Cohomogeneity-one $G_2$-Laplacian flow on 7-torus

TL;DR

This paper demonstrates the long-time existence and convergence of a specific geometric flow on a 7-torus, providing the first example of such behavior for a cohomogeneity-one $G_2$-Laplacian flow on a compact 7-manifold.

Contribution

It establishes the first example of a cohomogeneity-one $G_2$-Laplacian flow on a compact 7-manifold that exists globally and converges to a torsion-free $G_2$ structure.

Findings

Flow exists for all time on $ ext{T}^4$

Flow converges to the flat structure

First example of such flow on a compact 7-manifold

Abstract

We prove the hypersymplectic flow of simple type on standard torus $\mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one $G_2$-Laplacian flow on a compact $7$-manifold which exists for all time and converges to a torsion-free $G_2$ structure modulo diffeomorphisms.