Fast finite-energy planes in symplectizations and applications

Hryniewicz, Umberto

doi:10.1090/S0002-9947-2011-05387-0

Fast finite-energy planes in symplectizations and applications
HTML articles powered by AMS MathViewer

by Umberto Hryniewicz PDF

Trans. Amer. Math. Soc. 364 (2012), 1859-1931 Request permission

Abstract:

We define the notion of fast finite-energy planes in the symplectization of a closed $3$-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \mathbb {R}^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number $-1$.

References

Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
Yakov Eliashberg, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR 1171908
John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
Emmanuel Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), no. 4, 637–677 (French). MR 1129802, DOI 10.1007/BF02566670
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
Helmut Hofer and Markus Kriener, Holomorphic curves in contact dynamics, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 77–131. MR 1662750, DOI 10.1090/pspum/065/1662750
H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 3, 337–379 (English, with English and French summaries). MR 1395676, DOI 10.1016/s0294-1449(16)30108-1
H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995), no. 2, 270–328. MR 1334869, DOI 10.1007/BF01895669
H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, Topics in nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 35, Birkhäuser, Basel, 1999, pp. 381–475. MR 1725579
H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 78–117. MR 1432460
H. Hofer, K. Wysocki, and E. Zehnder, A characterisation of the tight three-sphere, Duke Math. J. 81 (1995), no. 1, 159–226 (1996). A celebration of John F. Nash, Jr. MR 1381975, DOI 10.1215/S0012-7094-95-08111-3
H. Hofer, K. Wysocki, and E. Zehnder, A characterization of the tight $3$-sphere. II, Comm. Pure Appl. Math. 52 (1999), no. 9, 1139–1177. MR 1692144, DOI 10.1002/(SICI)1097-0312(199909)52:9<1139::AID-CPA5>3.3.CO;2-C
H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2) 148 (1998), no. 1, 197–289. MR 1652928, DOI 10.2307/120994
H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), no. 1, 125–255. MR 1954266, DOI 10.4007/annals.2003.157.125
H. Hofer, K. Wysocki, and E. Zehnder, Finite energy cylinders of small area, Ergodic Theory Dynam. Systems 22 (2002), no. 5, 1451–1486. MR 1934144, DOI 10.1017/S0143385702001013
U. Hryniewicz. Systems of global surfaces of sections on dynamically convex energy levels. In preparation.
U. Hryniewicz and P. Salomão. On the existence of disk-like global sections for Reeb flows on the tight $3$-sphere. Preprint.
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062, DOI 10.1007/978-1-4612-4264-2_{1}2
Dusa McDuff, The local behaviour of holomorphic curves in almost complex $4$-manifolds, J. Differential Geom. 34 (1991), no. 1, 143–164. MR 1114456
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629, DOI 10.1090/coll/052
Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI 10.1002/cpa.3160310203
Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874, DOI 10.1016/0040-9383(93)90052-W
Dietmar Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22 (1990), no. 2, 113–140. MR 1045282, DOI 10.1112/blms/22.2.113
M. Schwarz. Cohomology operations from $S^1$-cobordisms in Floer homology. PhD Thesis (1995).
Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473, DOI 10.2140/gt.2007.11.2117
C. B. Thomas (ed.), Contact and symplectic geometry, Publications of the Newton Institute, vol. 8, Cambridge University Press, Cambridge, 1996. Papers from the Special Programme held at the University of Cambridge, Cambridge, July–December 1994. MR 1432455
Alan Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–518. MR 512430, DOI 10.2307/1971185
C. Wendl. Compactness for embedded pseudo-holomorphic curves in $3$-manifolds. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 2, 313-342.