Laplacian Flow of $G_{2}$-Structures on $S^3\!\times\!\mathbb{R}^4$

A 7-dimensional smooth manifold $M$ admits a $G_{2}$-structure if there is a reduction of the structure group of its frame bundle from $GL(7,\mathbb{R})$ to the group $G_{2}$, viewed as a subgroup of $SO(7,\mathbb{R})$. On a manifold with $G_{2}$-structure there exists a “non-degenerate” 3-form $\varphi$, which determines a Riemannian metric $g_{\varphi}$ in a non-linear fashion. Let $(M,\varphi)$ be a manifold with $G_{2}$-structure. If $\varphi$ is parallel with respect to Levi-Civita connection of the metric $g_{\varphi}$, $\nabla\varphi=0$, then $(M,\varphi)$ is called $G_{2}$-manifold. Such manifolds are always Ricci-flat and have holonomy contained in $G_{2}$. The condition $\nabla\varphi=0$ is equivallent to $\varphi$ to be closed, $d\varphi=0$, and co-closed, $\delta\varphi=0$, form. It is very interesting to understand how we can get a parallel $\varphi$ on a certain manifold with $G_{2}$-structure via the evolution of some specific quantities. I will tell about the flow $\frac{\partial\varphi(t)}{\partial t}=\Delta\varphi$ on a $S^3\!\times\!\mathbb{R}^4$, where $\varphi(t)$ is a continuous family of $G_{2}$-structures defined on this space and $\Delta=d\delta+\delta$$d$ is a Hodge-Laplacian operator.