On certain quantifications of Gromov’s nonsqueezing theorem

Sackel, Kevin¹ ; Song, Antoine² ; Varolgunes, Umut³ ; Zhu, Jonathan J⁴

¹ Department of Mathematics and Statistics, University of Massachusetts, Amherst, Amherst, MA, United States
² Mathematics Department, California Institute of Technology, Pasadena, CA, United States
³ Department of Mathematics, Boğaziçi University, İstanbul, Turkey
⁴ Department of Mathematics, University of Washington, Seattle, WA, United States

Geometry & topology, Tome 28 (2024) no. 3, pp. 1113-1152.

Résumé

Let $R>1$ and let $B$ be the Euclidean $4$–ball of radius $R$ with a closed subset $E$ removed. Suppose that $B$ embeds symplectically into the unit cylinder ${\mathbb{𝔻}}^2\times {ℝ}^2$. By Gromov’s nonsqueezing theorem, $E$ must be nonempty. We prove that the Minkowski dimension of $E$ is at least $2$, and we exhibit an explicit example showing that this result is optimal at least for $R\le \sqrt{2}$. In the appendix by Joé Brendel, it is shown that the lower bound is optimal for $R<\sqrt{3}$. We also discuss the minimum volume of $E$ in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

Détail

DOI : 10.2140/gt.2024.28.1113

Keywords: Gromov nonsqueezing, waist inequality, folding

Bibliographie

[1] A Akopyan, R Karasev, A tight estimate for the waist of the ball, Bull. Lond. Math. Soc. 49 (2017) 690 | DOI

[2] P Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407 | DOI

[3] F Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983) 305 | DOI

[4] J Brendel, G Mikhalkin, F Schlenk, Non-isotopic symplectic embeddings of cubes, in preparation

[5] R Casals, R Vianna, Full ellipsoid embeddings and toric mutations, Selecta Math. 28 (2022) 61 | DOI

[6] J D Evans, KIAS lectures on symplectic aspects of degenerations, lecture notes (2019)

[7] J D Evans, I Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143 | DOI

[8] S Galkin, A Usnich, Laurent phenomenon for Landau–Ginzburg potential, preprint (2010)

[9] M Golla, L Starkston, The symplectic isotopy problem for rational cuspidal curves, Compos. Math. 158 (2022) 1595 | DOI

[10] Y Groman, U Varolgunes, Locality of relative symplectic cohomology for complete embeddings, Compos. Math. 159 (2023) 2551 | DOI

[11] M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 | DOI

[12] M Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003) 178 | DOI

[13] P Hacking, Y Prokhorov, Smoothable del Pezzo surfaces with quotient singularities, Compos. Math. 146 (2010) 169 | DOI

[14] M Harada, K Kaveh, Integrable systems, toric degenerations and Okounkov bodies, Invent. Math. 202 (2015) 927 | DOI

[15] E Heintze, H Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. 11 (1978) 451 | DOI

[16] A B Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 539

[17] Y Lekili, M Maydanskiy, The symplectic topology of some rational homology balls, Comment. Math. Helv. 89 (2014) 571 | DOI

[18] E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247 | DOI

[19] E Lerman, S Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997) 4201 | DOI

[20] P Lisca, On symplectic fillings of lens spaces, Trans. Amer. Math. Soc. 360 (2008) 765 | DOI

[21] D Mcduff, Displacing Lagrangian toric fibers via probes, from: "Low-dimensional and symplectic topology" (editor M Usher), Proc. Sympos. Pure Math. 82, Amer. Math. Soc. (2011) 131 | DOI

[22] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2012)

[23] Y Memarian, On Gromov’s waist of the sphere theorem, J. Topol. Anal. 3 (2011) 7 | DOI

[24] J R Munkres, Elementary differential topology, 54, Princeton Univ. Press (1963)

[25] J Oakley, M Usher, On certain Lagrangian submanifolds of S2 × S2 and CPn, Algebr. Geom. Topol. 16 (2016) 149 | DOI

[26] E Opshtein, Maximal symplectic packings in P2, Compos. Math. 143 (2007) 1558 | DOI

[27] W D Ruan, Lagrangian torus fibration of quintic hypersurfaces, I : Fermat quintic case, from: "Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds" (editors C Vafa, S T Yau), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc. (2001) 297 | DOI

[28] F Schlenk, Embedding problems in symplectic geometry, 40, de Gruyter (2005) | DOI

[29] J Seade, On the topology of isolated singularities in analytic spaces, 241, Birkhäuser (2006) | DOI

[30] M Symington, Four dimensions from two in symplectic topology, from: "Topology and geometry of manifolds", Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 153 | DOI

[31] L Traynor, Symplectic packing constructions, J. Differential Geom. 42 (1995) 411

[32] R F D V Vianna, Infinitely many exotic monotone Lagrangian tori in CP2, J. Topol. 9 (2016) 535 | DOI

[33] R Vianna, Infinitely many monotone Lagrangian tori in del Pezzo surfaces, Selecta Math. 23 (2017) 1955 | DOI