Surveys in Differential Geometry
Volume 22 (2017)
Remarks on $G_2$-manifolds with boundary
Pages: 103 – 124
This article is based on the author’s 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 in the article is a connection with the maximal submanifold equation, in spaces of indefinite signature.
Published 13 September 2018