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Instability of the soliton for the focusing, mass-critical generalized KdV equation
Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $
1. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China |
2. | School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, Hubei, China |
In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex $ \mathbb{R}P^{2n-1} $, and if all the closed Reeb orbits are non-degenerate, then there are at least $ n $ closed Reeb orbits, where $ n\geq2 $, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in $ {{\bf R}}^{2n} $ to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.
References:
[1] |
M. Abreu and L. Macarini,
Dynamical convexity and elliptic periodic orbits for Reeb flows, Math. Ann., 369 (2017), 331-386.
doi: 10.1007/s00208-017-1532-4. |
[2] |
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. |
[3] |
P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer and O. van Koert,
Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal., 204 (2012), 273-284.
doi: 10.1007/s00205-011-0475-2. |
[4] |
P. Albers, U. Frauenfelder, O. van Koert and G. P. Paternain,
Contact geometry of the restricted three-body problem, Commun. Pure Appl. Math., 65 (2012), 229-263.
doi: 10.1002/cpa.21380. |
[5] |
H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf,
Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Commun. Pure Appl. Math., 38 (1985), 253-289.
doi: 10.1002/cpa.3160380302. |
[6] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993.
doi: 10.1007/978-1-4612-0385-8. |
[7] |
D. Cristofaro-Gardiner and M. Hutchings,
From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
|
[8] |
G. Dell'Antoio, B. D'Onofrio and I. Ekeland,
Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1413-1415.
|
[9] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in R2n, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp.
doi: 10.1007/s00526-017-1173-1. |
[10] |
H. Duan, H. Liu, Y. Long and W. Wang,
Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1-18.
doi: 10.1007/s10114-016-6019-9. |
[11] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[12] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[13] |
V. L. Ginzburg and B. Z. G$\ddot{u}$rel,
Lusternik-Schnirelmann theory and closed Reeb orbits, Math. Z., 295 (2020), 515-582.
doi: 10.1007/s00209-019-02361-2. |
[14] |
V. L. Ginzburg, B. Z. G$\ddot{u}$rel and L. Macarini,
Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math., 228 (2018), 407-453.
doi: 10.1007/s11856-018-1769-y. |
[15] |
M. Girardi,
Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 285-294.
doi: 10.1016/S0294-1449(16)30423-1. |
[16] |
J. Gutt and J. Kang,
On the minimal number of periodic orbits on some hypersurfaces in R2n, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.
doi: 10.5802/aif.3069. |
[17] |
H. Hofer, K. Wysocki and E. Zehnder,
The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.
doi: 10.2307/120994. |
[18] |
H. Hofer, K. Wysocki and E. Zehnder,
Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math., 157 (2003), 125-255.
doi: 10.4007/annals.2003.157.125. |
[19] |
U. L. Hryniewicz and P. A. S. Salomão, Elliptic bindings for dynamically convex Reeb flows on the real projective three-space, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 43, 57 pp.
doi: 10.1007/s00526-016-0975-x. |
[20] |
X. Hu and Y. Long,
Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Sci. China Ser. A, 45 (2002), 1038-1052.
doi: 10.1007/BF02879987. |
[21] |
H. Liu and Y. Long,
Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces, Calc. Var. Partial Differential Equations, 54 (2015), 3753-3787.
doi: 10.1007/s00526-015-0921-3. |
[22] |
H. Liu, Y. Long and W. Wang,
Resonance identities for closed characteristics on compact star-shaped hypersurfaces in R2n, J. Funct. Anal., 266 (2014), 5598-5638.
doi: 10.1016/j.jfa.2014.02.035. |
[23] |
H. Liu and Y. Xiao,
Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Adv. Math., 318 (2017), 158-190.
doi: 10.1016/j.aim.2017.07.024. |
[24] |
Y. Long,
Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[25] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., 207, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[26] |
Y. Long and C. Zhu,
Closed characteristics on compact convex hypersurfaces in R2n, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[27] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[28] |
C. Viterbo,
Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.
|
[29] |
C. Viterbo,
Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.
doi: 10.1090/S0002-9947-1989-0978370-5. |
[30] |
W. Wang,
Closed trajectories on symmetric convex Hamiltonian energy surfaces, Discrete Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[31] |
W. Wang,
Closed characteristics on compact convex hypersurfaces in R8, Adv. Math., 297 (2016), 93-148.
