Explicit Soliton for the Laplacian Co-Flow on a Solvmanifold

TL;DR

This paper constructs an explicit soliton solution for the Laplacian co-flow of invariant G2-structures on a specific 7-dimensional solvmanifold, advancing understanding of geometric flows on homogeneous spaces.

Contribution

It provides the first explicit soliton solution for the Laplacian co-flow on a particular almost Abelian 7-manifold, applying Lauret's Ansatz to homogeneous spaces.

Findings

Explicit soliton solution found for the Laplacian co-flow on a solvmanifold.

Demonstrates the applicability of Lauret's Ansatz to invariant G2-structures.

Advances the study of geometric flows on homogeneous spaces.

Abstract

We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant $G_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian $7$-manifold.

Equations

$$g_{\phi }(X,Y)\cdot d\mathrm{Vol}:=(X┘\phi )\wedge (Y┘\phi )\wedge \phi .$$

$$\frac{\partial {\psi }_t}{\partial t}=-{\Delta }_{{\psi }_t}{\psi }_t,$$

$${\Omega }^1$$

$${\Omega }^6$$

$${\Omega }^2\oplus {\mathrm{S}}^2(T^{\ast }M)={\Omega }^1\otimes {\Omega }_7^3\cong \mathrm{End}(TM)=\mathfrak{so}(TM)\oplus \mathrm{sym}(TM).$$

$$[{\mathbb{R}}^7]\cong {\Omega }_7^2\text{and}{\mathfrak{g}}_2\cong {\Omega }_{14}^2.$$

$$i(h)=\frac{1}{2}{h}_{il}{g}^{lm}{\phi }_{mjk}d{x}^{ijk}\text{and}j(\eta )(u,v)=\ast ((u┘\phi )\wedge (v┘\phi )\wedge \eta ),$$

$$\mathbb{R}{g}_{\phi }\cong {\Omega }_1^3\text{and}{\mathrm{S}}_0^2({\mathrm{T}}^{\ast }M)\cong {\Omega }_{27}^3.$$

$$d\phi ={\tau }_0\psi +3{\tau }_1\wedge \phi +\ast {\tau }_3\text{and}d\psi =4{\tau }_1\wedge \psi +\ast {\tau }_2,$$

$${\nabla }_l{\phi }_{abc}=:T_{lm}{g}^{mn}{\psi }_{nabc},$$

$$T=\frac{{\tau }_0}{4}{g}_{\phi }-{\tau }_{27}+({\tau }_1{)}^{♯}┘\phi -\frac{1}{2}{\tau }_2,$$

$${\Delta }_{\psi }\psi =d{d}^{\ast }\psi =d{\tau }_0\wedge \phi +{\tau }_0^2\psi +{\tau }_0\ast {\tau }_3+d{\tau }_3.$$

$$\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}\text{such that}\mathrm{Ad}(h)\mathfrak{m}\subset \mathfrak{m},\forall h\in H,$$

$$\frac{\partial }{\partial t}{\gamma }_t=q({\gamma }_t).$$

$$\frac{d}{dt}{\gamma }_t=q({\gamma }_t).$$

$$h\cdot {\psi }_0:=(h^{-1}{)}^{\ast }{\psi }_0={\psi }_0(h^{-1}\cdot ,h^{-1}\cdot ,h^{-1}\cdot ,h^{-1}\cdot ),h\in \mathrm{Gl}(\mathfrak{g}).$$

$$\theta (\mathfrak{gl}(\mathfrak{g})){\psi }_0={\Lambda }^4(\mathfrak{g}{)}^{\ast },$$

$$G_{{\psi }_0}:=\{h\in \mathrm{Gl}(\mathfrak{g});h\cdot {\psi }_0={\psi }_0\}$$

$${\mathfrak{g}}_{{\psi }_0}:=\mathrm{Lie}(G_{{\psi }_0})=\{A\in \mathfrak{gl}(\mathfrak{g});\theta (A){\psi }_0=0\}.$$

$$\theta ({\mathfrak{q}}_{{\psi }_0}){\psi }_0={\Lambda }^4(\mathfrak{g}{)}^{\ast }.$$

$$G_{\psi }=G_{h\cdot {\psi }_0}=h^{-1}{G}_{{\psi }_0}h\text{and}{\mathfrak{g}}_{\psi }={\mathfrak{g}}_{h\cdot {\psi }_0}=\mathrm{Ad}(h^{-1}){\mathfrak{g}}_{{\psi }_0},$$

