Abstract
We survey some symplectic embedding results focussing on the case when both domain and range are products of 4-dimensional ellipsoids or polydisks with Euclidean space. The stabilized problems have additional flexibility but some 4-dimensional obstructions persist.
References
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
K. Christianson, J. Nelson, Symplectic embeddings of four-dimensional polydisks into balls (2016). arXiv:1610.00566
D. Cristofaro-Gardiner, R. Hind, Symplectic embeddings of products. Commun. Math. Helv. (2015). arXiv:1508.02659
D. Cristofaro-Gardiner, D. Frenkel, F. Schlenk, Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs. Algebr. Geom. Topol. 17, 1189–1260 (2017)
D. Cristofaro-Gardiner, R. Hind, D. McDuff, The ghost stairs stabilize to sharp symplectic embedding obstructions (2017). arXiv:1702.03607
Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory, in Geometric and Functional Analysis, GAFA 2000 (Tel Aviv, 1999), Special volume, Part II (2000), pp. 560–673
D. Frenkel, D. Müller, Symplectic embeddings of 4-dimensional ellipsoids into cubes. J. Symplectic Geom. 13, 765–847 (2015)
M. Gromov, Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
L. Guth, Symplectic embeddings of polydisks. Invent. Math. 172, 477–489 (2008)
R. Hind, Some optimal embeddings of symplectic ellipsoids. Topology 8, 871–883 (2015)
R. Hind, E. Kerman, New obstructions to symplectic embeddings. Invent. Math. 196, 383–452 (2014)
R. Hind, E. Kerman, New obstructions to symplectic embeddings: erratum. Preprint (2017)
R. Hind, S. Lisi, Symplectic embeddings of polydisks. Sel. Math. 21, 1099–1120 (2015)
R. Hind, E. Opshtein, Lagrangian tori in the ball (in preparation)
M. Hutchings, Quantitative embedded contact homology. J. Differ. Geom. 88, 231–266 (2011)
D. McDuff, The Hofer conjecture on embedding symplectic ellipsoids. J. Differ. Geom. 88, 519–532 (2011)
D. McDuff, F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids. Ann. Math. 175, 1191–1282 (2012)
E. Opshtein, Maximal symplectic packings of \(\mathcal{P}^{2}\). Compos. Math. 143, 1558–1575 (2007)
A. Pelayo, S.V. Ngọc, The Hofer question on intermediate symplectic capacities. Proc. Lond. Math. Soc. 110, 787–804 (2015)
F. Schlenk, Embedding Problems in Symplectic Geometry. De Gruyter Expositions in Mathematics, vol. 40 (Walter de Gruyter, Berlin, 2005)
Acknowledgements
The author is partially supported by grant # 317510 from the Simons Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, a part of Springer Nature
About this chapter
Cite this chapter
Hind, R. (2017). Stabilized Symplectic Embeddings. In: Angella, D., Medori, C., Tomassini, A. (eds) Complex and Symplectic Geometry. Springer INdAM Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-62914-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-62914-8_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62913-1
Online ISBN: 978-3-319-62914-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)