Soliton solutions for the Laplacian co-flow of some [formula omitted]-structures with symmetry

Abstract

We consider the Laplacian “co-flow” of G2-structures: ∂∂tψ=−Δdψ where ψ is the dual 4-form of a G2-structure φ and Δd is the Hodge Laplacian on forms. Assuming short-time existence and uniqueness, this flow preserves the condition of the G2-structure being coclosed (dψ=0). We study this flow for two explicit examples of coclosed G2-structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold N which is taken to be either a nearly Kähler manifold or a Calabi–Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi–Yau case, we find all the soliton solutions explicitly. In the nearly Kähler case, we find several special soliton solutions, and reduce the general problem to a single third order highly nonlinear ordinary differential equation.