definition 0.0.1
A Liouville domain is a pair \((X,\lambda)\), where- \(X\) is a \(2n\)-manifold with boundary \(\partial X\) and,
- \(\lambda\in \Omega^1(X, \RR)\) is a one form on \(\Omega^1(X, \RR)\) so that \(\omega=d\lambda\) is a symplectic form for \(X\).
example 0.0.2
Let \(Q\) be a manifold. The cotangent bundle \(X=T^*Q\) is an exact symplectic manifold; call the primitive \(\lambda_{can}=\sum_{i=1}^np_idq_i\). In local coordinates, the Liouville vector field is \(Z= \sum_{i=1}^n p_i \partial p_i\). The flow of this vector field acts by scalar multiplication on the fibers of \(T^*Q\), \begin{align*}\phi^t: T^*Q\to T^*Q && (q, p) \mapsto (q, tp)\end{align*} which inflates the symplectic form. Now equip \(Q\) with a metric \(g\). The metric determines a unit sphere bundle \(S^*Q\subset T^*Q\), which is transverse to \(V\). Letting \(\alpha=\lambda|_{S^*Q}\) makes \((S^*Q, \alpha)\) a contact manifold. The time \(t\)-flow on \((S^*Q, \alpha)\) corresponds to the time \(t\) geodesic flow on the base ((geodesics and symplectic cohohomology of the cotangent bundle)). It is known ([Fet52]) that every closed manifold has at least 1 closed geodesic, so all of these examples of contact manifolds have a Reeb orbit.example 0.0.3
A Stein manifold is a complex manifold \(X\) which admits a proper embedding into \(\CC^N\) with \(N\) sufficiently large; therefore \(X\) is a symplectic manifold with symplectic form \(\omega_{\CC^N}|_X\). On \(\CC^n\) we have the function \(\phi=|z|^2:\CC^n\to \RR\) which is exhausting and strictly plurisubharmonic. The symplectic form on \(\CC^n\) can be written as \(dd^c\phi\). By restricting \(\phi\) to \(X\), we obtain a primitive \(d^c\phi|_X\) for \(\omega|_X\), which makes \(X\) a Liouville manifold. In particular, every affine variety is an example of a Liouville manifold.References
[Fet52] | Abram Il'ich Fet. Variational problems on closed manifolds. Matematicheskii Sbornik, 72(2):271--316, 1952. |