$$\ast =(h^{-1}{)}^{\ast }{\ast }_0{h}^{\ast }\text{and}{h}^{\ast }\circ \Delta ={\Delta }_0\circ h^{\ast },$$

$$\alpha \wedge \ast \alpha =$$

$$=$$

$$\ast \psi =(h^{-1}{)}^{\ast }{\ast }_0{h}^{\ast }\psi =(h^{-1}{)}^{\ast }{\ast }_0{\psi }_0=h\cdot {\phi }_0=\phi .$$

$$\theta (Q_{\psi })\psi =$$

$$=$$

$$Q_{\psi }=\mathrm{Ad}(h)Q_0.$$

$$\frac{d}{dt}{\psi }_t=-{\Delta }_t{\psi }_t.$$

$$\frac{d}{dt}{\psi }_t=$$

Full text

1. 1 Torsion forms of a $\mathrm{G}_{2}$-structure
2. 2 Invariant $\mathrm{G}_{2}$-structures on Lie groups
3. 3 Lie bracket flow
4. 4 Self Similar Solutions
5. 5 Example of a co-flow soliton

Introduction

Geometric flows in $\mathrm{G}_{2}$-geometry were first outlined by the seminal works of Bryant [Bryan2] and Hitchin [Hitchin2001], and have since been studied by several authors, e.g. [Bryan, Fino, Grigorian, Karigianis, Lauret]. These so-called $\mathrm{G}_{2}$-flows arise as a tool in the search for ultimately torsion-free $\mathrm{G}_{2}$-structures, by varying a non-degenerate $3$-form on an oriented and spin $7$–manifold $M$ towards some $\varphi\in\Omega^{3}:=\Omega^{3}(M)$ such that the *torsion * $\nabla^{g_{\varphi}}\varphi$ vanishes, where $g_{\varphi}$ is the natural Riemannian metric defined from $\varphi$ by

[TABLE]

Such pairs $(M^{7},\varphi)$ solving the nonlinear PDE problem $\nabla^{g_{\varphi}}\varphi\equiv 0$ are called $\mathrm{G}_{2}$–manifolds and are very difficult to construct, especially when $M$ is required to be compact. To this date, all known solutions stem from elaborate constructions in geometric analysis [Joyce, corti2015, Eguchi-Hanson].

Some weaker formulations of that problem can be obtained from the classical fact, first established by Fernández and Gray [Fernandez], that the torsion-free condition is equivalent to $\varphi$ being both closed and coclosed, in the sense that $d\varphi=0$ and $d\ast_{\varphi}\varphi=0$, respectively, and thus one may study each of these conditions separately. For instance, Grigorian [Grigorian] and Karigiannis and Tsui [Karigianis] considered the Laplacian co-flow of $\mathrm{G}_{2}$-structures $\{\varphi_{t}\}$ defined by

[TABLE]

where $\psi_{t}:=\ast_{t}\varphi_{t}$ is the Hodge dual and $\Delta_{\psi_{t}}\psi_{t}:=(dd^{\ast_{t}}+d^{\ast_{t}}d)\psi_{t}$ is the Hodge Laplacian of the metric $g_{\varphi_{t}}$ on $4$–forms. It is a natural process to consider among coclosed $\mathrm{G}_{2}$–structures, as it manifestly preserves that property, i.e., it flows $\psi_{t}$ in its de Rham cohomology class. Moreover, it is the gradient flow of Hitchin’s volume functional [Hitchin2001].

When $M^{7}=G$ is a Lie group, we propose to study this flow from the perspective introduced by Lauret [Lauret] in the general context of geometric flows on homogeneous spaces. As a proof of principle, we apply a natural Ansatz to construct an example of invariant self-similar solution, or soliton, of the Laplacian co-flow. Solitons are $\mathrm{G}_{2}$–structures which, under the flow, simply scale monotonically and move by diffeomorphisms. In particular, they provide potential models for singularities of the flow, as well as means for desingularising certain singular $\mathrm{G}_{2}$–structures, both of which are key aspects of any geometric flow. We follow in spirit the approach of Karigiannis et al. [Karigianis] to obtain solitons to the Laplacian coflow from a general Ansatz for a coclosed cohomogeneity one $\mathrm{G}_{2}$–structure on manifolds of the form $M^{7}=N^{6}\times L^{1}$, where $L^{1}=\mathbb{R}$ or $S^{1}$ and $N^{6}$ is compact and either nearly Kähler or a Calabi-Yau $3$-fold. In that case, as in ours, the symmetries of the space are exploited to reduce the soliton condition to a manageable ODE.

