Let $M$ be a surface with boundary and let $f\_t: M \to M, t \in [0,1]$ be a differentiable family of diffeomorphisms (I think this is usually called a diffeotopy). Suppose I have a Liouville form $\lambda\_0 \in \Omega\_1(M)$ such that $f\_0$ is exact, i.e. $f\_0^\*\lambda\_0 - \lambda\_0 = dF$ for some function $F: M \to \mathbb{R}$. Is it always possible to find a differentiable family of Liouville forms $\lambda\_t$ such that $f\_t$ is exact for all $t \in [0,1]$?



Deformation of a Liouville form with a diffeotopy
The statement is not true. For a function to be exact, it needs to preserve the symplectic form $d\lambda$. This puts strong conditions on the function, for instance it's differential should have determinant $1$ at fixed points. This is clearly not preserved in a diffeotopy.


