Finite energy foliations on overtwisted contact manifolds

Wendl, Chris¹

¹ Departement Mathematik, HG G38.1, Rämistrasse 101, 8092 Zürich, Switzerland

Geometry & topology, Tome 12 (2008) no. 1, pp. 531-616.

Résumé

We develop a method for preserving pseudoholomorphic curves in contact 3–manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3–manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1–parameter families of totally real surfaces. The technique is applied here to construct a finite energy foliation for every closed overtwisted contact 3–manifold.

Détail

DOI : 10.2140/gt.2008.12.531

Keywords: holomorphic curves, contact geometry, finite energy foliation, transverse surgery

Bibliographie

[1] C Abbas, Holomorphic open book decompositions

[2] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)", Astérisque 107, Soc. Math. France (1983) 87

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

[4] Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623

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

[6] J B Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. (2004) 4255

[7] H Geiges, Contact geometry, from: "Handbook of differential geometry. Vol. II", Elsevier/North-Holland, Amsterdam (2006) 315

[8] E Giroux, Lecture given at Georgia Topology conference (2001)

[9] E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)", Higher Ed. Press (2002) 405

[10] H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515

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

[12] 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

[13] 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

[14] 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

[15] H Hofer, K Wysocki, E Zehnder, A characterisation of the tight three-sphere, Duke Math. J. 81 (1995)

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

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

[18] C Hummel, Gromov's compactness theorem for pseudo-holomorphic curves, Progress in Mathematics 151, Birkhäuser Verlag (1997)

[19] W B R Lickorish, A representation of orientable combinatorial $3$–manifolds, Ann. of Math. $(2)$ 76 (1962) 531

[20] R Lutz, Sur quelques propriétés des formes différentielles en dimension trois, PhD thesis, Strasbourg (1971)

[21] R Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977) 1

[22] J Martinet, Formes de contact sur les variétés de dimension $3$, from: "Proceedings of Liverpool Singularities Symposium, II (1969/1970)", Springer (1971)

[23] D Mcduff, D Salamon, $J$–holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications 52, American Mathematical Society (2004)

[24] N Saveliev, Lectures on the topology of $3$–manifolds. An introduction to the Casson invariant, de Gruyter Textbook, Walter de Gruyter Co. (1999)

[25] M Seppälä, T Sorvali, Geometry of Riemann surfaces and Teichmüller spaces, North–Holland Mathematics Studies, North–Holland Publishing Co. (1992)

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

[27] A H Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503

[28] C Wendl, Punctured holomorphic curves with boundary in $3$–manifolds: Fredholm theory and embededdness

[29] C Wendl, Finite energy foliations and surgery on transverse links, PhD thesis, New York University (2005)

[30] R Ye, Filling by holomorphic curves in symplectic $4$–manifolds, Trans. Amer. Math. Soc. 350 (1998) 213