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 0.0.1

The flux of a Lagrangian homotopy \(\li_t:L\times I\to X\) is the cohomology class defined by \[\Flux_{\li_t}(c):=\int_{\li_t(c)}\omega.\] To show that this is a cohomology class, we need to show that the flux homomorphism \(\Flux_{\li_t}\) vanishes on boundaries \(\partial b\in C_1(L_0)\) Without loss of generality, suppose that \(I=[0,1]\), and let \(b\in C_2(L)\) be a 2-chain. Then \(\li_t(b)\) similarly gives a 3-chain in \(X\) whose boundary components satisfy the relation \(\li_t(\partial b)=i_0(b)-i_1(b)+\partial(\li_t(b))\). Applying the definition of the flux homomorphism to this relation yields: \[ \Flux_{\li_t}(\partial b)=\int_{\li_t(\partial b)}\omega=\int_{i_0(b)}\omega-\int_{i_1(b)}\omega + \int_{\partial(\li_t(b))}\omega\] The first two terms vanish because \(i_0(b)\) and \(i_1(b)\) are subsets of a Lagrangian submanifold so \(\omega|_{\li_0(L)}=\omega|_{\li_1(L)}=0\). Because \(\omega\) is closed, an application of Stoke's theorem shows that the last term vanishes as well.

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.

figure 0.0.3:A basic computation of flux swept out between two submanifolds of \(\CC^*\)

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 \([\Flux_{\li_t}]=[\eta]\in \Omega(L)\). This example highlights two important properties of Lagrangian submanifolds, and Lagrangian homotopy. First, there homotopies a stronger notion of equivalence for Lagrangian submanifolds beyond homotopy by looking at those homotopies which sweep out zero flux.

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, \[[\Flux_{i_{t}}|_J]=0\in H^1(L).\] Similar to the case of Hamiltonian isotopies, exact isotopies of Lagrangian submanifolds can be characterized in terms of the Lagrangian isotopy itself. Let \(\li_t: L\times I\to X\) be a Lagrangian homotopy. This is a exact homotopy if there exists a Hamiltonian function \(H_t:L\to \RR\) such that for all \(v\in TL\), \[\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v).\] We show one direction here, which is that \(\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v)\) implies \(\Flux_{\li_t}\) vanishes in cohomology. Let \(c:S^1\to L\) be any 1-cycle in \(L\). Parameterize a 2-chain \(\li_t\circ c: S^1\times I \to X\) with coordinates \((\theta, t)\). The flux class applied to \(c\) can be explicitly computed: \begin{align*} \Flux_{\li_t}(c)=&\int_{\li_t\circ c}\omega =\int_{I \times S^1} (\li_t\circ c )^* \omega\\ =&\int_I \int_{S^1} c^*\circ (\li_t)^* \omega =\int_I \int_{S^1} (c^* \iota_{\frac{d}{dt}\li_t}\omega ) dt\\ =&\int_I \left(\int_{S^1} (c^* dH_t) \right)dt =\int_I \left(\int_{S^1} d(c^*H_t)\right) dt \end{align*} By Stoke's theorem, the integral of an exact form over the circle is zero. For the reverse direction, fix a base point \(x_0\in L\). For every point \(x\in L\), pick a path \(\gamma_x: [0,1]\to L\) with \(\gamma_x(0)=x_0\) and \(\gamma_x(1)=x\). Define the function \(H_t: L\to \RR\) by \[dH_t(x_1):=\int_{\li_t\circ \gamma} \omega.\] Because the flux of the isotopy is zero, this integral does not depend on the choices of paths \(\gamma_x\) and gives a well defined function on \(L\). We now show that this function generates the Lagrangian isotopy. The vector field \(\frac{d}{dt}\li_t\) is determined by the form \(\iota_{\frac{d}{dt}\li_t}\omega\). Since \(\iota_{\frac{d}{dt}\li_t}\) is closed, \begin{align*} \int_{\gamma_x} \iota_{\frac{d}{dt}\li_t}\omega =& \end{align*} \todo{We now check that this these two things match up by computing the vector field.} This shows that when \(L\) is embedded, a time dependent Hamiltonian \(H_t: L\times (-\epsilon, \epsilon)\to \RR\) defines an isotopy of embeddings \(\li_t: L\times(-\epsilon, \epsilon)\to X\) for small values of \(t\). We say this is the isotopy generated by the Hamiltonian \(H\). Provided that \(L\) is compact, we immediately get the following statement on the non-displacibility of Lagrangian submanifolds by small exact Lagrangian homotopies.

