Soliton solutions for the Laplacian co-flow of some G 2-structures with symmetry
Journal
Differential Geometry and its Application
Journal Volume
30
Journal Issue
4
Pages
318-333
Date Issued
2012
Author(s)
Abstract
We consider the Laplacian "co-flow" of G 2-structures: ∂∂tψ=-δdψ where ψ is the dual 4-form of a G 2-structure φ and δ d is the Hodge Laplacian on forms. Assuming short-time existence and uniqueness, this flow preserves the condition of the G 2-structure being coclosed (dψ=0). We study this flow for two explicit examples of coclosed G 2-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. © 2012 Elsevier B.V.
Type
journal article