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Hofer–Zehnder symplectic capacity for two-dimensional manifolds*

Published online by Cambridge University Press:  14 November 2011

Mei-Yue Jiang
Mei-Yue Jiang
Affiliation:
Department of Mathematics, Peking University, Beijing, 100871, P.R. China
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Synopsis

In this paper, we show that the Hofer-Zehnder symplectic capacity of a two-dimensional manifold is same as the area of the manifold. Using this fact, we also get a result on the symplectic capacity of the cotangent bundle of the torus with its canonical symplectic form.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 5 , 1993 , pp. 945 - 950
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Gromov, M.. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307–347.CrossRefGoogle Scholar
2Ekeland, I. and Hofer, H., Symplectic topology and Hamiltonian dynamics. Part 1, Math. Z. 200 (1990), 355–378; Part 2, Math. Z. 203 (1990), 553–569.CrossRefGoogle Scholar
3Hofer, H. and Zehnder, E.. A new capacity for symplectic manifolds. In Analysis et cetera, eds Rabinowtz, P. H. and Zehnder, E., pp. 405–428 (New York: Academic Press, 1990).CrossRefGoogle Scholar
4Moser, J., On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–296.Google Scholar
5Dacorogna, B. and Moser, J., On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincare. Anal. Non Linéaire 7 (1990), 1–26.CrossRefGoogle Scholar
6Morgan, F.. Geometric Measure Theory (New York: Academic Press, 1988).CrossRefGoogle Scholar
7Sikorav, J.. Rigidite symplectique dans le cotangent bundle de T n. Duke Math. J. 59 (1989), 759–763.CrossRefGoogle Scholar
8Hofer, H. and Viterbo, C.. The Weinstein conjecture in cotangent bundle and related results. Ann. Scuola Norm. Sup. Pisa, Ser. 4 15 (1988), 411–445.Google Scholar