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Two closed orbits for non-degenerate Reeb flows

Part of: Symplectic geometry, contact geometry Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  21 February 2020

MIGUEL ABREU ,
JEAN GUTT ,
JUNGSOO KANG  and
LEONARDO MACARINI
MIGUEL ABREU
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001Lisboa, Portugal. e-mail: [email protected]
JEAN GUTT
Affiliation:
Institut de Mathématiques de Toulouse, Université Toulouse III Paul–Sabatier, 118, route de Narbonne, F-31062Toulouse, France. e-mail: [email protected]
JUNGSOO KANG
Affiliation:
Department of Mathematical Sciences, Research Institute in Mathematics, Seoul National University, Gwanak-Gu, Seoul08826, South Korea. e-mail: [email protected]
LEONARDO MACARINI
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001Lisboa, Portugal. e-mail: [email protected]
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Abstract

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

MSC classification

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 53D25: Geodesic flows 37J10: Symplectic mappings, fixed points 37J55: Contact systems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 625 - 660
Copyright
© Cambridge Philosophical Society 2020

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References

REFERENCES

Abouzaid, M.. Symplectic cohomology and Viterbo’s theorem. In Free loop spaces in geometry and topology. IRMA Lect. Math. Theor. Phys., 24, Eur. Math. Soc. (Zürich, 2015), 271–485.Google Scholar
Abreu, M. and Macarini, L.. Contact homology of good toric contact manifolds. Compositio Math. 148 (2012), 304–334.CrossRefGoogle Scholar
Abreu, M. and Macarini, L.. Multiplicity of periodic orbits for dynamically convex contact forms. J. Fixed Point Theory and Applications 19 (2017), 175–204.CrossRefGoogle Scholar
Abreu, M. and Macarini, L.. On the mean Euler characteristic of Gorenstein toric contact manifolds. To appear in Int. Math. Res. Not. Available online at https://doi.org/10.1093/imrn/rny151.CrossRefGoogle Scholar
Abreu, M., Macarini, L. and Moreira, M.. On contact invariants of non-simply connected Gorenstein toric contact manifolds. Preprint arXiv:1812.10361 (2018).Google Scholar
Asselle, L. and Benedetti, G.. The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles. J. Topol. Anal. 8 (2016), 545–570.CrossRefGoogle Scholar
Bangert, V. and Long, Y.. The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346 (2010), 335–366.CrossRefGoogle Scholar
Benedetti, G. and Ritter, A.. Invariance of symplectic cohomology and twisted cotangent bundles over surfaces. Preprint arXiv:1807.02086 (2018).Google Scholar
Biran, P. and Cieliebak, K.. Lagrangian embeddings into subcritical Stein manifolds. Israel J. Math. 127 (2002), 221–244.CrossRefGoogle Scholar
Bourgeois, F. and Oancea, A.. Fredholm theory and transversality for the parametrized and for the S1-invariant symplectic action. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1181–1229.CrossRefGoogle Scholar
Bourgeois, F. and Oancea, A.. The index of Floer moduli problems for parametrized action functionals. Geom. Dedicata 165 (2013), 5–24.CrossRefGoogle Scholar
Bourgeois, F. and Oancea, A.. The Gysin exact sequence for S1-equivariant symplectic homology. J. Topol. Anal. 5 (2013), no. 4, 361–407.CrossRefGoogle Scholar
Bourgeois, F. and Oancea, A.. S 1-equivariant symplectic homology and linearized contact homology. Int. Math. Res. Not. (IMRN) (2017), no. 13, 3849–3937.Google Scholar
Cieliebak, K. and Latschev, J.. The role of string topology in symplectic field theory. In New perspectives and challenges in symplectic field theory. CRM Proc. Lecture Notes Vol. 49 (Amer. Math. Soc., Providence, RI, 2009), 113–146.Google Scholar
Cieliebak, K. and Oancea, A.. Symplectic homology and the Eilenberg–Steenrod axioms. Appendix written jointly with Peter Albers. Algebr. Geom. Topol. 18 (2018), no. 4, 1953–2130.CrossRefGoogle Scholar
Cristofaro–Gardiner, D. and Hutchings, M.. From one Reeb orbit to two. J. Differential Geom. 102 (2016), no. 1, 25–36.CrossRefGoogle Scholar
Davis, J. and Kirk, P.. Lecture notes in algebraic topology. Graduate Studies in Mathematics, 35 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
Duan, H., Liu, H., Long, Y. and Wang, W.. Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ2n. Acta Math. Sin. (Engl. Ser.) 34 (2018), 1–18.CrossRefGoogle Scholar
Duan, H., Long, Y. and Wang, W.. The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds. Calc. Var. Partial Differential Equations 55 (2016), 55–145.CrossRefGoogle Scholar
Espina, J.. On the mean Euler characteristic of contact manifolds. Internat. J. Math. 25 (2014), no. 5, 1450046, 36 pp.CrossRefGoogle Scholar
Frauenfelder, U., Schlenk, F. and van Koert, O.. Displaceability and the mean Euler characteristic. Kyoto J. Math. 52 (2012), no. 4, 797–815.CrossRefGoogle Scholar
