The branched deformations of the special Lagrangian submanifolds

TL;DR

This paper investigates the deformation of special Lagrangian submanifolds with branch points, constructing new immersed submanifolds under certain conditions and exploring their convergence and existence constraints.

Contribution

It introduces a method to construct branched deformations of special Lagrangian submanifolds using nondegenerate  harmonic 1-forms, addressing a question posed by Donaldson.

Findings

Constructed a family of immersed special Lagrangian submanifolds t that converge to twice the original as current.

Established constraints for the existence of nondegenerate  harmonic 1-forms on special Lagrangian submanifolds.

Connected the deformation theory with prior work by Abouzaid and Imagi.

Abstract

The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of immersed special Lagrangian submanifolds $\tilde{L}_t$, that are diffeomorphic to a branched covering of $L$ and $\tilde{L}_t$ convergence to $2L$ as current. This answers a question suggested by Donaldson. Combining with the work of Abouzaid and Imagi, we obtain constraints for the existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms.