Target-local Gromov compactness (Q538531)

In the frame of the study of compactness of families of pseudoholomorphic curves, the author develops a ``sideways stretching'' operation in symplectic field theory. In this paper, he defines the robust \(\mathcal K\)-convergence in the sense of Gromov and proves the following important result: If \((M,J,g)\) is an almost Hermitian manifold and \((J_{k}, g_{k})\) is a sequence of almost Hermitian structures which converges in \(C^{\infty}\) to \((J,g)\), if \( \mathcal K \subset \) Int(\(M\)) is a compact region and \(u_{k}\) is a sequence of generally immersed \(J_{k}\)-curves which are robustly \( \mathcal K\)-proper, satisfying the assumptions (1) Area\(_{u^{\ast}_{k}g_{k}}(S_{k})\leq C_{A}< \infty\) and (2) Genus (\(S_{k}\)) \( \leq C_{G} < \infty\), then a subsequence robustly \(\mathcal K\)-converges in the Gromov sense. The proof is made in three steps: first by supposing that the curves are immersed and \(||B_{u_{k}}||_{L^{\infty}}\) is uniformly bounded, next by supposing that \(||B_{u_{k}}||_{L^{2}}\) is uniformly bounded and the numbers of critical points of \(u_{k}\) are uniformly bounded and, finally, in general. Several interesting examples concerning this subject are also presented.