Explicit soliton for the Laplacian co-flow on a solvmanifold

Abstract

We apply the general Ansatz proposed by Lauret (Rend Semin Mat Torino 74:55–93, 2016) for the Laplacian co-flow of invariant $\mathrm {G}_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7–manifold. Our methods and the example itself are different from those presented by Bagaglini and Fino (Ann Mat Pura Appl 197(6):1855–1873, 2018).

1 Introduction

Geometric flows in $\mathrm {G}_2$-geometry were first outlined by the seminal works of Bryant [5] and Hitchin [10], and have since been studied by several authors, e.g. [3, 4, 9, 14, 15]. 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

$$\begin{aligned} 6g_\varphi (X,Y) \cdot d \mathrm {Vol} :=(X\lrcorner \varphi )\wedge (Y\lrcorner \varphi )\wedge \varphi . \end{aligned}$$

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 [6, 11, 12].

Some weaker formulations of that problem can be obtained from the classical fact, first established by Fernández and Gray [8], that the torsion-free condition is equivalent to $\varphi$ being both closed and coclosed, in the sense that $d\varphi =0$ and $d*_\varphi \varphi =0$, respectively, and thus one may study each of these conditions separately. For instance, Grigorian [9] and Karigiannis and Tsui [14] considered the Laplacian co-flow of $\mathrm {G}_2$-structures $\{\varphi _t\}$ defined by

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

(1)

where $\psi _t:=*_t\varphi _t$ is the Hodge dual and $\Delta _{\psi _t}\psi _t:=(dd^{*_t}+d^{*_t}d)\psi _t$ 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 [10].

When $M^7=G$ is a Lie group, we propose to study this flow from the perspective introduced by Lauret [15] 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. [14] 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.

2 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 [5, 13]. The natural action of $\mathrm {G}_2\subset {{\,\mathrm{SO}\,}}(7)$ decomposes $\Omega ^\cdot (M)$ into $\mathrm {G}_2$-invariant irreducible subbundles:

$$\begin{aligned} \begin{aligned} \Omega ^1&=\Omega ^1_7,&\Omega ^2&= \Omega ^2_7\oplus \Omega ^2_{14},&\Omega ^3&=\Omega ^3_1\oplus \Omega ^3_7\oplus \Omega ^3_{27},\\ \Omega ^6&=\Omega ^6_7,&\Omega ^5&= \Omega ^5_7\oplus \Omega ^5_{14},&\Omega ^4&=\Omega ^4_1\oplus \Omega ^4_7\oplus \Omega ^4_{27}, \end{aligned} \end{aligned}$$

(2)

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 [13, Lemma 2.24]. Notice also that $\Omega ^3_7\cong \Omega ^1$, so, contracting the dual 4-form $\psi =*_\varphi \varphi$ by a frame of TM, then using the Riemannian metric, one has

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

Here $\mathrm {S}^2(T^*M)$ denotes the symmetric bilinear forms and ${{\,\mathrm{sym}\,}}(TM)$ 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

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

(3)

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

$$\begin{aligned} i(h)=\frac{1}{2}h_{il}g^{lm}\varphi _{mjk}dx^{ijk} \quad \text {and} \quad j(\eta )(u,v)=*((u\lrcorner \varphi )\wedge (v\lrcorner \varphi )\wedge \eta ), \end{aligned}$$

(4)

where we adopt the familiar implicit summation convention for repeated indices and the inverse of the metric. The map i is injective [13, Corollary 2.16] and, by the $\mathrm {G}_2$-decomposition $\mathrm {S}^2(T^*M)={\mathbb {R}}g_\varphi \oplus {{\,\mathrm{S^2_0(T^*M)}\,}}$, it identifies

$$\begin{aligned} {\mathbb {R}}g_\varphi \cong \Omega ^3_1 \quad \text {and} \quad {{\,\mathrm{S^2_0(T^*M)}\,}}\cong \Omega ^3_{27}. \end{aligned}$$

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

$$\begin{aligned} d\varphi =\tau _0\psi +3\tau _1\wedge \varphi +*\tau _3 \quad \text {and} \quad d\psi =4\tau _1\wedge \psi +\tau _2\wedge \varphi , \end{aligned}$$

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

$$\begin{aligned} \nabla _l\varphi _{abc}=:T_{lm}g^{mn}\psi _{nabc}, \end{aligned}$$

which is expressed in terms of the irreducible $\mathrm {G}_2$-decomposition of ${{\,\mathrm{End}\,}}(TM)$ by [13, Theorem 2.27]

$$\begin{aligned} T=\frac{\tau _0}{4}g_\varphi -\tau _{27}-(\tau _1)^\sharp \lrcorner \varphi -\frac{1}{2}\tau _2, \end{aligned}$$

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 [2]

$$\begin{aligned} \Delta _\psi \psi =dd^{*}\psi =d\tau _0\wedge \varphi +\tau _0^2\psi + \tau _0*\tau _3+d\tau _3. \end{aligned}$$

If moreover $\tau _3$ vanishes, then $\psi$ is a Laplacian eigenform and the $\mathrm {G}_2$-structure is called nearly parallel.