1. Torsion forms of a $\mathrm{G}_{2}$-structure

Let us briefly review some elementary representation theory underlying $\mathrm{G}_{2}$-geometry, following the setup from [Bryan2, Karigianis2]. The natural action of $G_{2}\subset SO(7)$ decomposes $\Omega^{\bullet}(M)$ into $\mathrm{G}_{2}$-invariant irreducible subbundles:

[TABLE]

where each $\Omega^{k}_{l}$ has rank $l$. Studying the symmetries of torsion one finds that $\nabla\varphi\in\Omega^{1}\otimes\Omega^{3}_{7}$, so that tensor lies in a bundle of rank $49$ [Karigianis2, Lemma 2.24]. Notice also that $\Omega^{3}_{7}\cong\Omega^{1}$, so, contracting the dual $4$-form $\psi=\ast_{\varphi}\varphi$ by a frame of $TM$, then using the Riemannian metric, one has

[TABLE]

Here $\mathrm{S}^{2}(T^{\ast}M)$ denotes the symmetric bilinear forms and $\operatorname{sym}(TM)$ the symmetric endomorphisms of $TM$. Both of the above splittings are $\mathrm{G}_{2}$-invariant, so, comparing the $\mathrm{G}_{2}$-irreducible decomposition $\mathfrak{so}(7)=\mathfrak{g}_{2}\oplus[\mathbb{R}^{7}]$ and (2), we get the following identification between $\mathrm{G}_{2}$-irreducible summands

[TABLE]

For $\mathrm{S}^{2}(T^{\ast}M)\cong\operatorname{sym}(TM)$, Bryant defines maps $i:\mathrm{S}^{2}(T^{\ast}M)\rightarrow\Omega^{3}$ and $j:\Omega^{3}\rightarrow\mathrm{S}^{2}(T^{\ast}M)$ by

[TABLE]

where we adopt the familiar implicit summation convention for repeated indices and the inverse of the metric. The map $i$ is injective [Karigianis2, Corollary 2.16] and, by the $\mathrm{G}_{2}$-decomposition $\mathrm{S}^{2}(T^{\ast}M)=\mathbb{R}g_{\varphi}\oplus\operatorname{S^{2}_{0}(T^{\ast}M)}$, it identifies

[TABLE]

Accordingly, we have a decomposition for the torsion components $d\varphi\in\Omega^{4}$ and $d\psi\in\Omega^{5}$ given by

[TABLE]

where $\tau_{0}\in\Omega^{0}$, $\tau_{1}\in\Omega^{1}$, $\tau_{2}\in\Omega^{2}_{14}$ and $\tau_{3}\in\Omega^{3}_{27}$ are called the torsion forms. Indeed, the torsion is completely encoded in the full torsion tensor $T$, defined in coordinates by

[TABLE]

which is expressed in terms of the irreducible $\mathrm{G}_{2}$-decomposition of $\operatorname{End}(TM)$ by [Karigianis2, Theorem 2.27]

[TABLE]

where $\tau_{3}:=i(\tau_{27})$ and ${}^{\sharp}:\Omega^{1}\rightarrow\mathcal{X}(M)$ the musical isomorphism induced by the $\mathrm{G}_{2}$-metric. If moreover the $\mathrm{G}_{2}$-structure is co-closed, the torsion tensor $T=\frac{\tau_{0}}{4}g_{\varphi}-\tau_{27}$ is totally symmetric, and the Hodge Laplacian of $\psi$ is given by [Heisenberg]

[TABLE]

If moreover $\tau_{3}$ vanishes, then $\psi$ is a Laplacian eigenform and the $\mathrm{G}_{2}$-structure is called nearly parallel.

2. Invariant $\mathrm{G}_{2}$-structures on Lie groups

Let us briefly survey Lauret’s approach to geometric flows on homogeneous spaces [Lauret]. for $X_{1},...,X_{r}\in\Gamma(TM)$ and $\alpha_{1},\dots,\alpha_{s}\in\Gamma(T^{\ast}M)$. In particular, when $M=G/H$ is a reductive homogeneous space, i.e.

