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*Axioms*
**2019**,
*8*(1),
7;
doi:10.3390/axioms8010007

Article

The Laplacian Flow of Locally Conformal Calibrated G

_{2}-Structures^{1}

Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

^{2}

Departamento de Matemáticas—IUMA, Facultad de Ciencias Humanas y de la Educación, Universidad de Zaragoza, 22003 Huesca, Spain

^{*}

Author to whom correspondence should be addressed.

Received: 8 November 2018 / Accepted: 3 January 2019 / Published: 11 January 2019

## Abstract

**:**

We consider the Laplacian flow of locally conformal calibrated ${\mathrm{G}}_{2}$-structures as a natural extension to these structures of the well-known Laplacian flow of calibrated ${\mathrm{G}}_{2}$-structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated ${\mathrm{G}}_{2}$ manifolds and, in both cases, we obtain a flow of locally conformal calibrated ${\mathrm{G}}_{2}$-structures, which are ancient solutions, that is they are defined on a time interval of the form $(-\infty ,T)$, where $T>0$ is a real number. Moreover, for each of these examples, we prove that the underlying metrics $g\left(t\right)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to $-\infty $, and they blow-up at a finite-time singularity.

Keywords:

locally conformal calibrated G_{2}-structures; Laplacian flow; solvable Lie algebras

## 1. Introduction

A ${\mathrm{G}}_{2}$-structure on a 7-manifold M can be characterized by the existence of a globally defined 3-form $\phi $ (the ${\mathrm{G}}_{2}$ form) on M, which can be written at each point as
with respect to some local coframe $\{{e}^{1},\dots ,{e}^{7}\}$ on M. Here, ${e}^{127}$ stands for ${e}^{1}\wedge {e}^{2}\wedge {e}^{7}$, and so on. A ${\mathrm{G}}_{2}$-structure $\phi $ induces a Riemannian metric ${g}_{\phi}$ and a volume form $d{V}_{{g}_{\phi}}$ on M given by
for any pair of vector fields $X,Y$ on M, where ${i}_{X}$ denotes the contraction by X.

$$\phi ={e}^{127}+{e}^{347}+{e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245},$$

$${g}_{\phi}(X,Y)\phantom{\rule{0.166667em}{0ex}}d{V}_{{g}_{\phi}}=\frac{1}{6}\phantom{\rule{0.166667em}{0ex}}{i}_{X}\phi \wedge {i}_{Y}\phi \wedge \phi ,$$

The classes of ${\mathrm{G}}_{2}$-structures can be described in terms of the exterior derivatives of the 3-form $\phi $ and the 4-form ${\star}_{\phi}\phi $ [1,2], where ${\star}_{\phi}$ is the Hodge operator defined from ${g}_{\phi}$ and $d{V}_{{g}_{\phi}}$. If the 3-form $\phi $ is closed and coclosed, then the holonomy group of ${g}_{\phi}$ is a subgroup of the exceptional Lie group ${\mathrm{G}}_{2}$ [2], and the metric ${g}_{\phi}$ is Ricci-flat [3]. When this happens, the ${\mathrm{G}}_{2}$-structure is said to be torsion-free [4]. This condition has a variational formulation, due to Hitchin [5,6]. The first compact examples of Riemannian manifolds with holonomy ${\mathrm{G}}_{2}$ were constructed first by Joyce [7,8], and then by Kovalev [9]. Recently, other examples of compact manifolds with holonomy ${\mathrm{G}}_{2}$ were obtained in [10,11]. Explicit examples on solvable Lie groups were also constructed in [12]. A ${\mathrm{G}}_{2}$-structure $\phi $ is called locally conformal parallel if $\phi $ satisfies the two following conditions
for some closed non-vanishing 1-form $\theta $, which is known as the Lee form of the ${\mathrm{G}}_{2}$-structure. Such a ${\mathrm{G}}_{2}$-structure is locally conformal to one which is torsion-free. Ivanov, Parton and Piccinni in [13] prove that a compact locally conformal parallel ${\mathrm{G}}_{2}$ manifold is a mapping torus bundle over the circle ${S}^{1}$ with fibre a simply connected nearly Kähler manifold of dimension six and finite structure group.

$$d\phi =\theta \wedge \phi ,\phantom{\rule{2.em}{0ex}}d\left({\star}_{\phi}\phi \right)=\frac{4}{3}\theta \wedge {\star}_{\phi}\phi ,$$

We remind that a ${\mathrm{G}}_{2}$-structure $\phi $ is called closed (or calibrated according to [14]) if $d\phi =0$. In this paper we will focus our attention on the class of locally conformal calibrated ${\mathrm{G}}_{2}$-structures, which are characterized by the condition
where $\theta $ is a closed non-vanishing 1-form, which is also known as the Lee form of the ${\mathrm{G}}_{2}$-structure. We will refer to a manifold equipped with such a structure as a locally conformal calibrated ${\mathrm{G}}_{2}$ manifold. Each point of such a manifold has an open neighborhood U where $\theta =df$, for some $f\in \mathcal{F}\left(U\right)$ with $\mathcal{F}\left(U\right)$ being the algebra of the real differentiable functions on U, and the 3-form ${e}^{-f}\phi $ defines a calibrated ${\mathrm{G}}_{2}$-structure on U. Hence, locally conformal calibrated ${\mathrm{G}}_{2}$-structures are locally conformal equivalent to calibrated ${\mathrm{G}}_{2}$-structures, and they can be considered analogous in dimension 7 to the locally conformal symplectic manifolds, which have been studied in [15,16,17,18,19,20,21] and the references therein. Some results of locally conformal calibrated ${\mathrm{G}}_{2}$ manifolds were given in [22,23,24,25]. In fact, in [24] the first author and Ugarte introduced a differential complex for locally conformal calibrated ${\mathrm{G}}_{2}$ manifolds, and such manifolds were characterized as the ones endowed with a ${\mathrm{G}}_{2}$-structure $\phi $ for which the space of differential forms annihilated by $\phi $ becomes a differential subcomplex of the de Rham’s complex. Moreover, in [23] it is proved that a similar result to that of Ivanov, Parton and Piccinni holds for compact 7-manifolds with a suitable locally conformal calibrated ${\mathrm{G}}_{2}$-structure. More recently, a structure result for Lie algebras with an exact locally conformal calibrated ${\mathrm{G}}_{2}$-structure was proved by Bazzoni and Raffero in [22], where it is also shown that none of the non-Abelian nilpotent Lie algebras with closed ${\mathrm{G}}_{2}$-structures admits locally conformal calibrated ${\mathrm{G}}_{2}$-structures.

$$d\phi =\theta \wedge \phi ,$$

Compact ${\mathrm{G}}_{2}$-calibrated manifolds have interesting curvature properties. As we mentioned before, a ${\mathrm{G}}_{2}$ holonomy manifold is Ricci-flat or, equivalently, both Einstein and scalar-flat. But on a compact calibrated ${\mathrm{G}}_{2}$ manifold, both the Einstein condition [26] and scalar-flatness [27] are equivalent to the holonomy being contained in ${\mathrm{G}}_{2}$. In fact, Bryant in [27] shows that the scalar curvature is always non-positive.

Locally conformal calibrated ${\mathrm{G}}_{2}$-structures $\phi $ whose underlying Riemannian metric ${g}_{\phi}$ is Einstein have been studied in [25], where it was shown that in the compact case the scalar curvature of ${g}_{\phi}$ can not be positive. Then, Fino and Raffero in [25] show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated ${\mathrm{G}}_{2}$-structure $\phi $ unless the underlying metric ${g}_{\phi}$ is flat. However, in contrast to the compact homogeneous case, a non-compact example of homogeneous manifold S endowed with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure whose associated Riemannian metric is Einstein and non Ricci-flat was given in [25]. The manifold S is a simply connected solvable Lie group which is not unimodular (see Section 4.2 for details).

On the other hand, in [23] it is given an example of a compact manifold N with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure. The manifold N is a compact solvmanifold, that is N is a compact quotient of a simply connected solvable Lie group K by a lattice, endowed with an invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structure.

