Symplectic embeddings of polydisks. (English) Zbl 1153.53060

Summary: If \(P\) and \(P^{'}\) are symplectic polydisks of radii \(R_{1}\leq \cdots\leq R_{n}\) and \(R_{1}^{'}\leq \cdots\leq R_{n}^{'}\), respectively, then we prove that \(P\) symplectically embeds in \(P^{'}\) provided that \(C(n) R_{1}\leq R_{1}^{'}\) and \(C (n) R_{1}\dots R_{n}\leq R_{1}^{'}\dots R_{n}^{'}\). Up to a constant factor, these conditions are optimal.


53D35 Global theory of symplectic and contact manifolds
53D05 Symplectic manifolds (general theory)
Full Text: DOI arXiv


[1] Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics. Math. Z. 200(3), 355–378 (1989) · Zbl 0641.53035 · doi:10.1007/BF01215653
[2] Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985) · Zbl 0592.53025 · doi:10.1007/BF01388806
[3] Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983) · Zbl 0515.53037
[4] Guth, L.: The width-volume inequality. arXiv:math/0609569 · Zbl 1141.53039
[5] Hofer, H.: Symplectic capacities. In: Geometry of Low-dimensional Manifolds 2 (Durham, 1989). Lond. Math. Soc. Lect. Note Ser., vol. 151, pp. 15–34. Cambridge Univ. Press, Cambridge (1990)
[6] Cieliebak, K., Hofer, H., Latschev, J., Schlenk, F.: Quantitative symplectic geometry. arXiv:math/0506191 · Zbl 1143.53341
[7] Moser, J.: On the volume element of a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965) · Zbl 0141.19407 · doi:10.1090/S0002-9947-1965-0182927-5
[8] Schlenk, F.: Embedding Problems in Symplectic Geometry. de Gruyter Expositions in Mathematics, vol. 40. Walter de Gruyter, Berlin (2005) · Zbl 1073.53117
[9] Traynor, L.: Symplectic packing constructions. J. Differ. Geom. 42(2), 411–429 (1995) · Zbl 0861.52008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.