definition 0.0.1
Let \((X, \omega)\) be a symplectic manifold. A Lagrangian submanifold is a \(n\)-dimensional submanifold \(L\subset X\) such that \[\omega|_L=0.\]1: examples of Lagrangian submanifolds
For symplectic surfaces, we have a complete classification of the Lagrangian submanifolds.example 1.0.1
If \(n=1\), every curve is Lagrangian.example 1.0.2
The product torus is the Lagrangian torus in \(\CC^n\) given by \[T^n=\{|z|\in \CC^n\;\text{ such that }\; |z_i|=1\}.\] To prove that this is a Lagrangian submanifold we first observe that whenever \(L_1\subset (X_1, \omega_1)\) and \(L_2\subset (X_2, \omega_2)\) are Lagrangian submanifolds, then \(L_1\times L_2\subset (X_1\times X_2, \omega_1+\omega_2)\) is a Lagrangian submanifold as well. The product torus comes from taking the product of circles \(S^1\subset \CC\), which are Lagrangian submanifolds from example 1.0.1.example 1.0.3
Let \(Q\) be a smooth manifold, let \(\alpha\) be a \(1\)-form on \(X\). Let \(L\) be the graph of \(\alpha\) in \(T^*Q\), \[\{L:=(q, \alpha(q))\}\subset T^*Q.\] \(L\) is a Lagrangian submanifold if and only if \(d\alpha=0\).example 1.0.4
Let \(V\subset Q\) be a smooth submanifold. The conormal bundle \(N*V\subset T^*Q\) consists of all covectors \((q, p)\in T^*Q\) with \(q\in V\) and \(p(v)=0\) for all \(v\in TV\). This is always an \(n\)-dimensional submanifold of \(T^*Q\). We can choose local coordinates \(q_1, \ldots, q_n, p_1, \ldots p_k\) so that \(V=\{(0, \ldots, 0, q_{k+1}, \ldots, q_n)\;|\; q_i\in \RR\}\). In these local coordinates, \(N^*V=\{(0, \ldots, q_{k+1}, \ldots, q_n, p_1, \ldots, p_k, 0, \ldots, 0)\}\); which is a Lagrangian linear subspace for the symplectic form \(\sum_{i=1}^n dq_i \wedge dp_i\). The intersection of the zero section and a conormal bundle is \(Q\cap N^*V=V.\)2: relations between Lagrangian submanifolds
In these notes, we will attempt to shed a little light on the construction and classification of Lagrangian submanifolds. Even in the simplest examples, this classification can become quite difficult. To give a meaningful answer to this question, we'll need a notion of equivalence between Lagrangian submanifolds. For submanifolds, a natural equivalence to consider is the isotopy class or the homotopy class. Let \(\li_t: L\times I\to X\) be a homotopy of Lagrangian submanifolds. We now describe the flux class of the homotopy, which is an element \(\Flux_{\li_t}\in H^1(L;\RR)\). To define this class we prescribe its values on chains of \(L\). Let \(c\in C_1(L)\) be a chain. Then the homotopy can be applied to \(c\) to give a 2-chain \(\li_t(c)\in C_2(X)\).definition 2.0.1
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 2.0.2
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 2.0.4
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 2.0.5
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 2.0.6
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 2.0.7 [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 2.0.8
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 \[\phi(L)\cap B^*_\epsilon Q=\bigcup_{p\in L\cap Q} T^*_pQ.\]3: exercises on Lagrangian submanifolds
exercise 3.0.1
Prove example 1.0.3 without using coordinates. Try showing that \(\omega|_L=\pm d\alpha\).exercise 3.0.2
Show that there are no Lagrangian spheres inside of \(\CP^2\).exercise 3.0.3
Prove the converse of , showing that every Hamiltonian Lagrangian isotopy is generated by the flow of a function \(H_t: L\times \RR\to \RR\).exercise 3.0.4
Let \(M\) be an oriented manifold, and \(H\subset M\) be an oriented hypersurface. Let \(X=T^*M\) be the cotangent bundle equipped with the canonical symplectic form.- Show that the conormal bundle \(N^*H\subset T^*M\) is a Lagrangian submanifold.
- Let \(B_\epsilon(H)\subset X\) be a small neighborhood of \(H\subset M\subset T^*M\) . Find an embedded Lagrangian submanifold \(L\subset T^*M\) which, outside of a small neighborhood \(B_\epsilon(H)\subset X\), agrees with \(N^*H\cup M\), \[ L\setminus (B_\epsilon(H))=(N^*H\cup M)\setminus H.\]
exercise 3.0.5
Let \(\Sigma\) be a symplectic manifold with \(\dim_\RR(\Sigma)=2\). Let \(L_0, L_1\subset \Sigma\) be two Lagrangian submanifolds which intersect transversely at points \(p_i\). Show that there exists a Lagrangian submanifold \(L'\subset X\) which agrees with \(L_0, L_1\) outside a small neighborhood of the \(p_i\): \[L'\setminus \left(\bigcup_{i} B_\epsilon(p_i)\right)=(L_0\cup L_1) \setminus \left(\bigcup_{i} B_\epsilon(p_i)\right).\]exercise 3.0.6
Suppose that \(\alpha\) is a closed one form. Find a Hamiltonian isotopy of \(T^*N\) for which sends the zero section to \(\alpha\).exercise 3.0.7
Let \(H_t: L\times [0, c]\to \RR\) be a time-dependent Hamiltonian. Suppose for each \(t_0\) the function \(H_{t_0}: L\to \RR\) is Morse. Let \(\li_t: L\to X\) be the induced isotopy. Show that for each critical point \(x\in \Crit(H_0)\), there is a sequence of points and times \(x_i\in L\) , \(t_i\in [0, c]\) so that \(\lim_{i\to\infty} t_i=0\), \(\li_{t_i}(x_i)\subset\li_{t_i}(L)\cap \li_{0}(L)\) and \(\lim_{i\to\infty}\li_{t_i}(x_i)=x\).exercise 3.0.8
Which circles on \(S^2\) can be displaced by Hamiltonian isotopy?References
[Wei71] | Alan Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Advances in Mathematics, 6(3):329--346, 1971. |