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

TL;DR

This paper proves that the L^p norm of the Poisson bracket on surfaces exhibits lower-semicontinuity with respect to the C^0 norm, extending known rigidity results from the case p = infinity to finite p.

Contribution

It extends the rigidity of the Poisson bracket's L^p norm from the case p = infinity to finite p on 2-dimensional symplectic manifolds.

Findings

L^p norm of the Poisson bracket is lower-semicontinuous on surfaces

Extension of rigidity results from p = infinity to finite p

Applicable to symplectic surfaces with dim M = 2

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.