The Compactness Theory Of Pseudo Holomorphic Curves

In1985M.Gromov invented the beautiful theory of pseudo holomorphic curves and made it a powerful tool in symplectic geometry. On a symplectic manifold he considered almost complex structures which are "compatible" with or more generally,"tamed" by the symplectic structure and studied the space of "pseudo" holomorphic curves defined in terms of one such almost complex structure. The "moduli space" of pseudo holomorphic curves then contains invariants of the symplectic structure. The first question one asks about a moduli space is whether it is compact. If it is not compact, then one would like to understand its compactification. The total moduli space of pseudo holomorphic curves is too big, and one need to restrict to fixed topological types and impose an area bound. But even under these restrictions, in general one still cannot expect compactness due to three reasons. First, holomorphic spheres may split a sequence of holomorphic curves. Second, the underlying conformal structures of a sequence of holomorphic curves may degenerate. Third, buddling-off of holomorphic disks may also occur if one is dealing with holomorphic curves with boundary. To deal with these degenerations, Gromov introduced the notion of "cusp-curves". Gromov’s compactness theorem says that a sequence of holomorphic curves of a fixed topological type and uniformly bounded area converges to a cusp-curve. We give a complete proof for Gromov’s compactness theorem for pseudo holomorphic curves.