Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The Maslov class of the Lagrange surfaces and Gromov's pseudo-holomorphic curves


Author: L. V. Polterovich
Journal: Trans. Amer. Math. Soc. 325 (1991), 241-248
MSC: Primary 58F05; Secondary 58G30
MathSciNet review: 992608
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For an immersed Lagrange submanifold $ W \subset {T^\ast }X$, one can define a nonnegative integer topologic invariant $ m(W)$ such that the image of $ {H_1}(W;{\mathbf{Z}})$ under the Maslov class is equal to $ m(W) \cdot {\mathbf{Z}}$. In this paper, the value of $ m(W)$ is calculated for the case of a two-dimensional oriented manifold $ X$ with the universal cover homeomorphic to $ {{\mathbf{R}}^2}$ and an embedded Lagrange torus $ W$. It is proved that if $ X = {{\mathbf{T}}^2}$ and $ W$ is homologic to the zero section, then $ m(W) = 0$. In all the other cases $ m(W) = 2$. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in $ {T^\ast }{{\mathbf{R}}^2}$. The proof is based on the Gromov's theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.


References [Enhancements On Off] (What's this?)

  • [Ar] V. I. Arnol′d, The first steps of symplectic topology, Uspekhi Mat. Nauk 41 (1986), no. 6(252), 3–18, 229 (Russian). MR 890489
  • [Au] Michèle Audin, Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagragiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), no. 4, 593–623 (French). MR 966952, 10.1007/BF02566781
  • [Bi] Misha Bialy, On the number of caustics for invariant tori of Hamiltonian systems with two degrees of freedom, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 273–278. MR 1116641, 10.1017/S0143385700006143
  • [BP] M. L. Bialy and L. V. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom, Invent. Math. 97 (1989), no. 2, 291–303. MR 1001842, 10.1007/BF01389043
  • [Fl] Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, 10.1002/cpa.3160410603
  • [Gr] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
  • [McD] Dusa McDuff, Examples of symplectic structures, Invent. Math. 89 (1987), no. 1, 13–36. MR 892186, 10.1007/BF01404672
  • [P1] L. V. Polterovich, New invariants of embedded totally real tori and one problem of hamiltonian mechanics, Methods of the Qualitative Theory and the Theory of Bifurcations, Gorky 1988. (Russian)
  • [P2] L. V. Polterovich, Strongly optical Lagrangian manifolds, Mat. Zametki 45 (1989), no. 2, 95–104, 143 (Russian); English transl., Math. Notes 45 (1989), no. 1-2, 152–158. MR 1002523, 10.1007/BF01158062
  • [P3] -, The Maslov class nontriviality and Gromov's pseudo-holomorphic curves, 1989.
  • [Pa] P. Pansu, Notes sur le pages 316 à 323 de l'article de M. Gromov 'Pseudo-holomorphic curves in symplectic manifolds, preprint, 1986.
  • [Si] J.-C. Sikorav, Corollaries symplectiques, preprint, 1988.
  • [Vi] C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math. 100 (1990), no. 2, 301–320. MR 1047136, 10.1007/BF01231188

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F05, 58G30

Retrieve articles in all journals with MSC: 58F05, 58G30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-0992608-9
Article copyright: © Copyright 1991 American Mathematical Society