doi: 10.1016/j.aim.2016.03.044. |
[32] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
show all references
References:
[1] |
M. Abreu and L. Macarini,
Dynamical convexity and elliptic periodic orbits for Reeb flows, Math. Ann., 369 (2017), 331-386.
doi: 10.1007/s00208-017-1532-4. |
[2] |
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. |
[3] |
P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer and O. van Koert,
Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal., 204 (2012), 273-284.
doi: 10.1007/s00205-011-0475-2. |
[4] |
P. Albers, U. Frauenfelder, O. van Koert and G. P. Paternain,
Contact geometry of the restricted three-body problem, Commun. Pure Appl. Math., 65 (2012), 229-263.
doi: 10.1002/cpa.21380. |
[5] |
H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf,
Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Commun. Pure Appl. Math., 38 (1985), 253-289.
doi: 10.1002/cpa.3160380302. |
[6] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993.
doi: 10.1007/978-1-4612-0385-8. |
[7] |
D. Cristofaro-Gardiner and M. Hutchings,
From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
|
[8] |
G. Dell'Antoio, B. D'Onofrio and I. Ekeland,
Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1413-1415.
|
[9] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in R2n, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp.
doi: 10.1007/s00526-017-1173-1. |
[10] |
H. Duan, H. Liu, Y. Long and W. Wang,
Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1-18.
doi: 10.1007/s10114-016-6019-9. |
[11] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[12] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[13] |
V. L. Ginzburg and B. Z. G$\ddot{u}$rel,
Lusternik-Schnirelmann theory and closed Reeb orbits, Math. Z., 295 (2020), 515-582.
doi: 10.1007/s00209-019-02361-2. |
[14] |
V. L. Ginzburg, B. Z. G$\ddot{u}$rel and L. Macarini,
Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math., 228 (2018), 407-453.
doi: 10.1007/s11856-018-1769-y. |
[15] |
M. Girardi,
Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 285-294.
doi: 10.1016/S0294-1449(16)30423-1. |
[16] |
J. Gutt and J. Kang,
On the minimal number of periodic orbits on some hypersurfaces in R2n, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.
doi: 10.5802/aif.3069. |
[17] |
H. Hofer, K. Wysocki and E. Zehnder,
The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.
doi: 10.2307/120994. |
[18] |
H. Hofer, K. Wysocki and E. Zehnder,
Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math., 157 (2003), 125-255.
doi: 10.4007/annals.2003.157.125. |
[19] |
U. L. Hryniewicz and P. A. S. Salomão, Elliptic bindings for dynamically convex Reeb flows on the real projective three-space, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 43, 57 pp.
doi: 10.1007/s00526-016-0975-x. |
[20] |
X. Hu and Y. Long,
Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Sci. China Ser. A, 45 (2002), 1038-1052.
doi: 10.1007/BF02879987. |
[21] |
H. Liu and Y. Long,
Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces, Calc. Var. Partial Differential Equations, 54 (2015), 3753-3787.
doi: 10.1007/s00526-015-0921-3. |
[22] |
H. Liu, Y. Long and W. Wang,
Resonance identities for closed characteristics on compact star-shaped hypersurfaces in R2n, J. Funct. Anal., 266 (2014), 5598-5638.
doi: 10.1016/j.jfa.2014.02.035. |
[23] |
H. Liu and Y. Xiao,
Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Adv. Math., 318 (2017), 158-190.
doi: 10.1016/j.aim.2017.07.024. |
[24] |
Y. Long,
Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[25] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., 207, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[26] |
Y. Long and C. Zhu,
Closed characteristics on compact convex hypersurfaces in R2n, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[27] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[28] |
C. Viterbo,
Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.
|
[29] |
C. Viterbo,
Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.
doi: 10.1090/S0002-9947-1989-0978370-5. |
[30] |
W. Wang,
Closed trajectories on symmetric convex Hamiltonian energy surfaces, Discrete Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[31] |
W. Wang,
Closed characteristics on compact convex hypersurfaces in R8, Adv. Math., 297 (2016), 93-148.
doi: 10.1016/j.aim.2016.03.044. |
[32] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
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