On the graph theorem for Lagrangian minimizing tori

Mario Jorge Dias Carneiro; Rafael O. Ruggiero

doi:10.3934/dcds.2018260

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2018, Volume 38, Issue 12: 6029-6045. Doi: 10.3934/dcds.2018260

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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
* Corresponding author: Rafael O. Ruggiero

Received: September 2017

Revised: April 2018

Published: December 2018

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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Abstract

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.

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

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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.