\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

Liouville manifolds are geometrically bounded

Let \(u: \RR\times S^1\to \RR\) be a solution to the Floer equation ((Floer equation)). Let \(\rho=\exp(r\circ u)\). By applying \(d(\exp(r))\) to the Floer equation, and using (contact type almost complex structure) 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.

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References

[Sei06]Paul Seidel. A biased view of symplectic cohomology. Current developments in mathematics, 2006(1):211--254, 2006.