Since Hamilton introduced the Ricci flow in 1982 [28], geometric flows have been an important tool in studying geometric structures on manifolds. In ${\mathrm{G}}_{2}$ geometry, geometric flows for different ${\mathrm{G}}_{2}$-structures have been proposed. Let M be a 7-manifold endowed with a calibrated ${\mathrm{G}}_{2}$-structure $\phi $. The Laplacian flow starting from $\phi $ is the initial value problem
where $\phi \left(t\right)$ is a closed ${\mathrm{G}}_{2}$ form on M, and ${\Delta}_{t}=d\phantom{\rule{0.166667em}{0ex}}{d}^{*}+{d}^{*}d$ is the Hodge Laplacian operator associated with the metric $g\left(t\right)={g}_{\phi \left(t\right)}$ induced by the 3-form $\phi \left(t\right)$. This flow was introduced by Bryant in [27] as a tool to find torsion-free ${\mathrm{G}}_{2}$-structures on compact manifolds. Short-time existence and uniqueness of the solution when M is compact were proved in [29]. The analytic and geometric properties of the Laplacian flow have been deeply investigated in the series of papers [30,31,32]. Non-compact examples where the flow converges to a flat ${\mathrm{G}}_{2}$-structure have been given in [33].

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right),\hfill \\ d\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right)=0,\hfill \\ \phi \left(0\right)=\phi ,\hfill \end{array}\right.$$

In [34], a flow evolving the 4-form $\psi ={\star}_{\phi}\phi $ in the direction of minus its Hodge Laplacian was introduced, and it is called Laplacian coflow of $\phi $. This flow preserves the condition of the ${\mathrm{G}}_{2}$-structure $\phi $ being coclosed, that is $\psi \left(t\right)$ is closed for any t, and it was studied in [34] for two explicit examples of coclosed ${\mathrm{G}}_{2}$-structures. But no general result is known about the short time existence of the coflow. A modified Laplacian coflow was introduced by Grigorian in [35] (see also [36]). There it was proved that for compact manifolds, the modified Laplacian coflow has a unique solution $\psi \left(t\right)$ for the short time period $t\in [0,\u03f5]$, for some $\u03f5>0$. Geometric properties of both coflows on the 7-dimensional Heisenberg group and on 7-dimensional almost-abelian Lie groups were proved in [37,38], respectively.

Some work has also been done on other related flows of ${\mathrm{G}}_{2}$-structures—such as the Laplacian flow and the Laplacian coflow, for locally conformal parallel ${\mathrm{G}}_{2}$-structures. These flows has been originally proposed by the second author with Otal and Villacampa in [39], and the first examples of long time solutions of the flows are given in [39].

In this note, for any locally conformal calibrated ${\mathrm{G}}_{2}$-structure $\phi $ on a manifold M, we consider the Laplacian flow of $\phi $ given by

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right),\hfill \\ d\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right)=\theta \left(t\right)\wedge \phi \left(t\right),\hfill \\ \phi \left(0\right)=\phi .\hfill \end{array}\right.$$

We do not known any general result on the short time existence of solution for this flow. Nevertheless, in Section 4 (Theorems 1 and 2), for each of the aforementioned examples of solvable Lie groups K and S with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure, we show that the solution of the before Laplacian flow is ancient, that is it is defined on a time interval of the form $(-\infty ,T)$, where $T>0$ is a real number. Moreover, for each of the two examples K and S, we show that the underlying metrics $g\left(t\right)={g}_{\phi \left(t\right)}$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to $-\infty $, and they blow-up in finite-time. As we mentioned before, the Lie group S has a locally conformal calibrated ${\mathrm{G}}_{2}$-structure inducing an Einstein metric. We prove that the solution $\phi \left(t\right)$ of the flow on S induces an Einstein metric for all time t where $\phi \left(t\right)$ is defined.

## 2. ${\mathrm{G}}_{\mathbf{2}}$-Structures

Let M be a 7-dimensional manifold with a ${\mathrm{G}}_{2}$-structure defined by a 3-form $\phi $. Denote by $\psi $ the 4-form $\psi ={\star}_{\phi}\phi $, where ${\star}_{\phi}$ is the Hodge star operator of the metric ${g}_{\phi}$ induced by $\phi $. Let $({\mathsf{\Omega}}^{*}\left(M\right),d)$ be the de Rham complex of differential forms on M. Then, Bryant in [27] proved that the forms $d\phi $ and $d\psi $ are such that
where ${\tau}_{0}\in {\mathsf{\Omega}}^{0}\left(M\right),{\tau}_{1}\in {\mathsf{\Omega}}^{1}\left(M\right),{\tau}_{2}\in {\mathsf{\Omega}}_{14}^{2}\left(M\right)$ and ${\tau}_{3}\in {\mathsf{\Omega}}_{27}^{3}\left(M\right)$. Here ${\mathsf{\Omega}}_{14}^{2}\left(M\right)$ and ${\mathsf{\Omega}}_{27}^{3}\left(M\right)$ are the spaces

$$\begin{array}{l}\left\{\begin{array}{l}d\phi ={\tau}_{0}\phantom{\rule{0.166667em}{0ex}}\psi +3\phantom{\rule{0.166667em}{0ex}}{\tau}_{1}\wedge \phi +{\star}_{\phi}{\tau}_{3},\\ d\psi =4{\tau}_{1}\wedge \psi -{\star}_{\phi}{\tau}_{2},\end{array}\right.\end{array}$$

$$\begin{array}{c}{\mathsf{\Omega}}_{14}^{2}\left(M\right)=\{\alpha \in {\mathsf{\Omega}}^{2}\left(M\right)\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{0.166667em}{0ex}}\alpha \wedge \phi =-{\star}_{\phi}\alpha \},\hfill \end{array}$$

$$\begin{array}{c}{\mathsf{\Omega}}_{27}^{3}\left(M\right)=\{\beta \in {\mathsf{\Omega}}^{3}\left(M\right)\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{0.166667em}{0ex}}\beta \wedge \phi =0=\beta \wedge {\star}_{\phi}\phi \}.\hfill \end{array}$$

The differential forms ${\tau}_{i}$ ($i=0,1,2,3$) that appear in (3), are called the intrinsic torsion forms of $\phi $.

In terms of the torsion forms, some classes of ${\mathrm{G}}_{2}$-structures with the defining conditions are recalled in the Table 1.

Note that if a manifold M has a locally conformal calibrated ${\mathrm{G}}_{2}$-structure $\phi $, then
with $\theta $ the Lee form of $\phi $. Thus, taking into account (3), the torsion form ${\tau}_{1}$ of the ${\mathrm{G}}_{2}$ form $\phi $ can be expressed in terms of the Lee form $\theta $ as ${\tau}_{1}=\frac{1}{3}\theta $. Moreover (see [24]), the torsion forms ${\tau}_{1}$ and ${\tau}_{2}$ of $\phi $ can be obtained as follows:

$$d\phi =\theta \wedge \phi ,$$

$$\begin{array}{l}{\tau}_{1}=-\frac{1}{12}{\star}_{\phi}\left({\star}_{\phi}d\phi \wedge \phi \right),\\ {\tau}_{2}={\star}_{\phi}\left(4{\tau}_{1}\wedge \left({\star}_{\phi}\phi \right)-d{\star}_{\phi}\phi \right).\end{array}$$

## 3. The Laplacian Flow of Locally Conformal Calibrated G_{2}-Structures

In this section, we introduce the Laplacian flow of a locally conformal calibrated ${\mathrm{G}}_{2}$-structure on a manifold M and, for its equations, we show some properties that help us solve the flow when M is a Lie group.

**Definition**

**1.**

Let M be a 7-manifold with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure φ. We say that a time-dependent ${\mathrm{G}}_{2}$-structure $\phi \left(t\right)$ on M, defined for t in some real open interval, satisfies the Laplacian flow system of φ if, for all times t for which $\phi \left(t\right)$ is defined, we have
where $\theta \left(t\right)$ is the Lee form of $\phi \left(t\right)$, and ${\Delta}_{t}=d\phantom{\rule{0.166667em}{0ex}}{d}^{*}+{d}^{*}d$ is the Hodge Laplacian operator associated with the metric $g\left(t\right)={g}_{\phi \left(t\right)}$ induced by the 3-form $\phi \left(t\right)$.

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right),\hfill \\ d\phantom{\rule{0.166667em}{0ex}}\phi \left(t\right)=\theta \left(t\right)\wedge \phi \left(t\right),\hfill \\ \phi \left(0\right)=\phi ,\hfill \end{array}\right.$$

In order to solve the first equation of the flow (5) for our examples, we follow the approach of [39].

