Gromov compactness theorem for \(J\)-complex curves with boundary (Q2708366)

The purpose of this paper is to give the proof of two related results. The first result is the Gromov compactness theorem for \(J\)-complex curves with boundary (but without boundary conditions on maps), and the second result is an improvement of the removal of a point singularity theorem. Let a sequence \(J_\mu\) of continuous almost-complex structures on a Riemannian manifold \((X, h)\) be given. Let \({C_{n}}\) be a sequence of complex nodal curves of fixed topological type. This means that all \(C_{n}\) can be parameterized by the same real surface \(\Sigma.\) Denote by \(\delta_n:\Sigma\rightarrow C_n\) some parameterizations. Let some sequence of \(J_{n}\)-holomorphic maps \(u_n : C_{n}\rightarrow X\) be given such that \((C_{n}, u_{n})\) are stable over \(X\) and suppose that, for some \(h\)-complete set \(A\subset X,\) NEWLINE\[NEWLINE\sup_{x\in A}{\|J_{n}(x) - J_{\infty}(x) \|}\rightarrow 0,NEWLINE\]NEWLINE where \(J_{\infty}\) is a continuous almost-complex structure on \(X,\) and \(u_{n}(C_{n})\subset A\) for all \(n.\) Say that the complex structures on \(C_{n}\) do not degenerate near the boundary if there exists \(R > 1\) such that, for every \(n\) and every boundary circle \(\gamma_{n,i}\) of \(C_{n},\) there exists an annulus \(A_{n} \subset C_{n}\) adjacent to \(\gamma_{n,i}\) such that all annuli \(A_{n,i}\) are mutually disjoint, do not contain nodal points of \(C_{n},\) and have conformal radii equal to \(R;\) that is, they are conformally isomorphic to \(A(1,R):=\{z\in\mathbb{C}:1 < |z |< R\}\).NEWLINENEWLINENEWLINETheorem 1. Suppose that there is a constant \(M\) such that \(\text{area}[u_{n}(C_{n})]\leq M \) for all \(n\) and that complex structures on \(C_{n}\) do not degenerate near the boundary. Then there is a subsequence \((C_{n_{k}}, U_{n_{k}})\) and parameterizations \(\sigma_{n_{k}}:\Sigma\rightarrow C_{n_{k}}\) such that \((C_{n_{k}}, U_{n_{k}}, \sigma_{n_{k}})\) converge to a stable \(J_{\infty}\)-complex curve \((C_{\infty}, u_{\infty}, \sigma_{\infty})\) over \(X\) parameterized by \(\Sigma\). Moreover, if the structures \(\delta^{*}_{n} j c_{n}\) are constant on the fixed annuli \(A_{i},\) each adjacent to a boundary circle \(\gamma_{i}\) of \(\Sigma,\) then the new parameterizations \(\sigma_{n_{k}}\) can be taken equal to \(\delta_{n_{k}}\) on some subannuli \(A^{'}_{i} \subset A_{i},\) also adjacent to \(\gamma_{i}.\) NEWLINENEWLINENEWLINELet \((X, J)\) be a compact almost-complex manifold, where again the tensor of almost-complex structure \(J\) is of class \(C^{0},\) that is, continuous only, and suppose that some Riemannian metric \(h\) on \(X\) is fixed. Denote by \(\check{\Delta}\) the punctured unit disk \(\{z\in \mathbb{C}:0 < |z |< 1\}\).NEWLINENEWLINENEWLINETheorem 2. There exists an \(\varepsilon = \varepsilon(X, J, h) > 0\) such that if a \(J\)-holomorphic mapping \( u:(\check{\Delta}, J_{st})\rightarrow (X,J)\) satisfies the condition \(\text{area}(u(R_{k}))\leq \varepsilon\) of ``slowgrowth'', for all annuli \(R_{k}=\{z\in \mathbb{C}: \frac{1}{e^{k+1}} \leq |z |\leq \frac{1}{e^{k}}\}\) for \(k\gg 1,\) then \(u\) extends to the origin.