Rigidity of the L^p norm of the Poisson bracket on surfaces

Abstract

For a symplectic manifold $M$ let $\{\cdot ,\cdot \}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G)\mapsto \Vert \{F,G\}{\Vert }_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C_c^{\infty }(M)$ when $\dim M=2$ and $p<\infty$, extending previous rigidity results for $p=\infty$ in arbitrary dimension.

12 pages, 2 figures

Cited by in corpus (1)

• Rigidity versus flexibility of the Poisson bracket with respect to the $L_p$-norm