[TABLE]

any $G$-invariant tensor $\gamma$ is completely determined by its value $\gamma_{x_{0}}$ at the point $x_{0}=[1_{G}]\in G/H$, where $\gamma_{x_{0}}$ is an $\operatorname{Ad}(H)$-invariant tensor at $\mathfrak{m}\cong T_{x_{0}}M$, i.e. $(\operatorname{Ad}(h))^{\ast}\gamma_{x_{0}}=\gamma_{x_{0}}$ for each $h\in H$. Given $x=[gx_{0}]\in G/H$, clearly $\gamma_{x}=(g^{-1})^{\ast}\gamma_{x_{0}}$. Consider now a geometric flow on $M$ of the general form

[TABLE]

Then, if $M=G/H$, requiring $G$-invariance of $\gamma_{t}$, for all $t$, reduces the flow to an ODE for a one-parameter family $\gamma_{t}$ of $\operatorname{Ad}(H)$-invariant tensors on the vector space $\mathfrak{m}$:

[TABLE]

Now, we fix $\dim G=7$ and $H=\{1\}$ the trivial subgroup. For any $\mathrm{G}_{2}$-structure $\varphi_{0}\in\Lambda^{3}(\mathfrak{g})^{\ast}$ on $\mathfrak{g}=\mathrm{Lie}(G)$, non-degeneracy means that, for each non-zero vector $v\in\mathfrak{g}$, the $2$-form $\iota(v)\varphi_{0}$ is symplectic on the vector space $\mathfrak{g}/\left\langle v\right\rangle$. We also know that the $\operatorname{Gl}(\mathfrak{g})$-orbit of the dual $4$-form $\psi_{0}=\ast_{0}\varphi_{0}$ is open in $\Lambda^{4}(\mathfrak{g})^{\ast}$ under the natural action

[TABLE]

Denoting by $\theta:\mathfrak{gl}(\mathfrak{g})\rightarrow\operatorname{End}(\Lambda^{4}(\mathfrak{g})^{\ast})$ the infinitesimal representation $\theta(A)\psi_{0}:=\frac{d}{dt}(e^{tA}\cdot\psi_{0})|_{t=0}$, we have

[TABLE]

and the Lie algebra of the stabilizer

[TABLE]

is characterised by

[TABLE]

Indeed, the orbit $\operatorname{Gl}(\mathfrak{g})\cdot\psi_{0}$ is parametrised by the homogeneous space $\operatorname{Gl}(\mathfrak{g})/G_{\psi_{0}}$. Using the reductive decomposition $\mathfrak{gl}(\mathfrak{g})=\mathfrak{g}_{\psi_{0}}\oplus\mathfrak{q}_{\psi_{0}}$ from equation (6), we have

[TABLE]

In particular, for the Laplacian $\Delta_{0}\psi_{0}$, there exists a unique $Q_{0}\in\mathfrak{q}_{\psi_{0}}$ such that $\theta(Q_{0})\psi_{0}=\Delta_{0}\psi_{0}$. Now, for any other $\psi=h\cdot\psi_{0}\in\operatorname{Gl}(\mathfrak{h})\cdot\psi_{0}$,

[TABLE]

where $\operatorname{Ad}:\operatorname{Gl}(\mathfrak{g})\rightarrow\operatorname{Gl}(\mathfrak{gl}(\mathfrak{g}))$. Moreover, we have the following relations.

Lemma 2.1**.**

Let $\psi=h\cdot\psi_{0}$ for $h\in\operatorname{Gl}(\mathfrak{g})$, denote $\ast$ the Hodge star and $\Delta$ the Laplacian operator of $\psi$, then

[TABLE]

where $\ast_{0}$ and $\Delta_{0}$ are the Hodge star and the Laplacian operator of $\psi_{0}$, respectively.

Proof.

