Surveys in Differential Geometry

Volume 22 (2017)

Remarks on $G_2$-manifolds with boundary

Pages: 103 – 124

DOI: https://dx.doi.org/10.4310/SDG.2017.v22.n1.a4

Author

Simon Donaldson (Department of Mathematics, Imperial College, London, United Kingdom; and Simons Center for Geometry and Physics, Stony Brook, New York, U.S.A.)

Abstract

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.

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Published 13 September 2018