Integral Gauss formula and the Poisson equation for the G_2-Laplacian

The study of special holonomy involves a nonlinear second-order operator on differential forms called the \(G_2\)-Laplacian. In the first half of the talk, we will discuss a formula linking this operator to a natural Hodge Laplacian on a hypersurface. This result bears intriguing resemblance to the Gauss–Codazzi equation for the scalar curvature. In the second half, we will explain how our formula provides an integrability condition for the Poisson equation associated with the \(G_2\)-Laplacian in the presence of cohomogeneity one symmetry.

Joint work with Timothy Buttsworth (The University of New South Wales) and Stepan Hudecek (The University of Queensland).