Let G be a simply connected solvable Lie group of dimension 7 with Lie algebra g. Let $\{{e}^{1},\cdots ,{e}^{7}\}$ be a basis of the dual space ${g}^{*}$ of g, and let ${f}_{i}={f}_{i}\left(t\right)(i=1,\cdots ,7)$ be some differentiable real functions depending on a parameter $t\in I\subset \mathbb{R}$ such that ${f}_{i}\left(0\right)=1$ and ${f}_{i}\left(t\right)\ne 0$, for any $t\in I$, where I is a real open interval. For each $t\in I$, we consider the basis $\{{x}^{1},\cdots ,{x}^{7}\}$ of ${g}^{*}$ defined by

$${x}^{i}={x}^{i}\left(t\right)={f}_{i}\left(t\right){e}^{i},\phantom{\rule{1.em}{0ex}}1\le i\le 7.$$

We consider the one-parameter family of left invariant ${\mathrm{G}}_{2}$-structures $\phi \left(t\right)$ on G given by
where ${f}_{ijk}={f}_{ijk}\left(t\right)$ stands for the product ${f}_{i}\left(t\right){f}_{j}\left(t\right){f}_{k}\left(t\right)$.

$$\begin{array}{ll}\phi \left(t\right)& ={x}^{127}+{x}^{347}+{x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}\\ & ={f}_{127}{e}^{127}+{f}_{347}{e}^{347}+{f}_{567}{e}^{567}+{f}_{135}{e}^{135}-{f}_{146}{e}^{146}-{f}_{236}{e}^{236}-{f}_{245}{e}^{245},\end{array}$$

Now, we introduce the function $\epsilon (i,j,k)$ on ordered indices $(i,j,k)$ as follows:

$$\epsilon (i,j,k)=\left\{\begin{array}{ll}1& \mathrm{if}\text{}(i,j,k)\in A=\left\{\right(1,2,7),(1,3,5),(3,4,7),(5,6,7\left)\right\};\\ -1& \mathrm{if}\text{}(i,j,k)\in B=\left\{\right(1,4,6),(2,3,6),(2,4,5\left)\right\};\\ 0& \mathrm{otherwise}.\end{array}\right.$$

Thus, the ${\mathrm{G}}_{2}$ form $\phi $ defined in (1), can be rexpressed as $\phi ={\sum}_{(i,j,k)\in A\cup B}\epsilon (i,j,k){e}^{ijk}$, and the ${\mathrm{G}}_{2}$ form $\phi \left(t\right)$ given by (6) becomes

$$\phi \left(t\right)=\sum _{(i,j,k)\in A\cup B}\epsilon (i,j,k){x}^{ijk}.$$

Therefore,

$$\begin{array}{ll}\frac{d}{dt}\phi \left(t\right)& ={\displaystyle \sum _{(i,j,k)\in A\cup B}}\epsilon (i,j,k)\frac{d{f}_{ijk}}{dt}{e}^{ijk}\\ & ={\displaystyle \sum _{(i,j,k)\in A\cup B}}\epsilon (i,j,k)\frac{{\left({f}_{ijk}\right)}^{\prime}}{{f}_{ijk}}{x}^{ijk}\\ & ={\displaystyle \sum _{(i,j,k)\in A\cup B}}\epsilon (i,j,k)\frac{d}{dt}\left(ln{f}_{ijk}\right){x}^{ijk}.\end{array}$$

Moreover, we have
where $\epsilon (i,j,k){\Delta}_{ijk}$ is the coefficient in ${x}^{ijk}$ of ${\Delta}_{t}\phi \left(t\right)$ if $(i,j,k)\in A\cup B$ (i.e., if $\epsilon (i,j,k)\ne 0$), and ${\Delta}_{lmn}$ is the coefficient in ${x}^{lmn}$ of ${\Delta}_{t}\phi \left(t\right)$ if $1\le l<m<n\le 7$ and $\epsilon (l,m,n)=0$. Consequently, the first equation of the flow (5) is equivalent to the system of differential equations
that is,

$${\Delta}_{t}\phi \left(t\right)=\sum _{(i,j,k)\in A\cup B}\epsilon (i,j,k){\Delta}_{ijk}\phantom{\rule{0.166667em}{0ex}}{x}^{ijk}+\sum _{1\le l<m<n\le 7,\phantom{\rule{0.166667em}{0ex}}(l,m,n)\notin A\cup B}{\Delta}_{lmn}\phantom{\rule{0.166667em}{0ex}}{x}^{lmn}\phantom{\rule{0.166667em}{0ex}},$$

$$\begin{array}{c}\left\{\begin{array}{ll}{\Delta}_{ijk}=\frac{{\left({f}_{ijk}\right)}^{\prime}}{{f}_{ijk}}& \mathrm{if}\text{}(i,j,k)\in A\cup B,\\ {\Delta}_{lmn}=0& \mathrm{if}\text{}1\le lmn\le 7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}(l,m,n)\notin A\cup B,\end{array}\right.\end{array}$$

$$\begin{array}{l}\left\{\begin{array}{ll}{\Delta}_{ijk}=\frac{d}{dt}ln\left({f}_{ijk}\right)& \mathrm{if}\text{}(i,j,k)\in A\cup B,\\ {\Delta}_{lmn}=0& \mathrm{if}\text{}1\le lmn\le 7\text{}\mathrm{and}\text{}(l,m,n)\notin A\cup B.\end{array}\right.\end{array}$$

We will also use the following properties of ${\Delta}_{ijk}$.

**Lemma**

**1.**

Let $\phi \left(t\right)$ be a family of left invariant ${\mathrm{G}}_{2}$-structures on the Lie group G solving the system (7), and such that $\phi \left(t\right)$ can be expressed as (6), for some functions ${f}_{i}={f}_{i}\left(t\right)$. For ordered indices $(i,j,k)$ and $(p,q,r)\in A\cup B$ (that is, $\epsilon (i,j,k)$ and $\epsilon (p,q,r)$ are both non-zero) we have

- i)
- if ${\Delta}_{ijk}={\Delta}_{pqr}$, then ${f}_{ijk}={f}_{pqr}$;
- ii)
- if ${f}_{ijk}{\Delta}_{ijk}={f}_{pqr}{\Delta}_{pqr}$, then ${f}_{ijk}={f}_{pqr}$;
- iii)
- if ${\Delta}_{ijk}+{\Delta}_{pqr}=0$, then ${f}_{ijk}{f}_{pqr}=1$;
- iv)
- if ${f}_{ijk}{\Delta}_{ijk}+{f}_{pqr}{\Delta}_{pqr}=0$, then ${f}_{ijk}+{f}_{pqr}=2$.

**Proof.**

The first statement of this Lemma was proved in [39]. Nevertheless, we point out how to prove it. Since ${\Delta}_{ijk}={\Delta}_{pqr}$, the system (8) implies that $\frac{d}{dt}ln{f}_{ijk}=\frac{d}{dt}ln{f}_{pqr}$. Hence, $ln{f}_{ijk}=ln{f}_{pqr}+C$, for some constant C. Now, using that ${f}_{i}\left(0\right)=1$, for $i=1,\dots ,7$, we have that $C=0$. So, ${f}_{ijk}={f}_{pqr}$, which proves i).

Now, let us suppose that ${f}_{ijk}{\Delta}_{ijk}={f}_{pqr}{\Delta}_{pqr}$, for some $i,j,k,p,q,r$ with $1\le i<j<k\le 7$ and $1\le p<q<r\le 7$. From (7), we get

$${\left({f}_{ijk}\right)}^{\prime}={\left({f}_{pqr}\right)}^{\prime}.$$

Integrating this equation, we obtain ${f}_{ijk}={f}_{pqr}+C$, for some constant C. Since ${f}_{i}\left(0\right)=1$, for all $i=1,\dots ,7$, we have $C=0$, and so ${f}_{ijk}={f}_{pqr}$. This proves ii).

To prove iii), we use (8), and we obtain
for some constant C. But ${f}_{i}\left(0\right)=1$, for all $i=1,\dots ,7$, imply that $C=0$, that is

$$ln({f}_{ijk}\xb7{f}_{pqr})=C,$$

$${f}_{ijk}\xb7{f}_{pqr}=1.$$

Finally, let us suppose that ${f}_{ijk}{\Delta}_{ijk}+{f}_{pqr}{\Delta}_{pqr}=0$, for some $i,j,k,p,q,r$ with $1\le i<j<k\le 7$ and $1\le p<q<r\le 7$. Then, using (7), we get ${\left({f}_{ijk}\right)}^{\prime}=-{\left({f}_{pqr}\right)}^{\prime}$. Integrating this equation, we obtain ${f}_{ijk}=-{f}_{pqr}+C$, for some constant C. But $C=2$ since ${f}_{i}\left(0\right)=1$, for all $i=1,\dots ,7$. Thus, ${f}_{ijk}+{f}_{pqr}=2$, which completes the proof. □

## 4. Solutions of the Laplacian Flow on Locally Conformal Calibrated ${\mathrm{G}}_{\mathbf{2}}$ Solvmanifolds

Lie groups admitting left invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structures constitute a convenient setting where it is possible to investigate the behaviour of the Laplacian flow (5) in the non-compact case.

