Deformations of Compact Coassociative 4-folds with Boundary
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.
Cite
@article{arxiv.0712.4325,
title = {Deformations of Compact Coassociative 4-folds with Boundary},
author = {Alexei Kovalev and Jason D. Lotay},
journal= {arXiv preprint arXiv:0712.4325},
year = {2009}
}
Comments
22 pages. v2: largely rewritten, correcting an error in the previous version, examples and references added