The annulus property of simple holomorphic discs

TL;DR

This paper proves that simple holomorphic discs have the annulus property, allowing interior points to be surrounded by small injective annuli, and demonstrates how to resolve interior singularities via perturbation of the almost complex structure.

Contribution

It introduces the annulus and half-annulus properties for simple holomorphic discs and applies these to resolve interior singularities through perturbations.

Findings

Interior points are surrounded by injective annuli

Interior singularities can be resolved by perturbing the almost complex structure

Half-annulus property is established for boundary points

Abstract

We show that any simple holomorphic disc admits the annulus property, i.e., each interior point is surrounded by an arbitrary small annulus consisting entirely of injective points. As an application we show that interior singularities of holomorphic discs can be resolved after slight perturbation of the almost complex structure. Moreover, for boundary points the analogue notion, the half-annulus property, is introduced and studied in detail.