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*} Let \(\Crit(H)\) be the set of critical points of \(H\). From , whenever \(x\in \Crit(H)\), \(\frac{d\li}{dt}(x)=0\) and so \(\li_{t_0}(x)=\li_{t_1}(x)\). Therefore, \(|\li_{t_0}(L)\cap \li_{t_1}(L)|\subset\Crit(H)\). The proposition follows from the Morse inequalities. In fact, this can extends to the case of time-dependent Hamiltonians for sufficiently small \(t\). Let \(H_t: L\to \RR\) generate a Hamiltonian isotopy. Then there exists \(\epsilon>0\) such that for all \(t_0, t_1\in [0,\epsilon)\), \[|\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L).\] Secondly, every cohomological class is realizable as a flux class in the cotangent bundle. This observation motivates the following theorem, which shows that this behavior occurs more generally.

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 \[\phi|_L=\li.\]In other words, a symplectic neighborhood of a Lagrangian submanifold is determined by the diffeomorphism class of the Lagrangian. This local model will be useful later, as many of our constructions involving Lagrangian submanifolds will involve restricting to a Weinstein neighborhood, performing the construction there, and then implanting our construction into the original symplectic manifold. An example of such a construction is the following.

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 \[\phi(L)\cap B^*_\epsilon Q=\bigcup_{p\in L\cap Q} T^*_pQ.\] At each \(q\in L\cap Q\), consider the Lagrangian submanifold \(T^*qQ\). Take local coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\) identifying \(q\) with the origin so that \(T^*qQ\) is the linear subspace in the \(p_i\) directions. We can take a Weinstein neighborhood \(T^*_\epsilon(T^*_qQ)\) of \(T^*_qQ\), whose cotangent bundle structure is \begin{align*} T^*_\epsilon(T^*_qQ)\to T^*qQ && (q_1, \ldots, q_n, p_1, \ldots, p_n)\mapsto (p_1, \ldots, p_n). \end{align*} Since the intersection \(L\cap Q\) is transverse, the projection \(T_qL\to T_q(T^*qQ)\) is surjective. Therefore when restricted to a small enough neighborhood of \(\in U\subset T^*_qQ\) the Lagrangian \(L|_{T^*_\epsilon U}\) presents itself as a section of \(T^*_\epsilon U\to U\). Therefore, there exists a one form \(\eta\in \Omega^1(U)\) so that \(L|_{T^*U}\) is parameterized by \((p, \eta_p)\). By taking an even smaller \(U\), we may assume that \(U\) is a contractible neighborhood, and \(\eta=dH\) is an exact one-form. Pick \(\rho\) a function which vanishes in a neighborhood of \(q\in U\), and takes the value \(1\) in a neighborhood of \(\partial U\). Consider the Lagrangian section of \(T^*U\) parameterized by \(d(\rho \cdot H)\). This section is Hamiltonian isotopic to \(L|_{T^*_\epsilon U}\) relative boundary. Additionally, \(d(\rho\cdot H)\) agrees with \(U=T^*_qQ\) in a small neighborhood of \(q\). The Lagrangian submanifold \(L\setminus (L|_{T^*U})\cup (d(\rho\cdot H))\) is Hamiltonian isotopic to \(L\).

References

[Wei71]

Alan Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Advances in Mathematics, 6(3):329--346, 1971.