Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Packing stability for symplectic $ 4$-manifolds


Authors: O. Buse, R. Hind and E. Opshtein
Journal: Trans. Amer. Math. Soc. 368 (2016), 8209-8222
MSC (2010): Primary 53D05, 57R17
DOI: https://doi.org/10.1090/tran/6802
Published electronically: April 15, 2016
MathSciNet review: 3546797
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The packing stability in symplectic geometry was first noticed by Biran (1997): the symplectic obstructions to embed several balls into a manifold disappear when their size is small enough. This phenomenon is known to hold for all closed manifolds with rational symplectic class, as well as for all ellipsoids. In this note, we show that packing stability holds for all closed, and several open, symplectic $ 4$-manifolds.


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

  • [BH11] Olguta Buse and Richard Hind, Symplectic embeddings of ellipsoids in dimension greater than four, Geom. Topol. 15 (2011), no. 4, 2091-2110. MR 2860988 (2012m:53188), https://doi.org/10.2140/gt.2011.15.2091
  • [BH13] O. Buse and R. Hind, Ellipsoid embeddings and symplectic packing stability, Compos. Math. 149 (2013), no. 5, 889-902. MR 3069365, https://doi.org/10.1112/S0010437X12000826
  • [Bir97] P. Biran, Symplectic packing in dimension $ 4$, Geom. Funct. Anal. 7 (1997), no. 3, 420-437. MR 1466333 (98i:57057), https://doi.org/10.1007/s000390050014
  • [Bir99] Paul Biran, A stability property of symplectic packing, Invent. Math. 136 (1999), no. 1, 123-155. MR 1681101 (2000b:57039), https://doi.org/10.1007/s002220050306
  • [EV16] Michael Entov and Misha Verbitsky, Unobstructed symplectic packing for tori and hyper-Kähler manifolds, Journal of Topology and Analysis , posted on (2016), 1650022., https://doi.org/10.1142/S1793525316500229
  • [LMS13] Janko Latschev, Dusa McDuff, and Felix Schlenk, The Gromov width of 4-dimensional tori, Geom. Topol. 17 (2013), no. 5, 2813-2853. MR 3190299
  • [McD98] Dusa McDuff, From symplectic deformation to isotopy, Topics in symplectic $ 4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 85-99. MR 1635697 (99j:57025)
  • [McD09] Dusa McDuff, Symplectic embeddings of 4-dimensional ellipsoids, J. Topol. 2 (2009), no. 1, 1-22. MR 2499436 (2010b:53155), https://doi.org/10.1112/jtopol/jtn031
  • [McD11] Dusa McDuff, The Hofer conjecture on embedding symplectic ellipsoids, J. Differential Geom. 88 (2011), no. 3, 519-532. MR 2844441 (2012j:53113)
  • [MO15] Dusa McDuff and Emmanuel Opshtein, Nongeneric $ J$-holomorphic curves and singular inflation, Algebr. Geom. Topol. 15 (2015), no. 1, 231-286. MR 3325737, https://doi.org/10.2140/agt.2015.15.231
  • [MS98] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616 (2000g:53098)
  • [Ops07] Emmanuel Opshtein, Maximal symplectic packings in $ \mathbb{P}^2$, Compos. Math. 143 (2007), no. 6, 1558-1575. MR 2371382 (2008j:53146), https://doi.org/10.1112/S0010437X07003041
  • [Ops13a] Emmanuel Opshtein, Polarizations and symplectic isotopies, J. Symplectic Geom. 11 (2013), no. 1, 109-133. MR 3022923, https://doi.org/10.4310/JSG.2013.v11.n1.a6
  • [Ops13b] Emmanuel Opshtein, Singular polarizations and ellipsoid packings, Int. Math. Res. Not. IMRN 11 (2013), 2568-2600. MR 3065088, https://doi.org/10.1093/imrn/rns137
  • [Ops15] Emmanuel Opshtein, Symplectic packings in dimension 4 and singular curves, J. Symplectic Geom. 13 (2015), no. 2, 305-342. MR 3338894, https://doi.org/10.4310/JSG.2015.v13.n2.a3

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53D05, 57R17

Retrieve articles in all journals with MSC (2010): 53D05, 57R17


Additional Information

O. Buse
Affiliation: Department of Mathematics, Indiana University – Purdue University, Indianapolis, Indiana 46202

R. Hind
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556

E. Opshtein
Affiliation: Institut de Recherche Mathematique Avancée UMR 7501, Universite de Strasbourg et CNRS, 7 rue Rene Descartes, 67000 Strasbourg, France

DOI: https://doi.org/10.1090/tran/6802
Received by editor(s): April 28, 2014
Received by editor(s) in revised form: June 23, 2015
Published electronically: April 15, 2016
Additional Notes: The first author was partially supported by NSF grant DMS-1211244
The second author was partially supported by Grant # 317510 from the Simons Foundation
The third author was partially supported by ANR project “hameo” ANR-116JS01-010-01
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society