A thick-thin decomposition of $J$-holomorphic curves

Abstract: We show the existence of a thick thin decomposition of the domain of a pseudo holomorphic curve with boundary. The geometry of the thick part is bounded uniformly in the energy. Furthermore, in the thick part, there is a uniform bound on the differential which is exponential in the energy. The thin part consists of annuli of small energy the number of which is at most linear in the energy and genus. The decomposition can be seen as a quantitative version of Gromov compactness which applies before passing to the limit.