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 $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C^\infty_c(M)$ when $\dim M = 2$ and $p < \infty$, extending previous rigidity results for $p = \infty$ in arbitrary dimension.