Modified Laplacian coflow of $G_{2}$-structures on manifolds with symmetry

TL;DR

This paper studies a modified Laplacian coflow of $G_{2}$-structures on 7-manifolds with symmetry, deriving new soliton solutions and analyzing their properties within warped product geometries involving Calabi-Yau or nearly Kähler six-manifolds.

Contribution

It introduces a modified Laplacian coflow for $G_{2}$-structures on symmetric manifolds and finds new compact soliton solutions, advancing understanding of geometric flows in special holonomy contexts.

Findings

Derived the form of $G_{2}$-structures on warped product manifolds.

Analyzed the soliton equations for the modified flow.

Obtained new compact soliton solutions.

Abstract

We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly K\"{a}hler manifold. We show that in these cases the $G_{2}$-structures are determined by their torsion components up to a phase factor. We then study the modified Laplacian coflow $\frac{d\psi }{dt}=\Delta _{\psi }\psi +2d\left( \left( C-Tr T\right) \varphi \right) $ of these $G_{2}$-structures, where $\varphi $ and $\psi $ are the fundamental $3$-form and $4 $-form which define the $G_{2}$-structure and $\Delta _{\psi }$ is the Hodge Laplacian associated with the $G_{2}$-structure. This flow is known to have short-time existence and uniqueness. We analyse the soliton equations for this flow and obtain new compact soliton solutions.