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

TL;DR

This paper investigates the Laplacian co-flow of certain $G_2$-structures with symmetry, deriving explicit soliton solutions in specific geometric settings, including Calabi-Yau and nearly K"ahler manifolds.

Contribution

It provides explicit soliton solutions for the Laplacian co-flow on symmetric $G_2$-structures, especially solving the flow in Calabi-Yau cases and reducing the nearly K"ahler case to a nonlinear ODE.

Findings

Explicit soliton solutions in Calabi-Yau case

Several special solitons in nearly K"ahler case

Reduction to a third-order nonlinear ODE

Abstract

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($d\psi =0$). 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.