Gromov compactness theorem for stable curves

TL;DR

This paper proves the Gromov compactness theorem for stable curves with variable complex structures and boundary conditions, using stable maps and providing new estimates for degenerations and bubbling phenomena.

Contribution

It extends Gromov compactness to more general settings with continuous almost complex structures and boundary conditions, introducing new estimates and moduli space structures.

Findings

Established a uniform description of degeneration and bubbling phenomena.

Proved compactness for curves with boundary on totally real submanifolds.

Provided a Hausdorff convergence result for the moduli space.

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.