Ginzburg, V. and Gürel, B.. Lusternik–Schnirelmann theory and closed Reeb orbits. Math. Z. (2019), DOI: 10.1007/s00209-019-02361-2.CrossRefGoogle Scholar
Ginzburg, V., Gürel, B. and Macarini, L.. On the Conley conjecture for Reeb flows. Internat. J. Math. 26 (2015), DOI: 10.1142/S0129167X15500470.CrossRefGoogle Scholar
Ginzburg, V., Gürel, B. and Macarini, L.. Multiplicity of Closed Reeb Orbits on Prequantization Bundles. Israel J. Math. 228 (2018), no. 1, 407–453.CrossRefGoogle Scholar
Ginzburg, V., Hein, D., Hryniewicz, U. and Macarini, L.. Closed Reeb orbits on the sphere and symplectically degenerate maxima. Acta Math. Vietnam 38 (2013), no. 1, 55–78.CrossRefGoogle Scholar
Ginzburg, V. and Shon, J.. On the filtered symplectic homology of prequantisation bundles. Internat. J. Math. 29 (2018), DOI: 10.1142/S0129167X18500714.CrossRefGoogle Scholar
Gromoll, D. and Meyer, W.. Periodic geodesics on compact Riemannian manifolds. J. Differential Geom. 3 (1969), 493–510.CrossRefGoogle Scholar
Gurel, B.. Perfect Reeb flows and action-index relations. Geom. Dedicata 174 (2015), 105–120.CrossRefGoogle Scholar
Gutt, J. and Kang, J.. On the minimal number of periodic orbits on some hypersurfaces in ℝ2n. Ann. inst. Fourier 66 (2016), 2485–2505.CrossRefGoogle Scholar
Gutt, J. and Usher, M.. Symplectically knotted codimension-zero embeddings of domains in ℝ4. Duke Math. J. 168 (2019), no. 12, 2299–2363.CrossRefGoogle Scholar
Hilton, P., Mislin, G. and Roitberg, J.. Localization of nilpotent groups and spaces. North-Holland Mathematics Studies, No. 15. Notas de Matemá tica, No. 55 (North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975).Google Scholar
Hryniewicz, U. and Macarini, L.. Local contact homology and applications. J. Topol. Anal. 7 (2015), 167–238.CrossRefGoogle Scholar
Kang, J.. Equivariant symplectic homology and multiple closed Reeb orbits. Internat. J. Math. 24 (2013), no. 13, 1350096, 30 pp.CrossRefGoogle Scholar
Klaus, S. and Kreck, M.. A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres. Math. Proc. Camb. Phil. Soc. 136 (2004), 617–623.CrossRefGoogle Scholar
Klingenberg, W.. Lectures on closed geodesics. Grundlehren der Mathematischen Wissenschaften, Vol. 230 (Springer-Verlag, Berlin-New York, 1978).CrossRefGoogle Scholar
Kwon, M. and van Koert, O.. Brieskorn manifolds in contact topology. Bull. Lond. Math. Soc. 48 (2016), 173–241.CrossRefGoogle Scholar
Lerman, E.. Contact toric manifolds. J. Symplectic Geom. 1 (2003), 785–828.CrossRefGoogle Scholar
Liu, H. and Long, Y.. The existence of two closed characteristics on every compact star-shaped hypersurface in ℝ4. Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 1, 40–53.CrossRefGoogle Scholar
Macarini, L. and Paternain, G.. Equivariant symplectic homology of Anosov contact structures. Bull. Braz. Math. Soc. (N.S.) 43 (2012), 513–527.CrossRefGoogle Scholar
May, J. and Ponto, K.. More concise algebraic topology. Chicago Lectures in Mathematics. (University of Chicago Press, Chicago, IL, 2012).Google Scholar
McLean, M.. Local Floer homology and infinitely many simple Reeb orbits. Algebr. Geom. Topol. 12 (2012), no. 4, 1901–1923.CrossRefGoogle Scholar
McLean, M.. Reeb orbits and the minimal discrepancy of an isolated singularity. Invent. Math. 204 (2016), no. 2, 505–594.Google Scholar
McLean, M. and Ritter, A.. The McKay correspondence via Floer theory. Preprint arXiv:1802.01534 (2018).Google Scholar
Neisendorfer, J.. Algebraic methods in unstable homotopy theory. New Mathematical Monographs, 12 (Cambridge University Press, Cambridge, 2010).Google Scholar
Rademacher, H.. Morse Theorie und geschlossene Geodatische. Bonner Math. Schriften 229 (1992).Google Scholar
Rademacher, H.. On the average indices of closed geodesics. J. Diff. Geom. 29 (1989), 65–83.Google Scholar
Ritter, A.. Deformations of symplectic cohomology and exact Lagrangians in ALE spaces. Geom. Funct. Anal. 20 (2010), 779–816.CrossRefGoogle Scholar
Ritter, A.. Topological quantum field theory structure on symplectic cohomology. J. Topol. 6 (2013), 391–489.CrossRefGoogle Scholar
Taimanov, I.. The type numbers of closed geodesics. Regul. Chaotic Dyn. 15 (2010), 84–100.CrossRefGoogle Scholar
Taubes, C.. The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11 (2007), 2117–2202.CrossRefGoogle Scholar
Ustilovsky, I.. Infinitely Many Contact Structures on S 4m+1 Int. Math. Res. Not. (IMRN) 14 (1999), 781–791.Google Scholar
Vigué–Poirrier, M. and Sullivan, D.. The homology theory of the closed geodesic problem. J. Differential Geometry 11 (1976), no. 4, 633–644.CrossRefGoogle Scholar
Viterbo, C.. Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9 (1999), no. 5, 985–1033.CrossRefGoogle Scholar
Whitehead, G.. Elements of homotopy theory. Graduate Texts in Mathematics, 61 (Springer-Verlag, New York-Berlin, 1978).CrossRefGoogle Scholar