Deformations of Compact Coassociative 4-folds with Boundary

TL;DR

This paper investigates the deformation theory of compact coassociative 4-folds with boundary in G_2 manifolds, demonstrating unobstructed deformations and stability under G_2-form variations, with results analogous to special Lagrangian submanifolds.

Contribution

It establishes that small deformations of coassociative 4-folds with boundary are unobstructed and form a smooth moduli space, also showing stability under G_2-form deformations.

Findings

Deformations form a finite-dimensional smooth moduli space.

Deformations are unobstructed for small changes.

N is stable under small G_2-form deformations.

Abstract

Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N with boundary contained in fixed, codimension 1 submanifold S of M with a compatible Hermitian symplectic structure. We show that `small' coassociative deformations of N with special Lagrangian boundary in S are unobstructed and form a smooth moduli space of finite dimension not greater than the first Betti number of the boundary of N. It is also shown that N is `stable' under small deformations of the closed G_2-form on the ambient 7-manifold M. The results can be compared to those for special Lagrangian submanifolds of Calabi--Yau manifolds proved by A.Butscher in math.DG/0110052.