Compactness for embedded pseudoholomorphic curves in 3-manifolds
Let \(M\) be a closed oriented \(3\)-manifold. A stable Hamiltonian structure on \(M\) is given by \(\mathcal{H}=(\xi,X,\omega,J)\) where {\parindent4mm \begin{itemize}\item[--] \(\xi\) is a smooth cooriented \(2\)-plane on \(M;\) \item[--] \(\omega\) is a smooth closed \(2\)-form on \(M\), which restricts to a symplectic structure on the vector bundle \(\xi\rightarrow M\); \item[--] \(X\) is a smooth vector field which is transverse to \(\xi\), satisfies \(\omega(X,\cdot)\equiv0\), and the flow of \(X\) preserves \(\xi\); \item[--] \(J\) is a smooth complex structure on the bundle \(\xi\rightarrow M\) compatible with \(\omega\) in the sense that \(\omega(\cdot,J\cdot)\) defines a bundle metric. \end{itemize}} An example of a stable Hamiltonian structure is obtained on a contact \(3\)-manifold \(M\); let \(\lambda\) be the contact form, then \(\text{d} \lambda\) is a symplectic structure on \(\xi=\ker\lambda\); if \(X_{\lambda}\) is the corresponding Reeb field and \(J\) any complex structure on \(\xi\) compatible with \(\text{d}\lambda\), then \((\xi,X_{\lambda},\text{d} \lambda,J)\) is a stable Hamiltonian structure. The symplectization of \(M\) is the \(4\)-manifold \(\mathbb{R}\times M\) with the almost complex structure \(\tilde{J}\) defined by \(\tilde{J}v=Jv\) if \(v\in\xi\), \(\tilde{J}\partial_{a}=X\), where \(\partial_{a}\) is the derivation in the unit direction of \(\mathbb{R}.\) A punctured pseudoholomorphic curve in \(\mathbb{R}\times M\) is \(\tilde{u}=(a,u):(\dot{\Sigma},j)\rightarrow(\mathbb{R}\times M,\tilde{J})\), where \(\dot{\Sigma}=\Sigma\setminus\Gamma\), \((\Sigma,j)\) is a closed Riemann surface, \(\Gamma\subset\Sigma\) is a finite set of ``punctures'', and \(\tilde{u}\) satisfies \(T\tilde{u}\circ j=\tilde{J}\circ T\tilde{u}\). The energy of the punctured holomorphic curve \(\tilde{u}=(a,u)\) is \[ E(\tilde{u})=E_{\omega}(\tilde{u})+E_{\lambda}(\tilde{u})=\int_{\dot{\Sigma} }u^{\ast}\omega+ \sup_{\phi\in\mathcal{T}}\int_{\dot{\Sigma}}u^{\ast }(\text{d}\phi\wedge\lambda), \] where \(\mathcal{T}=\{\phi\in C^{\infty}(\mathbb{R}, [0,1])\mid\phi^{\prime} \geq 0\}\). A finite energy surface is a punctured holomorphic curve \(\tilde{u}\) which is proper, connected and such that \(E(\tilde{u})<\infty\). A finite energy surface \(\tilde{u}=(a,u):\dot{\Sigma},j\rightarrow\mathbb{R}\times M\) is called nicely embedded if the map \(u:\dot{\Sigma}\rightarrow M\) is embedded. The author proves a compactness theorem for some families of nicely embedded finite energy surfaces, which strengthens the cylindrical case of the SFT compactness theorem of \textit{F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki} and \textit{E. Zehnder} [Geom. Topol. 7, 799--888 (2003; Zbl 1131.53312)].
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