# Laplacian coflow for warped $\mathrm{G}_2$-structures

## TL;DR

This paper studies the Laplacian coflow of G2-structures on warped products, providing explicit evolution equations, existence conditions, and new long-term solutions for specific types of SU(3)-structures.

## Contribution

It offers a new reinterpretation of the Laplacian coflow as evolution equations on the base manifold and identifies conditions for solutions, including long-term solutions for special SU(3)-structures.

## Key findings

- Derived explicit evolution equations for the flow.
- Established necessary and sufficient conditions for solution existence.
- Constructed new long-time solutions for nearly Kähler, symplectic half-flat, or balanced structures.

## Abstract

We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold endowed with an $\mathrm{SU}(3)$-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the $\mathrm{SU}(3)$-structure on $M^6$ and the warping function $f$. Necessary and sufficient conditions for the existence of solution for this flow are given. Finally we describe new long time solutions for this flow where the $\mathrm{SU}(3)$-structure on $M^6$ is nearly K\"ahler, symplectic half-flat or balanced.