2017, 24: 28-37. doi: 10.3934/era.2017.24.004

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

Received  September 29, 2016 Published  May 2017

Fund Project: We wish to thank Lev Buhovsky and Leonid Polterovich for reading a preliminary version of the paper and making useful comments and for their interest. We thank the anonymous referee for reviewing the paper. KS is partially supported by the Israel Science Foundation grant number 178/13, and by the European Research Council Advanced grant number 338809. FZ is partially supported by grant number 1281 from the GIF, the German–Israeli Foundation for Scientific Research and Development, and by grant number 1825/14 from the Israel Science Foundation

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.

Citation: Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004
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. MR 2605312

[5]

M. EntovL. 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. MR 2605312

[5]

M. EntovL. 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.

Figure 1.  Illustrating $K_n\subseteq U$ and an element $\Phi (Q') = Q$ in its subdivision
Figure 2.  Producing ${\tilde{F}}$ and ${\tilde{G}}$ (the dashed curves)
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