Beyond ECH capacities

Hutchings, Michael¹

¹ Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA

Geometry & topology, Tome 20 (2016) no. 2, pp. 1085-1126.

Résumé

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a “concave toric domain” and the target is a “convex toric domain” (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind and Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.

Détail

DOI : 10.2140/gt.2016.20.1085

Classification : 53D42
Keywords: embedded contact homology, symplectic embeddings

Bibliographie

[1] K Choi, D Cristofaro-Gardiner, D Frenkel, M Hutchings, V G B Ramos, Symplectic embeddings into four-dimensional concave toric domains, J. Topol. 7 (2014) 1054

[2] D Cristofaro-Gardiner, The absolute gradings on embedded contact homology and Seiberg–Witten Floer cohomology, Algebr. Geom. Topol. 13 (2013) 2239

[3] D Cristofaro-Gardiner, Symplectic embeddings from concave toric domains into convex ones, preprint (2014)

[4] D Cristofaro-Gardiner, A Kleinman, Ehrhart polynomials and symplectic embeddings of ellipsoids, preprint (2013)

[5] D Frenkel, D Müller, Symplectic embeddings of four-dimensional ellipsoids into cubes, preprint (2012)

[6] R Hind, S Lisi, Symplectic embeddings of polydisks, Selecta Mathematica 21 (2014) 1099

[7] M Hutchings, Embedded contact homology as a (symplectic) field theory, in preparation

[8] M Hutchings, The embedded contact homology index revisited, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 263

[9] M Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011) 231

[10] M Hutchings, Recent progress on symplectic embedding problems in four dimensions, Proc. Natl. Acad. Sci. USA 108 (2011) 8093

[11] M Hutchings, Lecture notes on embedded contact homology, from: "Contact and symplectic topology" (editors F Bourgeois, V Colin, A Stipsicz), Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc., Budapest (2014) 389

[12] M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of T3, Geom. Topol. 10 (2006) 169

[13] M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders, I, J. Symplectic Geom. 5 (2007) 43

[14] M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders, II, J. Symplectic Geom. 7 (2009) 29

[15] M Hutchings, C H Taubes, Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol. 17 (2013) 2601

[16] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007)

[17] M Landry, M Mcmillan, E Tsukerman, On symplectic capacities of toric domains, Involve 8 (2015) 665

[18] D Mcduff, The Hofer conjecture on embedding symplectic ellipsoids, J. Differential Geom. 88 (2011) 519

[19] D Mcduff, F Schlenk, The embedding capacity of 4–dimensional symplectic ellipsoids, Ann. of Math. 175 (2012) 1191

[20] F Schlenk, On symplectic folding, preprint (1999)

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

[22] R Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008) 1631

[23] C H Taubes, Pseudoholomorphic punctured spheres in R × (S1 × S2) : properties and existence, Geom. Topol. 10 (2006) 785

[24] C H Taubes, Pseudoholomorphic punctured spheres in R × (S1 × S2) : moduli space parametrizations, Geom. Topol. 10 (2006) 1855

[25] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, I, Geom. Topol. 14 (2010) 2497

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