Intersection theory of punctured pseudoholomorphic curves

Siefring, Richard¹

¹ Max Planck Institute for Math in Sci, Inselstraße 22, D-04103 Leipzig, Germany, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Geometry & topology, Tome 15 (2011) no. 4, pp. 2351-2457.

Résumé

We study the intersection theory of punctured pseudoholomorphic curves in $4$–dimensional symplectic cobordisms. Using the asymptotic results of the author [Comm. Pure Appl. Math. 61(2008) 1631–84], we first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings [JEMS 4(2002) 313–61].

We then turn our attention to curves in the symplectization $ℝ\times M$ of a $3$–manifold $M$ admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the $3$–manifold and we present conditions that will guarantee the projection of a curve to the $3$–manifold is an embedding.

Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition and an application concerning the existence of generalized pseudoholomorphic curves, as introduced by Hofer [Geom. Func. Anal. (2000) 674–704] .

Détail

DOI : 10.2140/gt.2011.15.2351

Keywords: pseudoholomorphic curves, symplectic field theory, Floer homology, intersection theory

Bibliographie

[1] A Abbondandolo, Morse theory for Hamiltonian systems, Research Notes in Mathematics 425, Chapman Hall, Boca Raton, FL (2001)

[2] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799

[3] D L Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004) 726

[4] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560

[5] Y Eliashberg, S S Kim, L Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol. 10 (2006) 1635

[6] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307

[7] H Hofer, Holomorphic curves and real three-dimensional dynamics, Geom. Funct. Anal. (2000) 674

[8] H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995) 270

[9] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies, from: "Contact and symplectic geometry (Cambridge, 1994)", Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 78

[10] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 337

[11] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, from: "Topics in nonlinear analysis", Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser (1999) 381

[12] H Hofer, K Wysocki, E Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. $(2)$ 157 (2003) 125

[13] H Hofer, E Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag (1994)

[14] M Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. $($JEMS$)$ 4 (2002) 313

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

[16] M Kriener, An intersection formula for finite energy half cylinders, PhD thesis, ETH Zurich (1998)

[17] D Mcduff, Singularities and positivity of intersections of $J$-holomorphic curves, from: "Holomorphic curves in symplectic geometry", Progr. Math. 117, Birkhäuser (1994) 191

[18] M J Micallef, B White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. $(2)$ 141 (1995) 35

[19] A Momin, Contact homology of orbit complements and implied existence

[20] E Mora-Donato, Pseudoholomorphic cylinders in symplectisations, PhD thesis, New York University (2003)

[21] R L Siefring, Intersection theory of finite energy surfaces, PhD thesis, New York University (2005)

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