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

On the graph theorem for Lagrangian minimizing tori

Mario Jorge Dias Carneiro 1, and  Rafael O. Ruggiero 2,,

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.

Keywords: Geodesic flow, Birkhoff theorems, global minimizers, Lagrangian graphs.
Mathematics Subject Classification: Primary: 53D25; Secondary: 53D12, 37J50.
Citation: Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete and Continuous Dynamical Systems, 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

[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. Contreras, J. 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. Contreras, J. M. Gambaudo, R. 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. Contreras, R. Iturriaga, G. 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.

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

[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. Contreras, J. 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. Contreras, J. M. Gambaudo, R. 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. Contreras, R. Iturriaga, G. 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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