Deformations of compact coassociative 4-folds with boundary

Let MM be a 7-manifold with a G2-structure induced by a closed ‘positive’ differential 3-form. We study deformations of a compact coassociative 4-submanifold N⊂MN⊂M with non-empty boundary ∂N∂N contained in a fixed, codimension 1 submanifold SS of MM with a compatible Hermitian symplectic structure. We show that ‘small’ coassociative deformations of NN with special Lagrangian boundary in SS form a smooth moduli space of finite dimension not greater than the Betti number b1(∂N)b1(∂N). It is also shown that NN is ‘stable’ under small deformations of the closed G2 3-form on the ambient 7-manifold MM. The results can be compared to those for minimal Lagrangian submanifolds of Calabi–Yau manifolds proved in [A. Butscher, Deformations of minimal Lagrangian submanifolds with boundary, Proc. Amer. Math. Soc. 131 (2002), 1953–1964]. Some examples are also discussed.