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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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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 Google Scholar |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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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