In this section, we consider two examples of solvable Lie groups K and S, each of them with a left invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structure, and we show that in both cases the solution is ancient (i.e., it is defined in some interval $(-\infty ,T)$, with $0<T<+\infty $) and the induced metrics blow-up at a finite-time singularity.

#### 4.1. The Laplacian Flow on K

Let K be the simply connected and solvable Lie group of dimension 7 whose Lie algebra k is defined by

$$k=\left({e}^{37},{e}^{47},-{e}^{17},-{e}^{27},{e}^{14}+{e}^{23},{e}^{13}-{e}^{24},0\right).$$

Here, ${e}^{37}$ stands for ${e}^{3}\wedge {e}^{7}$, and so on; and $\left({e}^{37},{e}^{47},-{e}^{17},-{e}^{27},{\mathrm{e}}^{14}+{\mathrm{e}}^{23},{\mathrm{e}}^{13}-{\mathrm{e}}^{24},0\right)$ means that there is a basis $\{{e}^{1},\cdots ,{e}^{7}\}$ of the dual space ${k}^{*}$ of k, satisfying
where d denotes the Chevalley-Eilenberg differential on ${k}^{*}$.

$$\begin{array}{l}d{e}^{1}={e}^{37},\phantom{\rule{2.em}{0ex}}d{e}^{2}={e}^{47},\phantom{\rule{2.em}{0ex}}d{e}^{3}=-{e}^{17},\phantom{\rule{2.em}{0ex}}d{e}^{4}=-{e}^{27},\\ d{e}^{5}={e}^{14}+{e}^{23},\phantom{\rule{1.em}{0ex}}d{e}^{6}={e}^{13}-{e}^{24},\phantom{\rule{1.em}{0ex}}d{e}^{7}=0,\end{array}$$

The 3-form $\phi $ on K given by
defines a left invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structure on the Lie group K, with Lee form $\theta ={e}^{7}$, and so with torsion form ${\tau}_{1}=\frac{1}{3}{e}^{7}$. In fact,

$$\phi ={e}^{127}+{e}^{347}+{e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245}$$

$$d\phi =-{e}^{1357}+{e}^{1467}+{e}^{2367}+{e}^{2457}={e}^{7}\wedge \phi .$$

In [23] it is proved that there exists a lattice $\mathsf{\Gamma}$ in K, so that the quotient space of right cosets $\mathsf{\Gamma}\setminus K$ is a compact solvmanifold endowed with an invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structure $\phi $, with Lee form $\theta ={e}^{7}$.

However, we should note that in the following Theorem, we will show a solution of the Laplacian flow (5) of the ${\mathrm{G}}_{2}$ form $\phi $ (defined by (10)) on the Lie group K. Such a solution does not solve the Laplacian flow of $\phi $ on the compact quotient $\mathsf{\Gamma}\setminus K$ since we will consider the Hodge Laplacian operator ${\Delta}_{t}$ on the Lie algebra k of K and we cannot check the Hodge Laplacian operator on the compact space $\mathsf{\Gamma}\setminus K$.

**Theorem**

**1.**

The family of locally conformal calibrated ${\mathrm{G}}_{2}$-structures $\phi \left(t\right)$ on K given by
is the solution for the Laplacian flow (5) of the ${\mathrm{G}}_{2}$ form φ given by (10), where $t\in \left(-\infty ,\frac{3}{8}\right)$. The Lee form $\theta \left(t\right)$ of $\phi \left(t\right)$ is $\theta \left(t\right)={e}^{7}$. Moreover, the underlying metrics $g\left(t\right)$ of this solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in K, as t goes to $-\infty $, and they blow-up as t goes to $\frac{3}{8}$.

$$\phi \left(t\right)={e}^{127}+{e}^{347}+{\left(1-{\textstyle \frac{8}{3}}t\right)}^{-3/2}\left({e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245}\right)$$

**Proof.**

As in Section 2, let ${f}_{i}={f}_{i}\left(t\right)\text{}(i=1,\cdots ,7)$ be some differentiable real functions depending on a parameter $t\in I\subset \mathbb{R}$ such that ${f}_{i}\left(0\right)=1$ and ${f}_{i}\left(t\right)\ne 0$, for any $t\in I$, where I is a real open interval. For each $t\in I$, we consider the basis $\{{x}^{1},\cdots ,{x}^{7}\}$ of left invariant 1-forms on K defined by
$${x}^{i}={x}^{i}\left(t\right)={f}_{i}\left(t\right){e}^{i},\phantom{\rule{1.em}{0ex}}1\le i\le 7.$$

Taking into account (9), the structure equations of K with respect to the basis $\{{x}^{1},\cdots ,{x}^{7}\}$ are

$$\begin{array}{l}d{x}^{1}=\frac{{f}_{1}}{{f}_{37}}{x}^{37},\phantom{\rule{2.em}{0ex}}d{x}^{2}=\frac{{f}_{2}}{{f}_{47}}{x}^{47},\phantom{\rule{2.em}{0ex}}d{x}^{3}=-\frac{{f}_{3}}{{f}_{17}}{x}^{17},\phantom{\rule{2.em}{0ex}}d{x}^{4}=-\frac{{f}_{4}}{{f}_{27}}{x}^{27},\\ d{x}^{5}=\frac{{f}_{5}}{{f}_{14}}{x}^{14}+\frac{{f}_{5}}{{f}_{23}}{x}^{23},\phantom{\rule{2.em}{0ex}}d{x}^{6}=\frac{{f}_{6}}{{f}_{13}}{x}^{13}-\frac{{f}_{6}}{{f}_{24}}{x}^{24},\phantom{\rule{2.em}{0ex}}d{x}^{7}=0.\end{array}$$

From now on, we write ${f}_{ij}={f}_{ij}\left(t\right)={f}_{i}\left(t\right){f}_{j}\left(t\right)$, ${f}_{ijk}={f}_{ijk}\left(t\right)={f}_{i}\left(t\right){f}_{j}\left(t\right){f}_{k}\left(t\right)$, and so forth. Then, for any $t\in I$, we consider the ${\mathrm{G}}_{2}$-structure $\phi \left(t\right)$ on K given by

$$\begin{array}{ll}\phi \left(t\right)& ={x}^{127}+{x}^{347}+{x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}\hfill \\ & ={f}_{127}{e}^{127}+{f}_{347}{e}^{347}+{f}_{567}{e}^{567}+{f}_{135}{e}^{135}-{f}_{146}{e}^{146}-{f}_{236}{e}^{236}-{f}_{245}{e}^{245}.\end{array}$$

Note that the 3-form $\phi \left(t\right)$ defined by (13) is such that $\phi \left(0\right)=\phi $ and, for any t, $\phi \left(t\right)$ determines the metric $g\left(t\right)$ on K such that the basis $\{{x}_{i}=\frac{1}{{f}_{i}}{e}_{i};\phantom{\rule{4pt}{0ex}}i=1,\cdots ,7\}$ of left invariant vector fields on K dual to $\{{x}^{1},\cdots ,{x}^{7}\}$ is orthonormal. So, $g\left(t\right)({e}_{i},{e}_{i})={f}_{i}^{2}$, and hence ${f}_{i}={f}_{i}\left(t\right)>0$.

To solve the flow (5) of $\phi $ we determine firstly the functions ${f}_{i}$ and the interval I so that $\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phi \left(t\right)$, for $t\in I$. We know that

$${\Delta}_{t}\phi \left(t\right)=({\star}_{t}d{\star}_{t}d-d{\star}_{t}d{\star}_{t})\phi \left(t\right).$$

