Periodic Reeb orbits on prequantization bundles

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2018, 12: 123-150. doi: 10.3934/jmd.2018005

Periodic Reeb orbits on prequantization bundles

Peter Albers ^1, , Jean Gutt ^2, and Doris Hein ^3,

1.	Mathematisches Institut, Universität Heidelberg, Mathematikon, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2.	Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
3.	Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, D-79104 Freiburg, Germany

Received January 17, 2017 Revised January 16, 2018 Published April 2018

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic orbit or carries at least cuplength $(M)+1$ simple periodic Reeb orbits.

Keywords: Periodic Reeb orbits, cuplength estimate, prequantization bundle, moduli space for a transfer map.

Mathematics Subject Classification: Primary: 53D10; Secondary: 37J45.

Citation: Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005

References:

[1]	P. Albers and D. Hein, Cuplength estimates in Morse cohomology, J. Topol. Anal., 8 (2016), 243-272. doi: 10.1142/S1793525316500102. Google Scholar
[2]	P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Math. Proc. Cambridge Philos. Soc., 149 (2010), 539-551. doi: 10.1017/S0305004110000435. Google Scholar
[3]	M. Abreu and L. Macarini, Multiplicity of periodic orbits for dynamically convex contact forms, J. Fixed Point Theory Appl., 19 (2017), 175-204. doi: 10.1007/s11784-016-0348-2. Google Scholar
[4]	H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math., 38 (1985), 253-289. doi: 10.1002/cpa.3160380302. Google Scholar
[5]	F. Bourgeois and A. Oancea, An exact sequence for contact-and symplectic homology, Invent. Math., 175 (2009), 611-680. doi: 10.1007/s00222-008-0159-1. Google Scholar
[6]	F. Bourgeois, A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, 55-77. Google Scholar
[7]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36. doi: 10.4310/jdg/1452002876. Google Scholar
[8]	I. Ekeland and J.-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319. doi: 10.2307/1971148. Google Scholar
[9]	U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269. doi: 10.1155/S1073792804133941. Google Scholar
[10]	H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008. doi: 10. 1017/CBO9780511611438. Google Scholar
[11]	V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092.Google Scholar
[12]	V. L. Ginzburg, D. Hein, U. Hryniewicz and L. Macarini, Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam, 38 (2013), 55-78. doi: 10.1007/s40306-012-0002-z. Google Scholar
[13]	V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows, Math. Z., 223 (1996), 397-409. doi: 10.1007/BF02621606. Google Scholar
[14]	J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${{\mathbb{R}}^{2n}}$, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505. doi: 10.5802/aif.3069. Google Scholar
[15]	J. Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds, J. Symplectic Geom., 15 (2017), 1019-1069. doi: 10.4310/JSG.2017.v15.n4.a3. Google Scholar
[16]	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), 270-328. doi: 10.1007/BF01895669. Google Scholar
[17]	H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185.Google Scholar
[18]	J. Kang, Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096. doi: 10.1142/S0129167X13500961. Google Scholar
[19]	E. Kerman, Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444. doi: 10.1112/S0010437X17007448. Google Scholar
[20]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${{\mathbf{R}}^{2n}}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120. Google Scholar
[21]	M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8577-5. Google Scholar
[22]	M. Schwarz, Cohomology Operations from S¹-cobordisms in Floer Homology, Ph. D. -thesis, Swiss Federal Inst. of Techn. Zurich, Diss. ETH No. 11182, 1995.Google Scholar
[23]	D. A. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360. doi: 10.1002/cpa.3160451004. Google Scholar
[24]	C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: 10.2140/gt.2007.11.2117. Google Scholar
[25]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462. doi: 10.1215/S0012-7094-07-13931-0. Google Scholar

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References:

[1]	P. Albers and D. Hein, Cuplength estimates in Morse cohomology, J. Topol. Anal., 8 (2016), 243-272. doi: 10.1142/S1793525316500102. Google Scholar
[2]	P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Math. Proc. Cambridge Philos. Soc., 149 (2010), 539-551. doi: 10.1017/S0305004110000435. Google Scholar
[3]	M. Abreu and L. Macarini, Multiplicity of periodic orbits for dynamically convex contact forms, J. Fixed Point Theory Appl., 19 (2017), 175-204. doi: 10.1007/s11784-016-0348-2. Google Scholar
[4]	H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math., 38 (1985), 253-289. doi: 10.1002/cpa.3160380302. Google Scholar
[5]	F. Bourgeois and A. Oancea, An exact sequence for contact-and symplectic homology, Invent. Math., 175 (2009), 611-680. doi: 10.1007/s00222-008-0159-1. Google Scholar
[6]	F. Bourgeois, A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, 55-77. Google Scholar
[7]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36. doi: 10.4310/jdg/1452002876. Google Scholar
[8]	I. Ekeland and J.-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319. doi: 10.2307/1971148. Google Scholar
[9]	U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269. doi: 10.1155/S1073792804133941. Google Scholar
[10]	H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008. doi: 10. 1017/CBO9780511611438. Google Scholar
[11]	V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092.Google Scholar
[12]	V. L. Ginzburg, D. Hein, U. Hryniewicz and L. Macarini, Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam, 38 (2013), 55-78. doi: 10.1007/s40306-012-0002-z. Google Scholar
[13]	V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows, Math. Z., 223 (1996), 397-409. doi: 10.1007/BF02621606. Google Scholar
[14]	J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${{\mathbb{R}}^{2n}}$, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505. doi: 10.5802/aif.3069. Google Scholar
[15]	J. Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds, J. Symplectic Geom., 15 (2017), 1019-1069. doi: 10.4310/JSG.2017.v15.n4.a3. Google Scholar
[16]	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), 270-328. doi: 10.1007/BF01895669. Google Scholar
[17]	H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185.Google Scholar
[18]	J. Kang, Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096. doi: 10.1142/S0129167X13500961. Google Scholar
[19]	E. Kerman, Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444. doi: 10.1112/S0010437X17007448. Google Scholar
[20]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${{\mathbf{R}}^{2n}}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120. Google Scholar
[21]	M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8577-5. Google Scholar
[22]	M. Schwarz, Cohomology Operations from S¹-cobordisms in Floer Homology, Ph. D. -thesis, Swiss Federal Inst. of Techn. Zurich, Diss. ETH No. 11182, 1995.Google Scholar
[23]	D. A. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360. doi: 10.1002/cpa.3160451004. Google Scholar
[24]	C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: 10.2140/gt.2007.11.2117. Google Scholar
[25]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462. doi: 10.1215/S0012-7094-07-13931-0. Google Scholar

Figure 1. The function $h$. The numbers at the graph indicate the slope at this point/section

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Figure 2. The moduli space at $\rho = 0$

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Figure 3. The moduli space at $\rho>0$

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