Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces

Karina Samvelyan; Frol Zapolsky

doi:10.3934/era.2017.24.004

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

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Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces

Karina Samvelyan^1, and
Frol Zapolsky^2,

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 28, 2016

Published: August 2017

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.

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Abstract

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.

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Mathematics Subject Classification: 53D99(Primary), 28A99(Secondary).

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Figure 1. Illustrating $K_n\subseteq U$ and an element $\Phi (Q') = Q$ in its subdivision

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Figure 2. Producing ${\tilde{F}}$ and ${\tilde{G}}$ (the dashed curves)

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References

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