We calculate separately each of the terms ${\star}_{t}d{\star}_{t}d\phi \left(t\right)$ and $-d{\star}_{t}d{\star}_{t}\phi \left(t\right)$ of ${\Delta}_{t}\phi \left(t\right)$. Taking into account (12) and the fact that the basis $\{{x}^{1}\left(t\right),\cdots ,{x}^{7}\left(t\right)\}$ is orthonormal, we have
and, on the other hand, we obtain

$$\begin{array}{ll}{\star}_{t}d{\star}_{t}d\phi \left(t\right)& =-\frac{\left({f}_{1}{f}_{4}-{f}_{2}{f}_{3}\right)\left({f}_{2}{f}_{3}+{f}_{1}{f}_{4}\right){f}_{5}}{{f}_{1}{f}_{2}{f}_{3}^{2}{f}_{4}^{2}{f}_{7}}{x}^{126}-\frac{\left({f}_{1}{f}_{4}-{f}_{2}{f}_{3}\right)\left({f}_{1}^{2}{f}_{2}^{2}+{f}_{3}^{2}{f}_{4}^{2}\right)}{{f}_{1}{f}_{2}^{2}{f}_{3}^{2}{f}_{4}{f}_{7}^{2}}{x}^{146}\hfill \\ & -\frac{\left({f}_{2}{f}_{3}-{f}_{1}{f}_{4}\right)\left({f}_{1}^{2}{f}_{2}^{2}+{f}_{3}^{2}{f}_{4}^{2}\right)}{{f}_{1}^{2}{f}_{2}{f}_{3}{f}_{4}^{2}{f}_{7}^{2}}{x}^{236}+\frac{\left({f}_{1}{f}_{4}-{f}_{2}{f}_{3}\right)\left({f}_{2}{f}_{3}+{f}_{1}{f}_{4}\right){f}_{5}}{{f}_{1}^{2}{f}_{2}^{2}{f}_{3}{f}_{4}{f}_{7}}{x}^{346}\\ & +\frac{\left({f}_{2}^{2}{f}_{3}^{2}{f}_{5}^{2}+{f}_{1}^{2}{f}_{4}^{2}{f}_{5}^{2}+{f}_{1}^{2}{f}_{3}^{2}{f}_{6}^{2}+{f}_{2}^{2}{f}_{4}^{2}{f}_{6}^{2}\right)}{{f}_{1}^{2}{f}_{2}^{2}{f}_{3}^{2}{f}_{4}^{2}}{x}^{567},\end{array}$$

$$\begin{array}{ll}d{\star}_{t}d{\star}_{t}\phi \left(t\right)& =\frac{\left({f}_{1}{f}_{2}-{f}_{3}{f}_{4}\right)\left({f}_{2}^{2}{f}_{3}^{2}+{f}_{1}^{2}{f}_{4}^{2}\right)}{{f}_{1}^{2}{f}_{2}^{2}{f}_{3}{f}_{4}{f}_{7}^{2}}{x}^{127}-\frac{{f}_{6}\left({f}_{2}{f}_{3}{f}_{5}+{f}_{1}{f}_{4}{f}_{5}+{f}_{1}{f}_{3}{f}_{6}+{f}_{2}{f}_{4}{f}_{6}\right)}{{f}_{1}^{2}{f}_{2}{f}_{3}^{2}{f}_{4}}{x}^{135}\\ & +\frac{{f}_{5}\left({f}_{2}{f}_{3}{f}_{5}+{f}_{1}{f}_{4}{f}_{5}+{f}_{1}{f}_{3}{f}_{6}+{f}_{2}{f}_{4}{f}_{6}\right)}{{f}_{1}^{2}{f}_{2}{f}_{3}{f}_{4}^{2}}{x}^{146}\\ & +\frac{{f}_{5}\left({f}_{2}{f}_{3}{f}_{5}+{f}_{1}{f}_{4}{f}_{5}+{f}_{1}{f}_{3}{f}_{6}+{f}_{2}{f}_{4}{f}_{6}\right)}{{f}_{1}{f}_{2}^{2}{f}_{3}^{2}{f}_{4}}{x}^{236}\\ & +\frac{{f}_{6}\left({f}_{2}{f}_{3}{f}_{5}+{f}_{1}{f}_{4}{f}_{5}+{f}_{1}{f}_{3}{f}_{6}+{f}_{2}{f}_{4}{f}_{6}\right)}{{f}_{1}{f}_{2}^{2}{f}_{3}{f}_{4}^{2}}{x}^{245}-\frac{\left({f}_{1}{f}_{2}-{f}_{3}{f}_{4}\right)\left({f}_{2}^{2}{f}_{3}^{2}+{f}_{1}^{2}{f}_{4}^{2}\right)}{{f}_{1}{f}_{2}{f}_{3}^{2}{f}_{4}^{2}{f}_{7}^{2}}{x}^{347}.\end{array}$$

Since $(1,2,6)$ and $(3,4,6)\notin A\cup B$, the system (7) implies that ${\Delta}_{126}=0={\Delta}_{346}$. Moreover, from (14) and (15) we have
and

$${\Delta}_{126}=\frac{{f}_{5}}{{f}_{7}}\left(\frac{{f}_{2}}{{f}_{1}{f}_{4}^{2}}-\frac{{f}_{1}}{{f}_{2}{f}_{3}^{2}}\right),$$

$${\Delta}_{346}=\frac{{f}_{5}}{{f}_{7}}\left(\frac{{f}_{4}}{{f}_{2}^{2}{f}_{3}}-\frac{{f}_{3}}{{f}_{1}^{2}{f}_{4}}\right).$$

Each of these equalities implies that ${f}_{14}^{2}={f}_{23}^{2}$, and so
since ${f}_{i}={f}_{i}\left(t\right)>0$.

$${f}_{14}={f}_{23}$$

Also (14) and (15) imply that the coefficients ${\Delta}_{ijk}$, with $(i,j,k)\in A\cup B$, are given by
where

$$\begin{array}{ll}{\Delta}_{127}=-\frac{{f}_{3}}{{f}_{1}}{B}_{23}+\frac{{f}_{4}}{{f}_{2}}{B}_{14},& {\Delta}_{347}=\frac{{f}_{2}}{{f}_{4}}{B}_{23}-\frac{{f}_{1}}{{f}_{3}}{B}_{14},\\ {\Delta}_{135}=\frac{{f}_{6}}{{f}_{13}}A,& {\Delta}_{245}=\frac{{f}_{6}}{{f}_{24}}A,\\ {\Delta}_{146}=\frac{{f}_{5}}{{f}_{14}}A-\frac{{f}_{1}}{{f}_{3}}{B}_{12}+\frac{{f}_{4}}{{f}_{2}}{B}_{34},& {\Delta}_{236}=\frac{{f}_{5}}{{f}_{23}}A+\frac{{f}_{2}}{{f}_{4}}{B}_{12}-\frac{{f}_{3}}{{f}_{1}}{B}_{34},\\ {\Delta}_{567}={A}_{2},\end{array}$$

$$\begin{array}{ll}A={f}_{5}\left(\frac{1}{{f}_{23}}+\frac{1}{{f}_{14}}\right)+{f}_{6}\left(\frac{1}{{f}_{13}}+\frac{1}{{f}_{24}}\right),& {A}_{2}={f}_{5}^{2}\left(\frac{1}{{f}_{23}^{2}}+\frac{1}{{f}_{14}^{2}}\right)+{f}_{6}^{2}\left(\frac{1}{{f}_{13}^{2}}+\frac{1}{{f}_{24}^{2}}\right),\\ {B}_{12}=\frac{1}{{f}_{7}^{2}}\left(\frac{{f}_{2}}{{f}_{4}}-\frac{{f}_{1}}{{f}_{3}}\right),& {B}_{34}=\frac{1}{{f}_{7}^{2}}\left(\frac{{f}_{4}}{{f}_{2}}-\frac{{f}_{3}}{{f}_{1}}\right),\\ {B}_{23}=\frac{1}{{f}_{7}^{2}}\left(\frac{{f}_{2}}{{f}_{4}}-\frac{{f}_{3}}{{f}_{1}}\right),& {B}_{14}=\frac{1}{{f}_{7}^{2}}\left(\frac{{f}_{4}}{{f}_{2}}-\frac{{f}_{1}}{{f}_{3}}\right).\end{array}$$

Using (17), one can check that ${f}_{135}{\Delta}_{135}={f}_{245}{\Delta}_{245}$. Thus, ${f}_{13}={f}_{24}$ by Lemma 1– ii). This equality and (16) imply

$${f}_{1}={f}_{2},\phantom{\rule{2.em}{0ex}}{f}_{3}={f}_{4}.$$

The equalities (19) imply that the functions ${B}_{12}$ and ${B}_{34}$ defined in (18) are such that ${B}_{12}=0={B}_{34}$. Hence, ${\Delta}_{146}=\frac{{f}_{5}}{{f}_{14}}A$. So, from (17), we have ${f}_{146}{\Delta}_{146}={f}_{245}{\Delta}_{245}$. Now, Lemma 1– ii) and (19) imply

$${f}_{5}={f}_{6}.$$

Moreover, from (18) and (19) we get ${B}_{14}=-{B}_{23}$. Then, from (17) we have ${f}_{127}{\Delta}_{127}+{f}_{347}{\Delta}_{347}=0$. Now, Lemma 1– iv) implies

$${f}_{12}+{f}_{34}=2/{f}_{7}.$$

Thus,

$${f}_{7}=\frac{2}{({f}_{1}^{2}+{f}_{3}^{2})}.$$

Using the equalities (19) and (21), we obtain that ${\Delta}_{135}={\Delta}_{567}$. Therefore, by Lemma 1– i) we have

$${f}_{13}={f}_{67}.$$

From this equality and (21), we obtain

$${f}_{6}=\frac{1}{2}{f}_{13}\phantom{\rule{0.166667em}{0ex}}({f}_{1}^{2}+{f}_{3}^{2}).$$

