Remarks on $G_{2}$-manifolds with boundary

This article is based on a lecture at the Journal of Differential Geometry Conference, Harvard 2017. We discuss closed and torsion-free $G_{2}$-structures on a 7-manifold with boundary, with prescribed $3$-form on the boundary. Much of the article is based on an observation that there is an intrinsic notion of "mean convexity" for such boundary data. When the boundary data is mean convex, classical arguments from Riemannian geometry can be applied. Another theme is a connection with the maximal submanifold equation, in spaces of indefinite signature.