The Laplacian Flow of Locally Conformal Calibrated G₂-Structures

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.

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

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

(1)

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

$$g_{\phi }(X,Y)d{V}_{g_{\phi }}=\frac{1}{6}{i}_X\phi \wedge i_Y\phi \wedge \phi ,$$

for any pair of vector fields $X,Y$ on M, where $i_X$ denotes the contraction by X.

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

$$d\phi =\theta \wedge \phi ,d\left({\star }_{\phi }\phi \right)=\frac{4}{3}\theta \wedge {\star }_{\phi }\phi ,$$

(2)

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.

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

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

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.

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

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta }_t\phi \left(t\right),\hfill \\ d\phi \left(t\right)=0,\hfill \\ \phi \left(0\right)=\phi ,\hfill \end{array}\right.$$

where $\phi \left(t\right)$ is a closed ${\mathrm{G}}_2$ form on M, and ${\Delta }_t=d{d}^{*}+d^{*}d$ 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].

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 $t\in [0,ϵ]$, for some $ϵ>0$. 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

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta }_t\phi \left(t\right),\hfill \\ d\phi \left(t\right)=\theta \left(t\right)\wedge \phi \left(t\right),\hfill \\ \phi \left(0\right)=\phi .\hfill \end{array}\right.$$

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

$$\begin{array}{l}\left\{\begin{array}{l}d\phi ={\tau }_0\psi +3{\tau }_1\wedge \phi +{\star }_{\phi }{\tau }_3,\\ d\psi =4{\tau }_1\wedge \psi -{\star }_{\phi }{\tau }_2,\end{array}\right.\end{array}$$

(3)

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

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

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

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. Some classes of ${\mathrm{G}}_2$-structures.

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	${\tau }_0,{\tau }_3=0$

.

Note that if a manifold M has a locally conformal calibrated ${\mathrm{G}}_2$-structure $\phi$, then

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

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:

$$\begin{array}{l}{\tau }_1=-\frac{1}{12}{\star }_{\phi }\left({\star }_{\phi }d\phi \wedge \phi \right),\\ {\tau }_2={\star }_{\phi }\left(4{\tau }_1\wedge \left({\star }_{\phi }\phi \right)-d{\star }_{\phi }\phi \right).\end{array}$$

(4)

3. The Laplacian Flow of Locally Conformal Calibrated G₂-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

$$\left\{\begin{array}{c}\frac{d}{dt}\phi \left(t\right)={\Delta }_t\phi \left(t\right),\hfill \\ d\phi \left(t\right)=\theta \left(t\right)\wedge \phi \left(t\right),\hfill \\ \phi \left(0\right)=\phi ,\hfill \end{array}\right.$$

(5)

where $\theta \left(t\right)$ is the Lee form of $\phi \left(t\right)$, and ${\Delta }_t=d{d}^{*}+d^{*}d$ is the Hodge Laplacian operator associated with the metric $g\left(t\right)=g_{\phi \left(t\right)}$ induced by the 3-form $\phi \left(t\right)$.

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

$$x^i=x^i\left(t\right)=f_i\left(t\right)e^i,1\le i\le 7.$$

We consider the one-parameter family of left invariant ${\mathrm{G}}_2$-structures $\phi \left(t\right)$ on G given by

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

(6)

where $f_{ijk}=f_{ijk}\left(t\right)$ stands for the product $f_i\left(t\right)f_j\left(t\right)f_k\left(t\right)$.

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

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

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

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

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

$$\begin{array}{c}\left\{\begin{array}{ll}{\Delta }_{ijk}=\frac{{\left(f_{ijk}\right)}^{\prime }}{f_{ijk}}& \mathrm{if}(i,j,k)\in A\cup B,\\ {\Delta }_{lmn}=0& \mathrm{if}1\le l<m<n\le 7\mathrm{and}(l,m,n)\notin A\cup B,\end{array}\right.\end{array}$$

(7)

that is,

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

(8)

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

$$ln(f_{ijk}·f_{pqr})=C,$$

for some constant C. But $f_i\left(0\right)=1$, for all $i=1,\dots ,7$, imply that $C=0$, that is

$$f_{ijk}·f_{pqr}=1.$$

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

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

(9)

where d denotes the Chevalley-Eilenberg differential on $k^{*}$.

The 3-form $\phi$ on K given by

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

(10)

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,

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

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

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

(11)

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 $\frac{3}{8}$.

Proof. 