In summary, from (19)–(22), we have

$${f}_{1}={f}_{2},\phantom{\rule{2.em}{0ex}}{f}_{3}={f}_{4},\phantom{\rule{2.em}{0ex}}{f}_{5}={f}_{6}=\frac{1}{2}{f}_{13}\phantom{\rule{0.166667em}{0ex}}({f}_{1}^{2}+{f}_{3}^{2}),\phantom{\rule{2.em}{0ex}}{f}_{7}=\frac{2}{{f}_{1}^{2}+{f}_{3}^{2}}.$$

Now, we can suppose that ${f}_{3}={f}_{1}=f$ (see below Lemma 2). Then, the previous conditions reduce to

$${f}_{1}={f}_{2}={f}_{3}={f}_{4}=f,\phantom{\rule{2.em}{0ex}}{f}_{5}={f}_{6}={f}^{4},\phantom{\rule{2.em}{0ex}}{f}_{7}={f}^{-2}.$$

Then, by (18), ${B}_{14}=0={B}_{23}$ since ${f}_{1}={f}_{2}={f}_{3}={f}_{4}$ by (23). So, ${\Delta}_{127}=0={\Delta}_{347}$.

This implies that the unique non-zero components ${\Delta}_{ijk}$ of the Laplacian of ${\Delta}_{t}\phi \left(t\right)$ are

$${\Delta}_{567}={\Delta}_{135}={\Delta}_{146}={\Delta}_{236}={\Delta}_{245}=4{f}^{4}.$$

Then, the system of differential Equations (7) reduces to

$${f}^{-5}{f}^{\prime}=\frac{2}{3}.$$

Integrating this equation, we obtain

$$f={\left(C-\frac{8}{3}t\right)}^{-\frac{1}{4}},\phantom{\rule{2.em}{0ex}}C=constant.$$

But $f\left(0\right)=1$ implies $C=1$. Hence,

$$f=f\left(t\right)={\left(1-\frac{8}{3}t\right)}^{-\frac{1}{4}}.$$

Therefore, the one-parameter family of 3-forms $\phi \left(t\right)$ given by (11) is the solution of the Laplacian flow of $\phi $ on K, and it exists for every $t\in \left(-\infty ,\frac{3}{8}\right)$.

A simple computation shows that
and so the Lee form $\theta \left(t\right)$ of $\phi \left(t\right)$ is $\theta \left(t\right)={e}^{7}$.

$$d\phi \left(t\right)={f}^{6}\left(-{e}^{1357}+{e}^{1467}+{e}^{2367}+{e}^{2457}\right)={e}^{7}\wedge \phi \left(t\right),$$

Now we study the behavior of the underlying metric $g\left(t\right)$ of such a solution in the limit for $t\to -\infty $. If we think of the Laplacian flow as a one parameter family of ${\mathrm{G}}_{2}$ manifolds with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure, it can be checked that, in the limit, the resulting manifold has vanishing curvature. For $t\in \left(-\infty ,\frac{3}{8}\right)$, let us consider the metric $g\left(t\right)$ on K induced by the ${\mathrm{G}}_{2}$ form $\phi \left(t\right)$ given by (11). Then,

$$\begin{array}{ll}g\left(t\right)& ={\left(1-\frac{8}{3}t\right)}^{-\frac{1}{2}}{\left({e}^{1}\right)}^{2}+{\left(1-\frac{8}{3}t\right)}^{-\frac{1}{2}}{\left({e}^{2}\right)}^{2}+{\left(1-\frac{8}{3}t\right)}^{-\frac{1}{2}}{\left({e}^{3}\right)}^{2}\\ & +{\left(1-\frac{8}{3}t\right)}^{-\frac{1}{2}}{\left({e}^{4}\right)}^{2}+{\left(1-\frac{8}{3}t\right)}^{-2}{\left({e}^{5}\right)}^{2}+{\left(1-\frac{8}{3}t\right)}^{-2}{\left({e}^{6}\right)}^{2}\\ & +{\left(1-\frac{8}{3}t\right)}^{-1}{\left({e}^{7}\right)}^{2}.\end{array}$$

Then, taking into account the symmetry properties of the Riemannian curvature $R\left(t\right)$ we obtain
where ${R}_{ijkl}=R\left(t\right)({e}_{i},{e}_{j},{e}_{k},{e}_{l})$. Therefore, ${lim}_{t\to -\infty}R\left(t\right)=0$.

$$\begin{array}{l}{R}_{1234}={R}_{1256}={R}_{3456}=-\frac{1}{2(1-\frac{8}{3}t)},\\ {R}_{1313}={R}_{1414}={R}_{2323}={R}_{2424}=\frac{3}{4(1-\frac{8}{3}t)},\\ {R}_{1515}={R}_{1616}={R}_{2525}={R}_{2626}={R}_{3535}={R}_{3636}={R}_{4545}={R}_{4646}\\ ={R}_{1324}={R}_{1432}={R}_{1526}={R}_{1652}={R}_{3546}={R}_{3654}=-\frac{1}{4(1-\frac{8}{3}t)},\\ {R}_{ijkl}=0\phantom{\rule{2.em}{0ex}}\mathrm{otherwise},\end{array}$$

Furthermore, the curvatures $R\left(g\left(t\right)\right)$ of $g\left(t\right)$ blow-up as t goes to $\frac{3}{8}$, and the finite-time singularity is of Type I since $R\left(g\left(t\right)\right)=\mathcal{O}{(1-\frac{8}{3}t)}^{-1}$ as $t\to \frac{3}{8}$; in fact,
□

$$\underset{t\to \frac{3}{8}}{lim}\frac{|R\left(g\left(t\right)\right)|}{{(1-\frac{8}{3}t)}^{-1}}<\infty .$$

To complete the proof of Theorem 1, we show that under the conditions (19)–(22) the assumption ${f}_{1}={f}_{3}$, that we made in its proof, is correct.

**Lemma**

**2.**

**Proof.**

Take $u={f}_{1}$ and $v={f}_{3}$. We know that if the 3-form $\phi \left(t\right)$ defined in (13) is the solution for the Laplacian flow (5) of the ${\mathrm{G}}_{2}$ form $\phi $, then the equalities (19)–(22) are satisfied. Now, taking into account (17), the equalities (19)–(22) imply that the Hodge Laplacian ${\Delta}_{t}\phi \left(t\right)$ of $\phi \left(t\right)$ has the following expression

$$\begin{array}{ll}{\Delta}_{t}\phi \left(t\right)& =-\frac{({u}^{2}-{v}^{2}){({u}^{2}+{v}^{2})}^{2}}{2{u}^{2}}{x}^{127}+\frac{({u}^{2}-{v}^{2}){({u}^{2}+{v}^{2})}^{2}}{2{v}^{2}}{x}^{347}+\\ & +{({u}^{2}+{v}^{2})}^{2}\left({x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}\right).\end{array}$$

Thus, for $(i,j,k)\in \left\{\right(1,2,7),(3,4,7\left)\right\}$, the equation ${\Delta}_{ijk}=\frac{{\left({f}_{ijk}\right)}^{\prime}}{{f}_{ijk}}$ of the system (7) becomes in both cases
while for $(i,j,k)\in A\cup B$ with $(1,2,7)\ne (i,j,k)\ne (3,4,7)$, the equation ${\Delta}_{ijk}=\frac{{\left({f}_{ijk}\right)}^{\prime}}{{f}_{ijk}}$ is expressed as

$$\frac{du}{dt}=-\frac{\left({u}^{2}-2{v}^{2}\right){\left({u}^{2}+{v}^{2}\right)}^{3}}{12u{v}^{2}},$$

$$\frac{dv}{dt}=\frac{\left(2{u}^{2}-{v}^{2}\right){\left({u}^{2}+{v}^{2}\right)}^{3}}{12{u}^{2}v}.$$

Therefore, the system (7) becomes

$$\left\{\begin{array}{c}\frac{du}{dt}=-\frac{\left({u}^{2}-2{v}^{2}\right){\left({u}^{2}+{v}^{2}\right)}^{3}}{12u{v}^{2}},\hfill \\ \frac{dv}{dt}=\frac{\left(2{u}^{2}-{v}^{2}\right){\left({u}^{2}+{v}^{2}\right)}^{3}}{12{u}^{2}v},\hfill \\ u\left(0\right)=v\left(0\right)=1.\hfill \end{array}\right.$$

