Deformations of Compact Coassociative 4-folds with Boundary

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Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G₂ structure given by a `positive' differential 3-form, sometimes called G₂-form. Assuming that a G₂-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 G₂-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.