Pseudoholomorphic quilts with figure eight singularity

TL;DR

This paper demonstrates that a specific type of singularity in pseudoholomorphic quilts, called the figure eight, can be smoothly removed under certain conditions, using advanced elliptic estimates and complex structures.

Contribution

It introduces a method to remove the figure eight singularity in pseudoholomorphic quilts when the Lagrangian correspondences are cleanly immersed, expanding understanding of quilt regularity.

Findings

Figure eight singularity can be removed under certain conditions.

Width-independent elliptic estimates are developed for non-standard complex structures.

The result applies to compositions of Lagrangian correspondences that are cleanly immersed.

Abstract

We show that the novel figure eight singularity in a pseudoholomorphic quilt can be continuously removed when composition of Lagrangian correspondences is cleanly immersed. The proof of this result requires a collection of width-independent elliptic estimates that allow for non-standard complex structures on the domain.