On the existence of symplectic barriers

• Pazit Haim-Kislev ^[1] ; Richard Hind ^[2] ; Yaron Ostrover ^[1]
  1. [1] Tel Aviv University

     Tel Aviv University

     Israel

     
  2. [2] University of Notre Dame

     University of Notre Dame

     Township of Portage, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-11
• Idioma: inglés
• DOI: 10.1007/s00029-024-00958-y
• Enlaces
  • Texto completo
• Resumen

  • In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. In particular, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

• Referencias bibliográficas
  • Biran, P.: Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal. 11, 407–464 (2001).
  • Biran, P., Cornea, O.: Rigidity and uniruling for Lagrangian submanifolds. Geom. Topol. 13, 2881–2989 (2009).
  • Brendel, J., Schlenk, F.: Pinwheels as Lagrangian barriers. Commun. Contemp. Math. 26, 2350020 (2024).
  • Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics. Math. Z. 200, 355–378 (1989).
  • Eliashberg, Y.M.: A theorem on the structure of wave fronts and its application in symplectic topology. Funktsional. Anal. i Prilozhen. 21,...
  • Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985).
  • Hind, R.: Symplectic capacities of domains in 𝐶2C 2 Int. Math. Res. Not. 2006, 37171 (2006).
  • Hofer, H., Zehnder, E.: A new capacity for symplectic manifolds. In: Analysis, et Cetera, pp. 405–427. Academic Press, Boston (1990).
  • Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel...
  • Katok, A.B.: Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. 37, 539–576 (1973).
  • McDuff, D.: Fibrations in symplectic topology. In: Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), No....
  • McDuff, D., Polterovich, L.: Symplectic packings and algebraic geometry. Invent. Math. 115, 405–434 (1994). (With an appendix by Yael Karshon).
  • McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Graduate Texts in Mathematics, 3rd edn. Oxford University Press, Oxford...
  • Sackel, K., Song, A., Varolgunes, U., Zhu, J.J.: On certain quantifications of Gromov’s non-squeezing theorem. To appear in Geometry and Topology...
  • Schlenk, F.: Embedding Problems in Symplectic Geometry, vol. 40 of De Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co....
  • Schlenk, F.: Symplectic embedding problems, old and new. Bull. Am. Math. Soc. (N.S.) 55, 139–182 (2018).
  • Tokieda, T.F.: Isotropic isotopy and symplectic null sets. Proc. Natl. Acad. Sci. U.S.A. 94, 13407–13408 (1997).
  • Traynor, L.: Symplectic packing constructions. J. Differ. Geom. 42, 411–429 (1995).
  • Lee, W., Oh, Y.-G., Vianna, R.: Asymptotic behavior of exotic Lagrangian tori Ta,b,c in CP2 as a + b + c → ∞. J. Symplectic Geom....