Ghost bubble censorship

TL;DR

This paper extends the understanding of degeneracy phenomena in holomorphic curves to approximately J-holomorphic maps, showing that certain degenerate behaviors occur at attaching points in the Gromov limit.

Contribution

It generalizes known results about holomorphic curves to the setting of approximately J-holomorphic maps, revealing similar degeneracy behaviors.

Findings

Degenerate behavior occurs at attaching points in Gromov limits of approximately J-holomorphic maps.

High multiplicity or vanishing derivatives characterize these degenerate behaviors.

Results apply to non-collapsed components in the limit of sequences of maps.

Abstract

When a Gromov limit of embedded holomorphic curves is constant on some component of the domain, the non-collapsed component must exhibit some degenerate behavior at the attaching points, such as high multiplicity or vanishing of the holomorphic derivative. Here we show the same holds for maps which are only approximately J-holomorphic.