# Laplacian coflow on the 7-dimensional Heisenberg group

## TL;DR

This paper investigates the behavior of Laplacian coflows of $G_2$-structures on the 7-dimensional Heisenberg group, revealing conditions under which solutions are ancient, finite, or eternal.

## Contribution

It provides a detailed analysis of the solution intervals for both the Laplacian and modified Laplacian coflows on the Heisenberg group, highlighting new solution behaviors.

## Key findings

- Laplacian coflow solutions are always ancient.
- Modified Laplacian coflow solutions can be finite, ancient, or eternal.
- The solution behavior depends on specific cases of the coflow.

## Abstract

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is ancient or eternal, that is it is defined on $(-\infty, \infty)$.