Thus,

$$\frac{dv}{du}=-\frac{v(2{u}^{2}-{v}^{2})}{u({u}^{2}-2{v}^{2})}.$$

To solve this differential equation, we consider the change of variable $w=v/u$. Then, (26) can be expressed as follows:

$$u\frac{dw}{du}+w=-w\frac{2-{w}^{2}}{1-2{w}^{2}}.$$

We solve this differential equation by applying separation of variables, and we get the following solution
for some constant C. This equation is equivalent to
for some constant $\tilde{C}$. Thus, $\tilde{C}=0$ since $u\left(0\right)=v\left(0\right)=1$. Therefore, since $v\left(t\right)={f}_{3}\left(t\right)\ne 0$ for all t, for the functions u and v we have three possibilities: $u=v$, $u=-v$ or $v=0$. But $u\left(0\right)=1=v\left(0\right)$, hence the only possibility is $u\left(t\right)=v\left(t\right)$, that is, ${f}_{1}\left(t\right)={f}_{3}\left(t\right)$. (Here, we would like to note that since $u\left(t\right)=v\left(t\right)$, the second differential equation of the system (25) reduces to $\frac{6}{u}\frac{du}{dt}=4{u}^{4}$, that is the differential Equation (24), which we have solved before.) □

$$lnu+C=-\frac{1}{6}\left(ln\left(1-{w}^{2}\right)+2lnw\right)=\frac{1}{6}ln\frac{{v}^{2}\left({u}^{2}-{v}^{2}\right)}{{u}^{4}},$$

$$\tilde{C}{u}^{2}={v}^{2}\left({u}^{2}-{v}^{2}\right),$$

**Remark**

**1.**

Note that proceeding in a similar way as Lauret did in [40] for the Ricci flow, we can evolve the Lie brackets $\mu \left(t\right)$ instead of the 3-form defining the ${\mathrm{G}}_{2}$-structure, and we can show that the corresponding bracket flow has a solution for every t. In fact, if we fix on ${\mathbb{R}}^{7}$ the 3-form ${x}^{127}+{x}^{347}+{x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}$, the basis $\{{x}_{1}\left(t\right),\dots ,{x}_{7}\left(t\right)\}$ defines, for every real number $t\in \left(-\infty ,\frac{3}{8}\right)$, a solvable Lie algebra with bracket $\mu \left(t\right)$ such that $\mu \left(0\right)$ is the Lie bracket of the Lie algebra k of K. Moreover, the solution of the bracket flow converges to the null bracket corresponding to the abelian Lie algebra as t goes to $-\infty $, and it blows-up as t goes to $\frac{3}{8}$.

**Remark**

**2.**

Taking into account (4) and (11), one can check that the torsion form ${\tau}_{2}\left(t\right)$ of $\phi \left(t\right)$ is given by

$${\tau}_{2}\left(t\right)=\frac{4}{3}{\left(1-{\textstyle \frac{8}{3}}t\right)}^{-1}\left({e}^{12}+{e}^{34}\right)-\frac{8}{3}{\left(1-{\textstyle \frac{8}{3}}t\right)}^{-5/2}{e}^{56}.$$

Thus, ${lim}_{t\to -\infty}{\tau}_{2}\left(t\right)=0$. However, the solution $\phi \left(t\right)$ does not converge to a locally conformal parallel ${\mathrm{G}}_{2}$-structure as t goes to $-\infty $ since, by (11), the ${\mathrm{G}}_{2}$ forms $\phi \left(t\right)$ degenerate when $t\to -\infty $. Moreover, $\phi \left(t\right)$ blows-up as t goes to $\frac{3}{8}$.

#### 4.2. The Laplacian Flow on S

Now we consider the simply connected and solvable Lie group S whose Lie algebra s is defined as follows:

$$s=\left(\frac{1}{2}{e}^{17},\frac{1}{2}{e}^{27},\frac{1}{2}{e}^{37},\frac{1}{2}{e}^{47},{e}^{14}+{e}^{23}+{e}^{57},{e}^{13}-{e}^{24}+{e}^{67},0\right).$$

Then, the 3-form $\phi $ given by
defines a left invariant locally conformal calibrated ${\mathrm{G}}_{2}$-structure on the Lie group S, with Lee form $\theta =-{e}^{7}$, and so with torsion form ${\tau}_{1}=-\frac{1}{3}{e}^{7}$. In fact,

$$\phi ={e}^{127}+{e}^{347}+{e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245}$$

$$d\phi ={e}^{1357}-{e}^{1467}-{e}^{2367}-{e}^{2457}=-{e}^{7}\wedge \phi .$$

Since S is a nonunimodular Lie group, S cannot admit a lattice $\mathsf{\Gamma}$ such that the quotient space $\mathsf{\Gamma}\setminus S$ is a compact solvmanifold. In fact, the linear map $s\to \mathbb{R}$, $X\to tr(adX)$ is such that $tr(ad{e}_{7})$ is non-zero, where $\{{e}_{1},\cdots ,{e}_{7}\}$ is the basis of s dual to the basis $\{{e}^{1},\cdots ,{e}^{7}\}$ of ${s}^{*}$.

**Theorem**

**2.**

The family of locally conformal calibrated ${\mathrm{G}}_{2}$-structures $\phi \left(t\right)$ on S given by
is the solution for the Laplacian flow (5) of the ${\mathrm{G}}_{2}$ form φ given by (28), where $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$. The Lee form $\theta \left(t\right)$ of $\phi \left(t\right)$ is $\theta \left(t\right)=-{e}^{7}$. Moreover, the underlying metrics $g\left(t\right)$ of this solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in S, as t goes to $-\infty $, and they blow-up as t goes to $\frac{1}{4}$.

$$\phi \left(t\right)={(1-4t)}^{3/4}\phantom{\rule{0.166667em}{0ex}}{e}^{127}+{(1-4t)}^{3/4}\phantom{\rule{0.166667em}{0ex}}{e}^{347}+{e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245}$$

**Proof.**

To study the flow (5) of the ${\mathrm{G}}_{2}$ form $\phi $ defined in (28), we should proceed as in Theorem 1. However, in order to short the proof, we will show directly that the one-parameter family of ${\mathrm{G}}_{2}$-structures given by (29) is the solution for the flow (5). For this, we consider the differentiable real functions ${f}_{i}={f}_{i}\left(t\right)$$(i=1,\cdots ,7)$ given by

$$\begin{array}{l}{f}_{i}\left(t\right)={\left(1-4t\right)}^{1/8},\phantom{\rule{2.em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i=1,2,3,4,\\ {f}_{5}\left(t\right)={f}_{6}\left(t\right)={\left(1-4t\right)}^{-1/4},\\ {f}_{7}\left(t\right)={\left(1-4t\right)}^{1/2}.\end{array}$$

These functions are defined for all $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$; moreover, ${f}_{i}\left(t\right)>0$, for $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$.

Now, for each $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$, we consider the basis $\{{x}^{1},\cdots ,{x}^{7}\}$ of left invariant 1-forms on S defined by
$${x}^{i}={x}^{i}\left(t\right)={f}_{i}\left(t\right){e}^{i},\phantom{\rule{1.em}{0ex}}1\le i\le 7.$$

Taking into account (30) and (27), the structure equations of S with respect to the basis $\{{x}^{1},\cdots ,{x}^{7}\}$ are

$$\begin{array}{ll}d{x}^{1}=\frac{1}{2}{\left(1-4t\right)}^{-1/2}{x}^{17},& d{x}^{2}=\frac{1}{2}{\left(1-4t\right)}^{-1/2}{x}^{27},\\ d{x}^{3}=\frac{1}{2}{\left(1-4t\right)}^{-1/2}{x}^{37},& d{x}^{4}=\frac{1}{2}{\left(1-4t\right)}^{-1/2}{x}^{47},\\ d{x}^{5}={\left(1-4t\right)}^{-1/2}\left({x}^{14}+{x}^{23}+{x}^{57}\right),& d{x}^{6}={\left(1-4t\right)}^{-1/2}\left({x}^{13}-{x}^{24}+{x}^{67}\right),\\ d{x}^{7}=0.& \end{array}$$

For any $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$, we consider the 3-form $\phi \left(t\right)$ on S given by

$$\phi \left(t\right)={x}^{127}+{x}^{347}+{x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}.$$

Then, this 3-form $\phi \left(t\right)$ defines a ${\mathrm{G}}_{2}$-structure on S, and it is equal to the 3-form $\phi \left(t\right)$ defined in (29). Note that the 3-form $\phi \left(t\right)$ is such that $\phi \left(0\right)=\phi $ and, for any t, $\phi \left(t\right)$ determines the metric $g\left(t\right)$ on S such that the basis $\{{x}_{i}=\frac{1}{{f}_{i}}{e}_{i};\phantom{\rule{4pt}{0ex}}i=1,\cdots ,7\}$ of left invariant vector fields on S dual to $\{{x}^{1},\cdots ,{x}^{7}\}$ is orthonormal. So, $g\left(t\right)({e}_{i},{e}_{i})={f}_{i}^{2}$.

Moreover, for every $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$, $\phi \left(t\right)$ defines a locally conformal calibrated ${\mathrm{G}}_{2}$-structure on S. In fact,
since on the right-hand side of (29) the terms ${e}^{127}$ and ${e}^{347}$ are both closed and $d\left({e}^{567}+{e}^{135}-{e}^{146}-{e}^{236}-{e}^{245}\right)={e}^{1357}-{e}^{1467}-{e}^{2367}-{e}^{2457}$. So, the Lee form $\theta \left(t\right)$ of $\phi \left(t\right)$ is $\theta \left(t\right)=-{e}^{7}$.

$$d\phi \left(t\right)={e}^{1357}-{e}^{1467}-{e}^{2367}-{e}^{2457}=-{e}^{7}\wedge \phi \left(t\right),$$

Next, we show that $\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phi \left(t\right)=({\star}_{t}d{\star}_{t}d-d{\star}_{t}d{\star}_{t})\phi \left(t\right)$. Using (31) and (32), we obtain

$$\frac{d}{dt}\phi \left(t\right)=-3{(1-4t)}^{-1}\left({x}^{127}+{x}^{347}\right).$$

On the other hand, we have
and

$$\left({\star}_{t}d{\star}_{t}d\right)\phi \left(t\right)=-4{(1-4t)}^{-1}{x}^{567}-2{(1-4t)}^{-1}\left({x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}\right),$$

$$\begin{array}{ll}(-d{\star}_{t}d{\star}_{t})\phi \left(t\right)=& -3{(1-4t)}^{-1}\left({x}^{127}+{x}^{347}\right)+4{(1-4t)}^{-1}{x}^{567}\\ & +2{(1-4t)}^{-1}\left({x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}\right).\end{array}$$

Therefore, (33), (34) and (35) imply $\frac{d}{dt}\phi \left(t\right)={\Delta}_{t}\phi \left(t\right)$.

To complete the proof, we study the behavior of the underlying metrics of such a solution in the limit for $t\to -\infty $. If we think of the Laplacian flow as a one parameter family of ${\mathrm{G}}_{2}$ manifolds with a locally conformal calibrated ${\mathrm{G}}_{2}$-structure, it can be checked that, in the limit, the resulting manifold has vanishing curvature. Denote by $g\left(t\right)$, $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$, the metric on S induced by the ${\mathrm{G}}_{2}$ form $\phi \left(t\right)$ given by (29). Then, $g\left(t\right)$ has the following expression

$$\begin{array}{ll}g\left(t\right)& ={\left(1-4t\right)}^{\frac{1}{4}}{\left({e}^{1}\right)}^{2}+{\left(1-4t\right)}^{\frac{1}{4}}{\left({e}^{2}\right)}^{2}+{\left(1-4t\right)}^{\frac{1}{4}}{\left({e}^{3}\right)}^{2}+{\left(1-4t\right)}^{\frac{1}{4}}{\left({e}^{4}\right)}^{2}\\ & +{\left(1-4t\right)}^{-\frac{1}{2}}{\left({e}^{5}\right)}^{2}+{\left(1-4t\right)}^{-\frac{1}{2}}{\left({e}^{6}\right)}^{2}+\left(1-4t\right){\left({e}^{7}\right)}^{2}.\end{array}$$

Now, one can check that every non-vanishing coefficient appearing in the expression of the Riemannian curvature $R\left(g\right(t\left)\right)$ of $g\left(t\right)$ is proportional to $\frac{1}{(1-4t)}$. Therefore, ${lim}_{t\to -\infty}R\left(t\right)=0$.

Furthermore, the curvatures $R\left(g\left(t\right)\right)$ of $g\left(t\right)$ blow-up as t goes to $\frac{1}{4}$, and the finite-time singularity is of Type I since $R\left(g\left(t\right)\right)=\mathcal{O}{(1-4t)}^{-1}$ as $t\to \frac{1}{4}$; in fact
□

$$\underset{t\to \frac{1}{4}}{lim}\frac{|R\left(g\left(t\right)\right)|}{{(1-4t)}^{-1}}<\infty .$$

**Remark**

**3.**

As we have noticed in Remark 1, we can also evolve the Lie brackets $\nu \left(t\right)$ instead of the 3-form defining the left invariant ${\mathrm{G}}_{2}$-structure on S, and we can show that the corresponding bracket flow has a solution for every $t\in \left(-\infty ,\frac{1}{4}\right)$. In fact, if we fix on ${\mathbb{R}}^{7}$ the 3-form ${x}^{127}+{x}^{347}+{x}^{567}+{x}^{135}-{x}^{146}-{x}^{236}-{x}^{245}$, the basis $\{{x}_{1}\left(t\right),\dots ,{x}_{7}\left(t\right)\}$ defines, for every real number $t\in \left(-\infty ,\frac{1}{4}\right)$, a solvable Lie algebra with bracket $\nu \left(t\right)$ such that $\nu \left(0\right)$ is the Lie bracket of the Lie algebra s of S. As for the Lie group K (see Remark 1), the solution of the bracket flow converges to the null bracket corresponding to the abelian Lie algebra as t goes to $-\infty $, and it blows-up as t goes to $\frac{1}{4}$.

**Remark**

**4.**

Taking into account (4) and (29), one can check that the torsion form ${\tau}_{2}\left(t\right)$ of $\phi \left(t\right)$ is given by

$${\tau}_{2}\left(t\right)=\frac{5}{3}{(1-4t)}^{-1/4}\left({e}^{12}+{e}^{34}\right)-\frac{10}{3}{(1-4t)}^{-1}{e}^{56}.$$

Thus, ${lim}_{t\to -\infty}{\tau}_{2}\left(t\right)=0$. However, the solution $\phi \left(t\right)$ does not converge to a locally conformal parallel ${\mathrm{G}}_{2}$-structure as t goes to $-\infty $ since, by (29), the ${\mathrm{G}}_{2}$ forms $\phi \left(t\right)$ blow-up when $t\to -\infty $, and $\phi \left(t\right)$ degenerate as t goes to $\frac{1}{4}$. Note that the metrics behaves differently for S than for K. Indeed, the induced metrics by the solution of the Laplacian flow on S blow-up at infinity and at the finite time, while the induced metrics by the solution of the Laplacian flow on K only blow-up as t goes to $\frac{3}{8}$.

**Remark**

**5.**

Note that, for every $t\in \left(-\infty ,{\textstyle \frac{1}{4}}\right)$, the metric $g\left(t\right)$ is an Einstein metric with negative scalar curvature on the Lie group S. In fact, with respect to the orthonormal basis $\{{x}_{1}\left(t\right),\cdots ,{x}_{7}\left(t\right)\}$, we have

$$Ric\left(g\left(t\right)\right)=-\frac{3}{1-4t}g\left(t\right)=-\frac{3}{1-4t}\sum _{1\le i\le 7}{\left({x}^{i}\right)}^{2}.$$

## Author Contributions

The three authors have contributed equally to the realization and writing of this article.

## Funding

The first and third authors were partially supported by MINECO-FEDER Grant MTM2014-54804-P and Gobierno Vasco Grant IT1094-16, Spain. The second author was partially supported by the project MTM2017-85649-P (AEI/Feder, UE) and Gobierno de Aragón/Fondo Social Europeo—Grupo Consolidado E22-17R Algebra y Geometría.

## Acknowledgments

We are grateful to the anonymous referees for useful comments and improvements. Moreover, we would like to thank Guest Editor.

## Conflicts of Interest

The authors declare no conflict of interest.

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Class | Type | Conditions |
---|---|---|

${\mathcal{X}}_{0}$ | parallel | ${\tau}_{0},{\tau}_{1},{\tau}_{2},{\tau}_{3}=0$ |

${\mathcal{X}}_{2}$ | closed, calibrated | ${\tau}_{0},{\tau}_{1},{\tau}_{3}=0$ |

${\mathcal{X}}_{4}$ | locally conformal parallel | ${\tau}_{0},{\tau}_{2},{\tau}_{3}=0$ |

${\mathcal{X}}_{2}\oplus {\mathcal{X}}_{4}$ | locally conformal calibrated | ${\tau}_{0},{\tau}_{3}=0$ |

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