Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry

We consider the Laplacian “co-flow” of $G_{2}$ -structures: $\frac{d}{dt} \psi =

\Delta_d \psi $w h e r e$ \psi $i s t h e d u a l 4 - f o r m o f a$ G_2 $- s t r u c t u r e$ \phi $a n d$ \Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_{2}$ -structure being coclosed ( $d ψ = 0$ ). 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 K"ahler 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 K"ahler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.

We consider the Laplacian “co-flow” of $G_{2}$ -structures: $\frac{d}{dt} \psi =

\Delta_d \psi $w h e r e$ \psi $i s t h e d u a l 4 - f o r m o f a$ G_2 $- s t r u c t u r e$ \phi $a n d$ \Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_{2}$ -structure being coclosed ( $d ψ = 0$ ). 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 K"ahler 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 K"ahler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.

Soliton solutions for the Laplacian coflow of some $G_{2}$ -structures with symmetry