The inner products on $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ induced by a $\mathrm{G}_{2}$-structure $\varphi=h\cdot\varphi_{0}$ are $g=(h^{-1})^{\ast}g_{0}$ and $g=h^{\ast}g_{0}$, respectively, where $g_{0}$ is the inner product induced by $\varphi_{0}$. So, for $\alpha\in\Lambda^{k}(\mathfrak{g})^{\ast}$ we have

[TABLE]

which gives the first claimed relation. In particular,

[TABLE]

Applying again the first relation to the operator $d^{\ast}=(-1)^{7k}\ast d\ast$, we have $d^{\ast}=(h^{-1})^{\ast}\circ d^{\ast_{0}}\circ h^{\ast}$, which yields the claim because $d$ commutes with the pullback $h^{\ast}$. ∎

As consequence of the above Lemma, we can relate $Q_{\psi}\in\mathfrak{q}_{\psi}$ to $Q_{0}\in\mathfrak{q}_{\psi_{0}}$:

[TABLE]

since $\mathfrak{g}_{\psi}\cap\mathfrak{q}_{\psi}=0$. Therefore,

[TABLE]

We will address the flow (5) in the particular case $(M,\gamma_{t})=(G,\psi_{t})$ and $q=-\Delta_{\psi_{t}}$, i.e. under the Laplacian co-flow (1). In particular, a $G$-invariant solution of the Laplacian co-flow is given by a 1-parameter family in $\mathfrak{g}$ solving

[TABLE]

Writing $\psi_{t}=:h_{t}^{-1}\cdot\psi_{0}$ for $h_{t}\in\operatorname{Gl}(\mathfrak{g})$, we have

[TABLE]

thus the evolution of $h_{t}$ under the flow (9) is given by

[TABLE]

3. Lie bracket flow

The Lie bracket flow is a dynamical system defined on the variety of Lie algebras, corresponding to an invariant geometric flow under a natural change of variables. It is introduced in [Lauret] as a tool for the study of regularity and long-time behaviour of solutions.

For each $h\in\operatorname{Gl}(\mathfrak{g})$, consider the following Lie bracket in $\mathfrak{g}$:

[TABLE]

Indeed, $(\mathfrak{g},[\cdot,\cdot])\xrightarrow{h}(\mathfrak{g},\mu)$ defines a Lie algebra isomorphism, and consequently an equivalence between invariant structures

[TABLE]

where $G_{\mu}$ is the $1$-connected Lie group with Lie algebra $(\mathfrak{g},\mu)$, $\eta$ is an automorphism such that $d\eta_{1}=h$ and $\psi_{\mu}=\eta^{\ast}\psi$. In particular, by Lemma 2.1, $\Delta_{\mu}\psi_{\mu}=\eta^{\ast}\Delta_{\psi}\psi$, or, equivalently, $Q_{\mu}=hQ_{\psi}h^{-1}$, by equation (8).

Lemma 3.1**.**

[Lauret, §4.1]* Let $\{h_{t}\}\subset\operatorname{Gl}(\mathfrak{g})$ be a solution of (10), then the bracket $\mu_{t}:=[\cdot,\cdot]_{h_{t}}$ evolves under the flow*

[TABLE]

in which $\delta_{\mu}:\operatorname{End}(\mathfrak{g})\rightarrow\Lambda^{2}(\mathfrak{g})^{\ast}\otimes\mathfrak{g}$ is the infinitesimal representation of the $\operatorname{Gl}(\mathfrak{g})$-action (11), defined by

[TABLE]

Proof.

Setting $Q_{\mu_{t}}:=h_{t}Q_{t}h_{t}^{-1}$, we compute:

[TABLE]

since $(h_{t}^{-1})^{\prime}=-h_{t}^{-1}h_{t}^{\prime}h_{t}^{-1}$. ∎

Remark**.**

Notice that, if $\{h_{t}\}\subset\operatorname{Gl}(\mathfrak{g})$ solves

[TABLE]

then $\mu_{t}$ solves the bracket flow (12).

4. Self Similar Solutions

We say that a $4$-form $\psi$ flows self-similarly along the flow (1) if the solution $\psi_{t}$ starting at $\psi$ has the form $\psi_{t}=b_{t}f_{t}^{\ast}\psi$, for some one-parameter families $\{f_{t}\}\subset\mathrm{Diff}(G)$ and time-dependent non-vanishing functions $\{b_{t}\}$. This is equivalent to the relation

[TABLE]

for some constant $\lambda\in\mathbb{R}$ and a complete vector field $X$. Suppose that the infinitesimal operator defined by $\Delta\psi=\theta(Q_{\psi})\psi$ had the particular form

[TABLE]

Then we have

[TABLE]

where $X_{D}$ is a vector field On $\mathfrak{g}$ defined by the $1$-parameter group of automorphisms $e^{tD}\in\operatorname{Aut}(\mathfrak{g})$.

