Holomorphic curves at one point

TL;DR

This paper establishes bounds on the pre-images of a point for J-holomorphic curves in symplectic manifolds and uses these bounds to compactify the moduli space by adding holomorphic buildings at that point.

Contribution

It provides uniform bounds on pre-images and energy for J-holomorphic curves, enabling a new compactification of the moduli space with holomorphic buildings.

Findings

Bound on the number of pre-images of a point for J-holomorphic curves.

Uniform energy bounds for curves in the complement of a point.

Compactification of the moduli space via holomorphic buildings.

Abstract

Let M be a closed symplectic manifold with a compatible almost complex structure J. We prove that for a point p in M and E>0, if v is a non-constant J-holomorphic curve with symplectic area smaller than E, then the number of the pre-images of p is bounded, and the bound is independent of v. We also provide a uniform Hofer's energy bound for J-holomorphic curves in M\p based on the symplectic area. Using these two results we compactify the moduli space of J-holomorphic curves in M by adding holomorphic buildings at the point p.