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

TL;DR

This paper introduces a quantitative thick-thin decomposition for pseudo-holomorphic curves with boundary, providing uniform geometric bounds on the thick part and controlling the thin part's annuli, enhancing Gromov compactness analysis.

Contribution

It establishes a new quantitative decomposition of pseudo-holomorphic curves, with explicit bounds on geometry and topology before taking limits.

Findings

Uniform bounds on the geometry of the thick part

Exponential bounds on the differential in the thick part

Linear bound on the number of thin annuli with small energy

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.