An elliptic boundary value problem for $G_{2}$-structures

Abstract

We show that the G 2 holonomy equation on a seven-dimensional manifold with boundary, with prescribed 3-form on the boundary and modulo the action of diffeomorphisms, is elliptic. The main point is to set up a suitable linear elliptic theory. This result leads to a deformation theory, governed by a finite-dimensional obstruction space. We discuss conditions under which this obstruction space vanishes and as one application we establish the existence of certain G 2 cobordisms between two small deformations of a Calabi–Yau 3-fold.