Given \(H_t: X\times \RR\to \RR\) a time dependent Hamiltonian, we obtain an action on the free loop space \begin{equation} A_{H_t}(\gamma)=-\int_\gamma \lambda + \int_\gamma H_t dt \end{equation}whose critical points are the time one periodic orbits of \(V_{H_t}\). Given a \(\omega\)-compatible almost complex structure, we observe that cylinders \(u: \RR_s\times S^1_t\to X\) which satisfy the \(H_t\) perturbed Floer equation \begin{equation} \partial_s u + J(\partial_t u - V_{H_t})=0 \end{equation}parameterize the negative gradient flow lines of \(A_{H_t}\). For curves \(\gamma_+, \gamma_-\), let \(\mathcal M(\gamma_+, \gamma_-)\) denote the moduli space of solutions to Floer's equation with ends limiting to \(\gamma_+, \gamma_-\). Supposing that \(X\) is an exact symplectic manifold, and that our time-dependent Hamiltonian is chosen in a generically, the Floer cochains \(\CF(X, H_t)\) are the graded vector space generated on the time one orbits of \(H_t\). We take a slightly different convention (following Equation 5.2 of [Abo10]) and give each orbit the grading \(\deg(\gamma)=n-CZ(\gamma)\), where \(CZ(\gamma)\) is the Conley-Zehnder index. The structure coefficients of the differential are given by counts of solutions to Floer's equation. The theory becomes powerful when it satisfies the following properties:

\(\CF(X, H_t)\) is a chain complex. The key step is to show that when \(\dim(\mathcal M(\gamma_+, \gamma_-))=1\), there exists a compactification of this moduli space by including broken cylinders. A compactification of the space comes from applying Gromov compactness, while additional requirements on \(X\) are sometimes required ensure that the only configurations which appear in the compactification are broken cylinders. In our setting, the only breaking configurations which may occur are broken cylinders, as \(X\) is exact (so \(\omega(\pi_2(X))=0)\).
Given \(H_{t, 0}\) and \(H_{t, 1}\) two time-dependent Hamiltonians, there exists a continuation map \(\CF(X, H_{t, 0})\to \CF(X, H_{t, 1})\). Furthermore, this map is a homotopy equivalence.
Finally, we need some way to compute \(\CF(X, H_{t, 1})\). One way to do this is to observe that for \(C^2\) small Hamiltonians the Floer cochains agree with the Morse cochains (and only consist of constant orbits). By either using the PSS-isomorphism or by analyzing Floer trajectories, the Floer cohomology can be compared to the Morse cohomology of \(X\).

The major difference in defining the Hamiltonian Floer cohomology for Liouville domains \(X\) (as opposed to compact symplectic manifolds) comes from the proof of Gromov-compactness. The first step in the proof of Gromov-compactness is to apply Arzel\'a-Ascoli to out sequence of pseudoholomorphic maps. Because \(\hat X\) is not compact, we cannot apply the Arzel\'a-Ascoli theorem to a sequence of pseudoholomorphic cylinders \(u: S^1\times \RR \to \hat X\). We now give an example of where we can solve the issue of non-compactness.

example 0.0.1

The maximum modulus principle states that if \(\phi: D^2\to \CC\) is a holomorphic function from the disk to \(\CC\), that the maximum of \(|\phi|: D^2\to \RR_{\geq 0}\) is achieved on \(\partial D^2\). Let \(\hat X\) be a non-compact symplectic manifold with compatible almost complex structure \(J\), along with a \(J-\jmath\)-holomorphic projection \(W: \hat X\to \CC\). Suppose that the fibers of \(W\) are compact. Pick two loops \(\gamma_-, \gamma_+\subset \hat X\) and \(r_0\in \RR\) large enough so that \(U:=W^{-1}(\{z \st |z|\leq r\})\) contains \(\gamma_-, \gamma_+\). We will prove that every pseudoholomorphic cylinder \(u: S^1\times \RR\to \hat X\) with ends limiting to \(\gamma_-, \gamma_+\) has image contained within the compact subset \(U\). The composition \(W\circ u: S^1\times \RR\to \CC\) is a holomorphic map, with ends limiting to \(W(\gamma_\pm)\), and therefore satisfies the maximum modulus principle. Since the boundary is sent to \(W(\gamma_\pm)\), we obtain that \(|W|\) achieves a value no greater than \(r_0\) on \(u\); therefore \(\Im(u)\subset U\). It follows that the image of \(u\) is contained within a compact set. In order to extend example 0.0.1 to the setting of \(\hat X\), we will use the maximum principle. First, we will need to assume that we have chosen our almost complex structure for \(\hat X\) so that the sub-bundle spanned by the vector fields \(\partial_r, R\) form an almost complex subspace.