In that case, $(G,\psi)$ is a soliton for the Laplacian co-flow with

[TABLE]

also, $X_{D}$ denotes the invariant vector field on $G$ induced by the $1$-parameter subgroup $=e^{tD}\in\operatorname{Aut}(\mathfrak{g})$.

A $\mathrm{G}_{2}$-structure whose underlying $4$-form $\psi$ satisfies (13) is called an algebraic soliton, and we say that it is expanding, steady, or shrinking if $\lambda$ is positive, zero, or negative, respectively.

Lemma 4.1**.**

Given $\psi_{2}=c\psi_{1}$ with $c\in\mathbb{R}^{\ast}$, the Laplacian operator satisfies the scaling property

[TABLE]

Proof.

Notice that $c\psi_{1}=(c^{1/4})^{4}\psi_{1}$, then $\varphi_{2}=c^{3/4}\varphi_{1}$, $g_{2}=c^{1/2}g_{1}$ and $\operatorname{vol}_{2}=c^{7/4}\operatorname{vol}_{1}$. For a $k$-form $\alpha$ we have

[TABLE]

So, for a $k$-form $\ast_{2}\alpha=c^{\frac{1}{4}(7-2k)}\ast_{1}\alpha$. And for the Hodge Laplacian operator we have

[TABLE]

∎

The following Lemma appears in [Fino, Lauret], we write the proof for the specific case of the Laplacian co-flow.

Lemma 4.2**.**

If $\psi$ is an algebraic soliton with $Q_{\psi}=cI+D$, then $\psi_{t}=b_{t}h_{t}^{\ast}\psi$ is a self-similar solution for the Laplacian co-flow (9), with

[TABLE]

Moreover,

[TABLE]

Proof.

Applying Lemmata 2.1 and 4.1, we have

[TABLE]

On the other hand,

[TABLE]

Replacing the above expressions in (9) and comparing terms we obtain the ODE system

[TABLE]

the solutions of which are as claimed.

Finally, we have

[TABLE]

so $Q_{t}=b_{t}^{-1/2}h_{t}^{-1}Q_{\psi}h_{t}$, which yields the second claim, since $Q_{\psi}h_{t}=h_{t}Q_{\psi}$. ∎

In terms of the bracket flow, we have $Q_{\mu_{t}}=h_{t}Q_{t}h_{t}^{-1}=b_{t}^{-1/2}Q_{\psi}$. Then, replacing in (12) the Ansatz

[TABLE]

we obtain $c_{t}^{\prime}=cb_{t}^{-1/2}c_{t}$, which has solution $c_{t}=e^{c.s_{t}}$, with $s_{t}$ as above.

Indeed, there is an equivalence between the time-dependent Lie bracket given in (16) and the corresponding soliton given in Lemma 4.2:

Theorem 4.3**.**

[Lauret, Theorem 6]* Let $(G,\varphi)$ be a $1$-connected Lie group with an invariant $\mathrm{G}_{2}$-structure. The following conditions are equivalent:*

1. (i)

The bracket flow solution starting at $[\cdot,\cdot]$ is given by

[TABLE] 2. (ii)

The operator $Q_{t}\in\mathfrak{q}_{\psi}\subset\operatorname{End}(\mathfrak{g})$, such that $\Delta_{\psi}\psi=\theta(Q_{\psi})\psi$, satisfies

[TABLE]

5. Example of a co-flow soliton

We now apply the previous theoretical framework to construct an explicit co-flow soliton from a natural Ansatz. Let $\mathfrak{g}=\mathbb{R}\times_{\rho}\mathbb{R}^{6}$ be the Lie algebra defined by $\rho(t)=\operatorname{exp}(tA)\in\operatorname{Aut}(\mathfrak{g})$, with

[TABLE]

The canonical $\operatorname{SU}(3)$-structure on $\mathbb{R}^{6}$ with respect to the orthonormal basis $\{e_{1},e_{6},e_{2},e_{5},e_{3},e_{4}\}$ is

[TABLE]

and the standard complex structure of $\mathbb{R}^{6}$ is

[TABLE]

We also have the natural $3$-form

[TABLE]

The structure equations of $\mathfrak{g}^{\ast}$ with respect to the dual basis of $\{e_{1},e_{6},e_{2},e_{5},e_{3},e_{4},e_{7}\}$ are

