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Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces
1. | School of Mathematical Sciences, Faculty of Exact Sciences, Tel Aviv University |
2. | Department of Mathematics, Faculty of Natural Sciences, University of Haifa |
For a symplectic manifold $(M,ω)$, let $\{·,·\}$ be the corresponding Poisson bracket. In this note we prove that the functional $ (F,G) \mapsto \|\{F,G\}\|_{L^p(M)} $ is lower-semicontinuous with respect to the $C^0$-norm on $C^∞_c(M)$ when $\dim M = 2$ and $p < ∞$, extending previous rigidity results for $p = ∞$ in arbitrary dimension.
References:
[1] |
L. Buhovsky,
The $2/3$
-convergence rate for the Poisson bracket, Geom. Funct. Anal., 19 (2010), 1620-1649.
doi: 10.1007/s00039-010-0045-z. |
[2] |
S. S. Cairns,
A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc., 67 (1961), 389-390.
doi: 10.1090/S0002-9904-1961-10631-9. |
[3] |
F. Cardin and C. Viterbo,
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[4] |
M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in Symplectic Topology and
Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence,
RI, 2010, 25–32.
doi: 10.1090/conm/512/10058. |
[5] |
M. Entov, L. Polterovich and F. Zapolsky,
Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), 1037-1055.
doi: 10.4310/PAMQ.2007.v3.n4.a9. |
[6] |
H. Federer,
Geometric Measure Theory Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. |
[7] |
K. Samvelyan,
Rigidity Versus Flexibility of the Poisson Bracket with Respect to the ${L}_p$
-Norm Master's thesis, Tel Aviv University, 2015. |
[8] |
F. Zapolsky,
Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.
doi: 10.3934/jmd.2007.1.465. |
show all references
References:
[1] |
L. Buhovsky,
The $2/3$
-convergence rate for the Poisson bracket, Geom. Funct. Anal., 19 (2010), 1620-1649.
doi: 10.1007/s00039-010-0045-z. |
[2] |
S. S. Cairns,
A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc., 67 (1961), 389-390.
doi: 10.1090/S0002-9904-1961-10631-9. |
[3] |
F. Cardin and C. Viterbo,
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[4] |
M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in Symplectic Topology and
Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence,
RI, 2010, 25–32.
doi: 10.1090/conm/512/10058. |
[5] |
M. Entov, L. Polterovich and F. Zapolsky,
Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), 1037-1055.
doi: 10.4310/PAMQ.2007.v3.n4.a9. |
[6] |
H. Federer,
Geometric Measure Theory Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. |
[7] |
K. Samvelyan,
Rigidity Versus Flexibility of the Poisson Bracket with Respect to the ${L}_p$
-Norm Master's thesis, Tel Aviv University, 2015. |
[8] |
F. Zapolsky,
Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.
doi: 10.3934/jmd.2007.1.465. |
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