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2018, 12: 123-150.
doi: 10.3934/jmd.2018005
Periodic Reeb orbits on prequantization bundles
| 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 |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
U. Frauenfelder,
The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269.
doi: 10.1155/S1073792804133941. |
| [10] |
H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008.
doi: 10. 1017/CBO9780511611438. |
| [11] |
V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185. |
| [18] |
J. Kang,
Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096.
doi: 10.1142/S0129167X13500961. |
| [19] |
E. Kerman,
Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444.
doi: 10.1112/S0010437X17007448. |
| [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. |
| [21] |
M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993.
doi: 10. 1007/978-3-0348-8577-5. |
| [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. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
P. Albers and D. Hein,
Cuplength estimates in Morse cohomology, J. Topol. Anal., 8 (2016), 243-272.
doi: 10.1142/S1793525316500102. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
U. Frauenfelder,
The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269.
doi: 10.1155/S1073792804133941. |
| [10] |
H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008.
doi: 10. 1017/CBO9780511611438. |
| [11] |
V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185. |
| [18] |
J. Kang,
Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096.
doi: 10.1142/S0129167X13500961. |
| [19] |
E. Kerman,
Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444.
doi: 10.1112/S0010437X17007448. |
| [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. |
| [21] |
M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993.
doi: 10. 1007/978-3-0348-8577-5. |
| [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. |
| [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. |
| [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. |
| [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. |
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