[TABLE]

From the above, we have

[TABLE]

There is a natural co-closed $\mathrm{G}_{2}$-structure on $\mathfrak{g}$, given by

[TABLE]

with dual $4$-form

[TABLE]

Clearly $\tau_{1}=0$ and $\tau_{2}=0$, and

[TABLE]

since $d\varphi\wedge\varphi=0$, i.e. $\tau_{0}=0$. Therefore, using (4),

[TABLE]

The Laplacian of $\psi$ is

[TABLE]

Consider the derivation $D=\operatorname{diag}(a,b,c,c,d,a,0)\in\operatorname{Der}(\mathfrak{g})$, and take the vector field on $\mathfrak{g}$

[TABLE]

Then we have

[TABLE]

From the soliton equation $-\Delta\psi=\mathcal{L}_{X_{D}}\psi+\lambda\psi$, we obtain a system of linear equations

[TABLE]

which has solution $D=\operatorname{diag}(2,4,2,2,0,2,0)$ and $\lambda=-8$. In particular, for the matrix $Q_{\psi}=D+\frac{\lambda}{4}I_{7}$, we have $\Delta\psi=\theta(Q_{\psi})\psi$. By Lemma 4.2, the functions

[TABLE]

yield the family of $4$-forms $\{\psi_{t}=c(t)(f(t)^{-1})^{\ast}\psi\}$, where

[TABLE]

Hence,

[TABLE]

defines a soliton of the Laplacian co-flow:

[TABLE]

Corollary 5.1**.**

The relevant geometric structures associated to the $4$-form given in (19) are:

1. (i)

the $\mathrm{G}_{2}$-structure

[TABLE] 2. (ii)

the $\mathrm{G}_{2}$-metric

[TABLE] 3. (iii)

the volume form

[TABLE] 4. (iv)

the torsion form and the full torsion tensor

[TABLE] 5. (v)

the Ricci tensor and the scalar curvature

[TABLE] 6. (vi)

the bracket flow solution

[TABLE]

Remark 5.2**.**

1. (1)

From Corollary 5.1 $\rm{(iv)}$ and $\rm{(v)}$, if $t\rightarrow-\infty$ then $\operatorname{Ric}(g_{t})\rightarrow 0$, $T(t)\rightarrow 0$ and $\mu_{t}\rightarrow 0$. Since $G_{\mu_{t}}$ is solvable for each $t$ [Lauret, Proposition 6], $(G_{\mu_{t}},\psi)$ smoothly converges to the flat $\mathrm{G}_{2}$-structure $(\mathbb{R}^{7},\varphi_{0})$. 2. (2)

Since $\operatorname{Ric}(g_{\psi})=\operatorname{diag}(0,0,0,0,0,0,-4)+4I_{7}\in\operatorname{Der}(\mathfrak{g})$, the metric $g_{\psi}$ is a shrinking Ricci soliton (cf. [arroyo]).

Afterword

We would like to conclude with two questions for future work.

1. (1)

If the cover Lie group $G$ admits a co-compact discrete subgroup $\Gamma$, it would be interesting to determine whether the corresponding co-closed $\mathrm{G}_{2}$-structure on the compact quotient $G/\Gamma$ is also a Laplacian co-flow soliton. 2. (2)

When the full torsion tensor $T=-\tau_{27}$ is traceless symmetric, the scalar curvature of the corresponding $\mathrm{G}_{2}$-metric is nonpositive, and it vanishes if, and only if, the structure is torsion-free (c.f. [Bryan2, (4.28)] or [Karigianis2, (4.21)]). This fact was first pointed out by Bryant for a closed $\mathrm{G}_{2}$-structure, in order to explain the absence of closed Einstein $\mathrm{G}_{2}$-structures (other than Ricci-flat ones) on compact $7$-manifolds, giving rise to the concept of extremally Ricci-pinched closed $\mathrm{G}_{2}$-structure [Bryan2, Remark 13]. Later on, Fernández et al. showed that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant co-closed $\mathrm{G}_{2}$-structure $\varphi$ such that $g_{\varphi}=g$ [Fernandez2].

In that context, it would be interesting to study pinching phenomena for the Ricci curvature of solvmanifolds with a co-closed (non-flat) left-invariant $\mathrm{G}_{2}$-structure and traceless torsion. In our present construction, for instance, we can see from Corollary 5.1 that

[TABLE]