As in Section 2, 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 left invariant 1-forms on K defined by

$$x^i=x^i\left(t\right)=f_i\left(t\right)e^i,1\le i\le 7.$$

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

$$\begin{array}{l}d{x}^1=\frac{f_1}{f_{37}}{x}^{37},d{x}^2=\frac{f_2}{f_{47}}{x}^{47},d{x}^3=-\frac{f_3}{f_{17}}{x}^{17},d{x}^4=-\frac{f_4}{f_{27}}{x}^{27},\\ d{x}^5=\frac{f_5}{f_{14}}{x}^{14}+\frac{f_5}{f_{23}}{x}^{23},d{x}^6=\frac{f_6}{f_{13}}{x}^{13}-\frac{f_6}{f_{24}}{x}^{24},d{x}^7=0.\end{array}$$

(12)

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

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

(13)

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 $\{x_i=\frac{1}{f_i}{e}_i;i=1,\cdots ,7\}$ 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

$$\begin{array}{ll}{\star }_{t}d{\star }_{t}d\phi \left(t\right)& =-\frac{\left(f_1{f}_4-f_2{f}_3\right)\left(f_2{f}_3+f_1{f}_4\right)f_5}{f_1{f}_2{f}_3^2{f}_4^2{f}_7}{x}^{126}-\frac{\left(f_1{f}_4-f_2{f}_3\right)\left(f_1^2{f}_2^2+f_3^2{f}_4^2\right)}{f_1{f}_2^2{f}_3^2{f}_4{f}_7^2}{x}^{146}\hfill \\ & -\frac{\left(f_2{f}_3-f_1{f}_4\right)\left(f_1^2{f}_2^2+f_3^2{f}_4^2\right)}{f_1^2{f}_2{f}_3{f}_4^2{f}_7^2}{x}^{236}+\frac{\left(f_1{f}_4-f_2{f}_3\right)\left(f_2{f}_3+f_1{f}_4\right)f_5}{f_1^2{f}_2^2{f}_3{f}_4{f}_7}{x}^{346}\\ & +\frac{\left(f_2^2{f}_3^2{f}_5^2+f_1^2{f}_4^2{f}_5^2+f_1^2{f}_3^2{f}_6^2+f_2^2{f}_4^2{f}_6^2\right)}{f_1^2{f}_2^2{f}_3^2{f}_4^2}{x}^{567},\end{array}$$

(14)

and, on the other hand, we obtain

$$\begin{array}{ll}d{\star }_{t}d{\star }_t\phi \left(t\right)& =\frac{\left(f_1{f}_2-f_3{f}_4\right)\left(f_2^2{f}_3^2+f_1^2{f}_4^2\right)}{f_1^2{f}_2^2{f}_3{f}_4{f}_7^2}{x}^{127}-\frac{f_6\left(f_2{f}_3{f}_5+f_1{f}_4{f}_5+f_1{f}_3{f}_6+f_2{f}_4{f}_6\right)}{f_1^2{f}_2{f}_3^2{f}_4}{x}^{135}\\ & +\frac{f_5\left(f_2{f}_3{f}_5+f_1{f}_4{f}_5+f_1{f}_3{f}_6+f_2{f}_4{f}_6\right)}{f_1^2{f}_2{f}_3{f}_4^2}{x}^{146}\\ & +\frac{f_5\left(f_2{f}_3{f}_5+f_1{f}_4{f}_5+f_1{f}_3{f}_6+f_2{f}_4{f}_6\right)}{f_1{f}_2^2{f}_3^2{f}_4}{x}^{236}\\ & +\frac{f_6\left(f_2{f}_3{f}_5+f_1{f}_4{f}_5+f_1{f}_3{f}_6+f_2{f}_4{f}_6\right)}{f_1{f}_2^2{f}_3{f}_4^2}{x}^{245}-\frac{\left(f_1{f}_2-f_3{f}_4\right)\left(f_2^2{f}_3^2+f_1^2{f}_4^2\right)}{f_1{f}_2{f}_3^2{f}_4^2{f}_7^2}{x}^{347}.\end{array}$$

(15)

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

$${\Delta }_{126}=\frac{f_5}{f_7}\left(\frac{f_2}{f_1{f}_4^2}-\frac{f_1}{f_2{f}_3^2}\right),$$

and

$${\Delta }_{346}=\frac{f_5}{f_7}\left(\frac{f_4}{f_2^2{f}_3}-\frac{f_3}{f_1^2{f}_4}\right).$$

Each of these equalities implies that $f_{14}^2=f_{23}^2$, and so

$$f_{14}=f_{23}$$

(16)

since $f_i=f_i\left(t\right)>0$.

