American Institute of Mathematical Sciences

Advanced
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 & 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: 1603.06966v4Google 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. 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. Google Scholar

[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. Google Scholar

[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. 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

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: 1603.06966v4Google 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. 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. Google Scholar

[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. Google Scholar

[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. 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

[1]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643
[2]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[3]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[4]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158
[5]

Fabio Camilli, Adriano Festa, Silvia Tozza. A discrete Hughes model for pedestrian flow on graphs. Networks & Heterogeneous Media, 2017, 12 (1) : 93-112. doi: 10.3934/nhm.2017004
[6]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173
[7]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[8]

Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933
[9]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[10]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687
[11]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[12]

Laurence Guillot, Maïtine Bergounioux. Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1333-1349. doi: 10.3934/cpaa.2009.8.1333
[13]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375
[14]

Salvador Villegas. Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1817-1821. doi: 10.3934/cpaa.2011.10.1817
[15]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035
[16]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[17]

G. A. Enciso, E. D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 549-578. doi: 10.3934/dcds.2006.14.549
[18]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355
[19]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
[20]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (51)
  • HTML views (82)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences