Gromov compactness theorem for stable curves

Abstract

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', in particular they can be non-closed and the complex structures on them can vary arbitrarily. In connection with this, we study in \S 2 moduli spaces of nodal curves with boundary and define a natural complex structure for such moduli spaces. We obtain an apriori estimate for pseudoholomorphic maps of ``long cylinders'' (see \S 3), which gives a uniform description for degeneration of complex structure on the curves and for the ``bubbling'' phenomenon. It also implies the Hausdorff convergence of the curves. We also prove in \S 5 the compactness theorem for curves with boundary on totally real submanifolds. For this ``boundary'' case we give appropriate generalizations of all ``inner'' constructions and estimates.