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

$$\begin{array}{ll}{\Delta }_{127}=-\frac{f_3}{f_1}{B}_{23}+\frac{f_4}{f_2}{B}_{14},& {\Delta }_{347}=\frac{f_2}{f_4}{B}_{23}-\frac{f_1}{f_3}{B}_{14},\\ {\Delta }_{135}=\frac{f_6}{f_{13}}A,& {\Delta }_{245}=\frac{f_6}{f_{24}}A,\\ {\Delta }_{146}=\frac{f_5}{f_{14}}A-\frac{f_1}{f_3}{B}_{12}+\frac{f_4}{f_2}{B}_{34},& {\Delta }_{236}=\frac{f_5}{f_{23}}A+\frac{f_2}{f_4}{B}_{12}-\frac{f_3}{f_1}{B}_{34},\\ {\Delta }_{567}=A_2,\end{array}$$

(17)

where

$$\begin{array}{ll}A=f_5\left(\frac{1}{f_{23}}+\frac{1}{f_{14}}\right)+f_6\left(\frac{1}{f_{13}}+\frac{1}{f_{24}}\right),& A_2=f_5^2\left(\frac{1}{f_{23}^2}+\frac{1}{f_{14}^2}\right)+f_6^2\left(\frac{1}{f_{13}^2}+\frac{1}{f_{24}^2}\right),\\ B_{12}=\frac{1}{f_7^2}\left(\frac{f_2}{f_4}-\frac{f_1}{f_3}\right),& B_{34}=\frac{1}{f_7^2}\left(\frac{f_4}{f_2}-\frac{f_3}{f_1}\right),\\ B_{23}=\frac{1}{f_7^2}\left(\frac{f_2}{f_4}-\frac{f_3}{f_1}\right),& B_{14}=\frac{1}{f_7^2}\left(\frac{f_4}{f_2}-\frac{f_1}{f_3}\right).\end{array}$$

(18)

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

$$f_1=f_2,f_3=f_4.$$

(19)

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

$$f_5=f_6.$$

(20)

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,

$$f_7=\frac{2}{(f_1^2+f_3^2)}.$$

(21)

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

$$f_6=\frac{1}{2}{f}_{13}(f_1^2+f_3^2).$$

(22)

In summary, from (19)–(22), we have

$$f_1=f_2,f_3=f_4,f_5=f_6=\frac{1}{2}{f}_{13}(f_1^2+f_3^2),f_7=\frac{2}{f_1^2+f_3^2}.$$

Now, we can suppose that $f_3=f_1=f$ (see below Lemma 2). Then, the previous conditions reduce to

$$f_1=f_2=f_3=f_4=f,f_5=f_6=f^4,f_7=f^{-2}.$$

(23)

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

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

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

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

Integrating this equation, we obtain

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

(24)

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

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

and so the Lee form $\theta \left(t\right)$ of $\phi \left(t\right)$ is $\theta \left(t\right)=e^7$.

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

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

where $R_{ijkl}=R\left(t\right)(e_i,e_j,e_k,e_l)$. Therefore, ${lim}_{t\to -\infty }R\left(t\right)=0$.

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.

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 φ given by (10), then $f_1\left(t\right)=f_3\left(t\right)$.

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

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

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

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

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

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

Therefore, the system (7) becomes

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

(25)

Thus,

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

(26)

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

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

We solve this differential equation by applying separation of variables, and we get the following solution

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

for some constant C. This equation is equivalent to

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

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 $\frac{6}{u}\frac{du}{dt}=4u^4$, that is the differential Equation (24), which we have solved before.) □

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

$${\tau }_2\left(t\right)=\frac{4}{3}{\left(1-\frac{8}{3}t\right)}^{-1}\left(e^{12}+e^{34}\right)-\frac{8}{3}{\left(1-\frac{8}{3}t\right)}^{-5/2}{e}^{56}.$$

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:

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

(27)

Then, the 3-form $\phi$ given by

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

(28)

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,

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

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

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

(29)

is the solution for the Laplacian flow (5) of the ${\mathrm{G}}_2$ form φ given by (28), where $t\in \left(-\infty ,\frac{1}{4}\right)$. 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 $\frac{1}{4}$.

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

$$\begin{array}{l}{f}_i\left(t\right)={\left(1-4t\right)}^{1/8},i=1,2,3,4,\\ f_5\left(t\right)=f_6\left(t\right)={\left(1-4t\right)}^{-1/4},\\ f_7\left(t\right)={\left(1-4t\right)}^{1/2}.\end{array}$$

(30)

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

Now, for each $t\in \left(-\infty ,\frac{1}{4}\right)$, we consider the basis $\{x^1,\cdots ,x^7\}$ of left invariant 1-forms on S defined by

$$x^i=x^i\left(t\right)=f_i\left(t\right)e^i,1\le i\le 7.$$

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

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

(31)

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

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

(32)

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 $\{x_i=\frac{1}{f_i}{e}_i;i=1,\cdots ,7\}$ 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 $t\in \left(-\infty ,\frac{1}{4}\right)$, $\phi \left(t\right)$ defines a locally conformal calibrated ${\mathrm{G}}_2$-structure on S. In fact,

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

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 $\theta \left(t\right)=-e^7$.

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

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

(33)

On the other hand, we have

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

(34)

and

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

(35)

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 $g\left(t\right)$, $t\in \left(-\infty ,\frac{1}{4}\right)$, 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 $t\in \left(-\infty ,\frac{1}{4}\right)$, 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

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