scholar

[5]

V. I. Arnold, Sturm theorems and symplectic geometry, Functional Anal. and Its Appl., 19 (1985), 1-10.   Google Scholar

[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  Google Scholar

[7]

V. Bangert, Minimal geodesics, Erg. Th. Dyn. Sys., 10 (1990), 263-286.  doi: 10.1017/S014338570000554X.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[12]

M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627.  doi: 10.1007/BF01444639.  Google Scholar

[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.  Google Scholar

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A. Candel and L. Conlon, Foliations I. Graduate Studies in Mathematics, vol. 23. Amer. Math. Soc. Providence, Rhode Island, 2000.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Th. Dyn. Sys., 19 (1999), 901-952.  doi: 10.1017/S014338579913387X.  Google Scholar

[22]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 1999.  Google Scholar

[23]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, J. Differential Geometry, 8 (1973), 437-463.  doi: 10.4310/jdg/1214431801.  Google Scholar

[24]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Lyon (notes), 2000. Google Scholar

[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.  Google Scholar

[26]

L. W. Green, A theorem of E. Hopf, Michigan Math. J., 5 (1958), 31-34.  doi: 10.1307/mmj/1028998009.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.  doi: 10.1088/0951-7715/5/3/001.  Google Scholar

[31]

R. Mañé, Global variational methods in conservative dynamics, IMPA, Rio de Janeiro, 1993. Google Scholar

[32]

J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar

[33]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Claredon Press, Oxford, 1995.  Google Scholar

[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.  Google Scholar

[35]

J. M. Morse, Calculus of variations in the large, AMS Colloquium Publications XVIII, Providence, RI, 1996.  Google Scholar

[36]

G. Paternain, Geodesic Flows, Progress in Mathematics vol. 180, Birkhauser, Boston, Mass. 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[37]

J. Pesin, Geodesic flows on closed Riemmanian manifolds without focal points, Math. USSR Izvestija, 11 (1977), 1195-1228.   Google Scholar

[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.  Google Scholar

[39]

R. Ruggiero, Manifolds admitting continuous foliations by geodesics, Geometriae Dedicata, 78 (1999), 161-170.  doi: 10.1023/A:1005228901975.  Google Scholar

[40]

C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math., 100 (1990), 301-320.  doi: 10.1007/BF01231188.  Google Scholar

show all references

References:
[1]

L. Amorim, Y.-G. Oh and J. O. Dos Santos, Exact Lagrangian submanifolds, Lagrangian Spectral Invariants and Aubry-Mather Theory, arXiv: 1603.06966v4 Google Scholar

[2]

M. C. Arnaud, On a Theorem Due to Birkhoff, GAFA, 20 (2010), 1307-1316.  doi: 10.1007/s00039-010-0091-6.  Google Scholar

[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.  Google Scholar

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978, Berlin, Heidelberg, New York.  Google Scholar

[5]

V. I. Arnold, Sturm theorems and symplectic geometry, Functional Anal. and Its Appl., 19 (1985), 1-10.   Google Scholar

[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  Google Scholar

[7]

V. Bangert, Minimal geodesics, Erg. Th. Dyn. Sys., 10 (1990), 263-286.  doi: 10.1017/S014338570000554X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[12]

M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627.  doi: 10.1007/BF01444639.  Google Scholar

[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.  Google Scholar

[14]

A. Candel and L. Conlon, Foliations I. Graduate Studies in Mathematics, vol. 23. Amer. Math. Soc. Providence, Rhode Island, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Th. Dyn. Sys., 19 (1999), 901-952.  doi: 10.1017/S014338579913387X.  Google Scholar

[22]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 1999.  Google Scholar

[23]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, J. Differential Geometry, 8 (1973), 437-463.  doi: 10.4310/jdg/1214431801.  Google Scholar

[24]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Lyon (notes), 2000. Google Scholar

[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.  Google Scholar

[26]

L. W. Green, A theorem of E. Hopf, Michigan Math. J., 5 (1958), 31-34.  doi: 10.1307/mmj/1028998009.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.  doi: 10.1088/0951-7715/5/3/001.  Google Scholar

[31]

R. Mañé, Global variational methods in conservative dynamics, IMPA, Rio de Janeiro, 1993. Google Scholar

[32]

J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar

[33]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Claredon Press, Oxford, 1995.  Google Scholar

[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.  Google Scholar

[35]

J. M. Morse, Calculus of variations in the large, AMS Colloquium Publications XVIII, Providence, RI, 1996.  Google Scholar

[36]

G. Paternain, Geodesic Flows, Progress in Mathematics vol. 180, Birkhauser, Boston, Mass. 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[37]

J. Pesin, Geodesic flows on closed Riemmanian manifolds without focal points, Math. USSR Izvestija, 11 (1977), 1195-1228.   Google Scholar

[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.  Google Scholar

[39]

R. Ruggiero, Manifolds admitting continuous foliations by geodesics, Geometriae Dedicata, 78 (1999), 161-170.  doi: 10.1023/A:1005228901975.  Google Scholar

[40]

C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math., 100 (1990), 301-320.  doi: 10.1007/BF01231188.  Google Scholar

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