Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Conformal invariance of symplectic packing numbers and relationship with symplectic capacities

Document Type : Original Article

Authors
1 -Faculty of Mathematics, Kim Hyong Jik University of Education, Pyongyang, Democratic People’s Republic of Korea
2 Faculty of Mathematics, Kim Hyong Jik University of Education, Pyongyang, Democratic People’s Republic of Korea
Abstract
In this paper, we proved conformal invariance of symplectic packing numbers and its consequences. And we also found out the relationship with symplectic capacities and finally, studied the existence of symplectic embedding from symplectic ellipsoid into polydisc.
Keywords
Subjects

1. M. Gosson et al., Symplectic capacities and the geometry
of uncertainty: The irruption of symplectic topology
in classical and quantum, Phys. Rep. 484, 131 (2009),
doi: 10.1016/j.physrep.2009.08.001.

2. D. McDuff, Introduction to symplectic topology, (Oxford
University Press, 1998).

3. D. McDuff, Hofer conjecture on embedding symplectic
ellipsoids, J. Diff. Geom. 88, 519 (2011), doi:
10.4310/jdg/1321366358.

4. D. McDuff et al., The embedding capacity of 4-
dimensional symplectic ellipsoids, Ann. Math. 175,
1191 (2012), doi: 10.4007/annals.2012.175.3.5.

5. M. Hutchings, An elementary alternative to ECH capacities,
PNAS 119(35), e2203090119 (2022), doi:
10.1073/pnas.2203090119.

6. B. Wormleighton, ECH capacities, EHRHART theory,
and toric varieties, J. Symplectic Geom. 19, 475 (2021),
doi: 10.4310/JSG.2021.v19.n2.a5.

7. D. Frenkel et al., Symplectic embeddings of 4-
dimensional ellipsoids into cubes, J. Symplectic Geom.

13, 765 (2015), doi: 10.4310/JSG.2015.v13.n4.a2.
8. L. Guth, Symplectic embeddings of polydisks, Invent.
Math. 172, 477 (2008), doi: 10.1007/s00222-007-0103-
9.

9. D. Cristofaro-Gardiner, Symplectic embeddings from
concave toric domains into convex ones, J. Diff.
Geom. 112, 199 (2019), arXiv:1409.4378 [math.SG],
doi: 10.4310/jdg/1559786421.

10. F. Schlenk, Embedding problems in symplectic geometry,
(Walter de Gruyter, Berlin, New York, 2005), doi:
10.1515/9783110199697.

11. P. Biran, Symplectic packing in dimension
4, Geom. Funct. Anal. 7, 420 (1997), doi:
10.1007/s000390050014.

12. D. Cristofaro-Gardiner, R. Hind, Boundaries of
open symplectic manifolds and the failure of packing
stability, arXiv:2307.01140 [math. SG], doi:
10.4171/JEMS/1707.

13. O. Buse et al., Packing stability for symplectic 4-
manifolds, Trans. Amer. Math. Soc. 368, 8209 (2016),
doi: 10.1090/tran/6802.

14. L. Traynor, Symplectic packing constructions, J. Diff.
Geom. 41, 735 (1995).

15. F. Schlenk, Symplectic Embedding Problems, Old and
New, Bull. Amer. Math. Soc. (N.S.) 55, 139 (2018), doi:
10.1090/bull/1587.

16. J. Basto Gonçalves, Symplectic rigidity and flexibility
of ellipsoids, Indag. Math. 24, 264 (2013), doi:
10.1016/j.indag.2012.09.011.

17. L. Baracco et al., On the Ekeland-Hofer symplectic capacities
of the real bidisc, Pacific J. Math. 305, 423
(2020), doi: 10.2140/pjm.2020.305.423
 

Articles in Press, Accepted Manuscript
Available Online from 24 October 2026

  • Receive Date 13 July 2026
  • Accept Date 14 October 2026
  • First Publish Date 24 October 2026
  • Publish Date 24 October 2026