Definition (Liouville Vector Field): Let \((W, \lambda)\) be exact symplectic. The Liouville vector field on \((W, \omega = d\lambda)\) is the (unique) vector field \(X\) such that \(\iota_x \omega = \lambda\).
Remark: \(X\) induces a flow \(\psi^{X, t}\), and for any compact embedded surface \(\Sigma_g \hookrightarrow W\) we have \begin{align*} \psi^{X, t *} d \lambda&= e^{t} d \lambda \implies \\ \operatorname{Area}_{d \lambda}\left(\psi^{X, t}(S)\right) &:=\int_{\psi^{X, t}(S)} d \lambda\\ &=\int_{S} (\psi^{X, t})^* d \lambda\\ &=e^{t} \operatorname{Area}_{d \lambda}(S) \end{align*}
This says that the flow lines of \(X\) “dilate” the areas of surfaces at an exponential rate, or that \(X\) is an “infinitesimal generator” of a canonical dilation..
Remark: This is useful because even if \(W\) isn’t compact, we can obtain \(W\) as the “limit” of compact submanifolds where we inflate along this flow.
Theorem: A Liouville vector field \(X\) satisfies \(\mathcal{L}_X \omega = \omega\), where \(\mathcal{L}_X\) is the Lie Derivative.
Proof: \begin{align*} \mathcal{L}_X \omega = [d, \iota_X] \omega = \iota_X(d\omega) + d(\iota_x \omega) = \iota_x(d\omega) + d\lambda = \iota_X(0) + d\lambda = d\lambda = \omega .\end{align*}
Use the fact that \(\omega\) is closed, so \(d\omega = 0\).
Definition (Liouville Domain): \((W, \lambda)\) is a Liouville domain iff \(W\) is a compact exact symplectic manifold with boundary such that the Liouville vector field \(X\) points outwards on \({{\partial}}W\) transversally.
Remark: This condition implies that \({{\partial}}W\) is a contact manifold with contact form $\alpha = {\left.{{\lambda}} \right|_{{{{\partial}}W}} } $.