Geometric flow of $G_{2}$-structures on $C(S^3\times S^3)$

TL;DR

This paper introduces a new first order flow for $G_{2}$-structures, explicitly solves it on a cone over $S^3 imes S^3$, and shows the flow deforms initial structures to a conic metric up to scaling.

Contribution

It constructs an explicit solution to a novel flow of $G_{2}$-structures on a specific cone and proves convergence to a conic metric from certain initial data.

Findings

Explicit solution for the flow on the cone over $S^3 imes S^3$

Flow deforms initial $G_2$-structure to a conic metric

Convergence holds up to homotheties

Abstract

We introduce a first order flow of $G_{2}$-structures and construct its explicit solution in case of a cone over $S^3\!\times\! S^3$. Also we prove for this situation that starting from certain initial datum the flow deforms corresponding to $G_2$-structure metric to a conic metric up to homotheties.