definition 0.0.1
The flux of a Lagrangian homotopy \(\li_t:L\times I\to X\) is the cohomology class defined by \[\Flux_{\li_t}(c):=\int_{\li_t(c)}\omega.\]example 0.0.2
Let \(\CC^*\) be equipped with the symplectic form \(\omega=\frac{1}{2\pi} d(\log r) \wedge d\theta\). Let \(L_r\) be the Lagrangian \[L_r:=\{z \text{ such that } \log|z|=r\}.\] Let \(\li_r:L\times [r_0, r_1]\to X\) be the isotopy between \(L_{r_0}\) and \(L_{r_1}\). Let \(e\in H_1(L_r)\) be the fundamental class of \(L\). The amount of flux swept out between these two Lagrangian submanifolds is given by the difference of their \(r\)-values. \[\Flux_{i_r}(e)=\int_{S^1} \int_{r_0}^{r^1} \frac{1}{2\pi} d(\log |z|)\wedge d\theta = r_1-r_0.\] For this reason, the value \(r\) is sometimes called the ``flux coordinate'' of the fibration \(\CC^*\to \RR\). The flux class therefore provides a nice parameterization of the space of Lagrangian submanifolds in this example.example 0.0.4
Consider the symplectic manifold \(X=T^*L\). Consider a closed one form \(\eta \in \Omega^1(L, \RR)\). Use this to create a Lagrangian isotopy \begin{align*} \li_t: L\times [0, 1]\to& T^*L && (q,t) \mapsto& t\cdot \eta_p \end{align*} between the zero section and the graph of \(\eta\). To each loop \(\gamma\in L\), look at the cylinder \(\li_t\circ \gamma:S^1\times I\to X\). This can be explicitly parameterized by \begin{align*} c: S^1\times I \to& T^*L&& (\theta, t) \mapsto& (\gamma(\theta), t\cdot \eta_{\gamma(\theta)}). \end{align*} The cotangent bundle has an exact symplectic form, \(\omega=d\lambda\), so computing the flux can be simplified by using Stoke's theorem: \begin{align*} \Flux_{\li_t}(\gamma)=&\int_{c}\omega=\int_{c}d\lambda \\ =&\int_{S^1\times \{1\}}c^*\lambda - \int_{S^1\times\{0\}}c^*\lambda \\ =&\int_{S^1}\gamma^*\eta=\eta(\gamma) \end{align*} So \([\Flux_{\li_t}]=[\eta]\in \Omega(L)\).definition 0.0.5
We call \(i_t: L\times I \to X\) an exact homotopy if for every subinterval \(J\subset I\), the flux of the restriction is exact, \[[\Flux_{i_{t}}|_J]=0\in H^1(L).\]proposition 0.0.6
Let \(H: L\to \RR\) be a non-time dependent Hamiltonian . Let \(\li_t: L\times[0, c]\to X\) be the isotopy generated by \(H\). There exists \(\epsilon>0\) such that for all \(t_0, t_1\in [0, \epsilon]\), \begin{align*} |\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L). \end{align*}theorem 0.0.7 [Wei71]
Let \(\li: L\to X\) be a compact Lagrangian submanifold of \((X, \omega)\). Then there exists a neighborhood of the zero section \(T^*_\epsilon L\subset TL\) and a symplectic embedding \(\phi: T^*_\epsilon L\to X\) which agrees on the zero section, in the sense that \[\phi|_L=\li.\]lemma 0.0.8
Let \(L\subset T^*Q\) be a Lagrangian submanifold. Suppose that the intersections between \(L\) and \(Q\) are transverse. Then there exists a Hamiltonian isotopy \(\phi\) of \(L\) so that \[\phi(L)\cap B^*_\epsilon Q=\bigcup_{p\in L\cap Q} T^*_pQ.\]References
[Wei71] | Alan Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Advances in Mathematics, 6(3):329--346, 1971. |