definition 0.0.2

Let \(\hat X\) be the completion of a Liouville domain. A choice of almost complex structure for \(\hat X\) is of contact type if \[d(\exp(r))\circ J = -\alpha.\] We will also need to assume that we have chosen our Hamiltonian so that over the symplectization it only depends on the \(r\)-coordinate. For such a contact type almost complex structure and Hamiltonian there exists a version of example 0.0.1.

proposition 0.0.3

Let \(H: \hat X\to \RR\) be a Hamiltonian which on the symplectization takes the form of \(h(\exp(r))\). Let \(\gamma_+, \gamma_-\) be time 1 orbits of \(V_{H_{t}}\). For a contact type almost complex structure, every solution \(u: \RR\times S^1\to \hat X\) of the Floer equation with ends limiting to \(\gamma_+, \gamma_-\) has image contained in the subset \(\hat X|_{\exp(r)\leq C}\), where \(C\) is the maximum value of \(\exp(r)\) on the orbits \(\gamma_+, \gamma_-\). Let \(u: \RR\times S^1\to \RR\) be a solution to the Floer equation (). Let \(\rho=\exp(r\circ u)\). By applying \(d(\exp(r))\) to the Floer equation, and using definition 0.0.2 we obtain : \begin{align*} 0=d(\exp(r))\circ \left(\partial_s u + J(\partial_t u - V_{H})\right)=& \partial_s(\rho) - \alpha(\partial_t u)+ \alpha(V_H) \end{align*} Because \(H= h(\rho)\), the Hamiltonian vector field associated to \(H\) is \(h'(\rho) V_\alpha\), where \(V_\alpha\) is the Reeb flow. From (orbits of an increasing Hamiltonian), we see that \(\alpha(V_H)=h'(\rho).\) \begin{align*} =& \partial_s(\rho) - \alpha(\partial_t u)+ h'(\rho) . \end{align*} Similarly, applying \(\alpha\) to the Floer equation: \begin{align*} 0=\alpha \left(\partial_s u + J(\partial_t u - V_{H})\right) =& \alpha(\partial_su)+ \partial_t(\exp(\rho))+ V_{H}(\exp(\rho)) \end{align*}Because \(H= h(\rho)\) has the same level sets as \(\rho\), \(V_H(\rho)=0\).\begin{align*} =& \alpha(\partial_s u) + \partial_t(\rho) \end{align*} Differentiating the first line with respect to \(s\), the second line with respect to \(t\), and summing the lines together we obtain \begin{align*} 0=& (\partial_s^2 + \partial_t^2)\circ \rho- \partial_s\alpha(\partial_t u)+\partial_s \rho h'(\rho) +\partial_t\alpha(\partial_s u)\\ \end{align*}As \([\partial_s, \partial_t]=0\), we can substitue \(-\partial_t\alpha(\partial_s u)+ \partial_s\alpha (\partial_t u)= -u^*\omega(\partial_s, \partial_t)\)\begin{align*} =& \Delta \rho- u^*\omega(\partial_s, \partial_t)+ \rho h'(\rho)\partial_s\rho + \rho h''(\rho) \partial_s\rho\\ \end{align*}By again applying Floer's equation, and using the compatibility of almost complex structure with \(J\), we may substitue \(u^*\omega(\partial_s, \partial_t)= u^*\omega(\partial_s, J\partial_t-X_H)=|\partial_s u^2|-dh'(\rho)\partial_s\)\begin{align*} =& \Delta \rho-|\partial_s|^2+\rho h''(\rho)\partial_s\rho \end{align*} We therefore obtain that \(\Delta\rho+\rho\cdot h''(\rho) \partial_s\rho\geq 0\). Observe now that where \(z\in S^1\times \RR\) is a proposed maximum for \(\rho\) that \(\partial_s\rho=0\), allowing us to write \((\partial^2_s \rho + \partial^2_t \rho)|_z \geq 0\). This implies that at least one of \(\partial^2_s, \partial^2_s\) has to be non-negative --- in particular, the second derivative test does not detect the maximum. A more general argument --- the maximum principle --- states that \(\rho\) achieves no local maxima; therefore \(\sup_{S^1\times \RR} \rho \leq \max_{t\in S^1} \exp(r\circ \gamma_\pm)=:C.\) It follows that the image of \(u\) is contained in \(\hat X|_{\rho< C}\), which is a compact set.

References

[Abo10]	Mohammed Abouzaid. A geometric criterion for generating the Fukaya category. Publications Mathématiques de l'IHÉS, 112:191--240, 2010.
[Sei06]	Paul Seidel. A biased view of symplectic cohomology. Current developments in mathematics, 2006(1):211--254, 2006.