3 Invariant $\mathrm {G}_2$-structures on Lie groups

Let us briefly survey Lauret’s approach to geometric flows on homogeneous spaces [15]. Consider the action of a Lie group G on a manifold M. A (r, s)-tensor $\gamma$ on M is G-invariant if $g^*\gamma =\gamma$, for each $g\in G$, where

$$\begin{aligned} g^*\gamma (X_1,\ldots ,X_r,\alpha _1,\ldots ,\alpha _s ) := \gamma (g_*X_1,\ldots ,g_*X_r,(g^{-1})^*\alpha _1,\ldots ,(g^{-1})^*\alpha _s), \end{aligned}$$

for $X_1,\ldots ,X_r\in \Gamma (TM)$ and $\alpha _1,\dots ,\alpha _s\in \Gamma (T^*M)$. In particular, when $M=G/H$ is a reductive homogeneous space, i.e.

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

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 ${{\,\mathrm{Ad}\,}}(H)$-invariant tensor at ${\mathfrak {m}}\cong T_{x_0}M$, i.e. $({{\,\mathrm{Ad}\,}}(h))^*\gamma _{x_0}=\gamma _{x_0}$ for each $h\in H$. Given $x=[gx_0]\in G/H$, clearly $\gamma _x=(g^{-1})^*\gamma _{x_0}$. Consider now a geometric flow on M of the general form

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

(5)

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 ${{\,\mathrm{Ad}\,}}(H)$-invariant tensors on the vector space ${\mathfrak {m}}$:

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

Now, we fix $\dim G=7$ and $H=\{1\}$ the trivial subgroup. For any $\mathrm {G}_2$-structure $\varphi _0\in \Lambda ^3({\mathfrak {g}})^*$ 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 ${{\,\mathrm{Gl}\,}}({\mathfrak {g}})$-orbit of the dual 4-form $\psi _0=*_0\varphi _0$ is open in $\Lambda ^4({\mathfrak {g}})^*$ under the natural action

$$\begin{aligned} h\cdot \psi _0:=(h^{-1})^*\psi _0=\psi _0(h^{-1}\cdot ,h^{-1} \cdot ,h^{-1}\cdot ,h^{-1}\cdot ), \quad h\in {{\,\mathrm{Gl}\,}}({\mathfrak {g}}). \end{aligned}$$

Denoting by $\theta : \mathfrak {gl}({\mathfrak {g}})\rightarrow {{\,\mathrm{End}\,}}(\Lambda ^4({\mathfrak {g}})^*)$ the infinitesimal representation $\theta (A)\psi _0:=\frac{d}{dt}(e^{tA}\cdot \psi _0)|_{t=0}$, we have

$$\begin{aligned} \theta (\mathfrak {gl}({\mathfrak {g}}))\psi _0=\Lambda ^4({\mathfrak {g}})^*, \end{aligned}$$

(6)

and the Lie algebra of the stabilizer

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

is characterised by

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

Indeed, the orbit ${{\,\mathrm{Gl}\,}}({\mathfrak {g}})\cdot \psi _0$ is parametrised by the homogeneous space ${{\,\mathrm{Gl}\,}}({\mathfrak {g}})/G_{\psi _0}$. Using the reductive decomposition $\mathfrak {gl}({\mathfrak {g}})={\mathfrak {g}}_{\psi _0}\oplus {\mathfrak {q}}_{\psi _0}$ from Eq. (6), we have

$$\begin{aligned} \theta ({\mathfrak {q}}_{\psi _0})\psi _0=\Lambda ^4({\mathfrak {g}})^*. \end{aligned}$$

(7)

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 $\psi =h\cdot \psi _0\in {{\,\mathrm{Gl}\,}}({\mathfrak {h}})\cdot \psi _0$,

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

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

Lemma 3.1

Let $\psi = h\cdot \psi _0$ for $h\in {{\,\mathrm{Gl}\,}}({\mathfrak {g}})$, denote $*$ the Hodge star and $\Delta$ the Laplacian operator of $\psi$, then

$$\begin{aligned} *=(h^{-1})^**_0h^*\quad \text {and} \quad h^*\circ \Delta =\Delta _0\circ h^*, \end{aligned}$$

where $*_0$ and $\Delta _0$ are the Hodge star and the Laplacian operator of $\psi _0$, respectively.

Proof

The inner products on ${\mathfrak {g}}$ and ${\mathfrak {g}}^*$ induced by a $\mathrm {G}_2$-structure $\varphi =h\cdot \varphi _0$ are $g=(h^{-1})^*g_0$ and $g=h^*g_0$, respectively, where $g_0$ is the inner product induced by $\varphi _0$. So, for $\alpha \in \Lambda ^k({\mathfrak {g}})^*$ we have

$$\begin{aligned} \alpha \wedge *\alpha =\,&g(\alpha ,\alpha ){{\,\mathrm{vol}\,}}\\ =\,&(h^*g_0)(\alpha ,\alpha )(h^{-1})^*{{\,\mathrm{vol}\,}}_0\\ =\,&(h^{-1})^*(g_0(h^*\alpha ,h^*\alpha ){{\,\mathrm{vol}\,}}_0)\\ =\,&\alpha \wedge (h^{-1})^**_0h^*\alpha , \end{aligned}$$

which gives the first claimed relation. In particular,

$$\begin{aligned} *\psi =(h^{-1})^**_0 h^*\psi =(h^{-1})^**_0\psi _0=h\cdot \varphi _0=\varphi . \end{aligned}$$

Applying again the first relation to the operator $d^*=(-1)^{7k}*d*$, we have $d^*=(h^{-1})^*\circ d^{*_0}\circ h^*$, which yields the claim because d commutes with the pullback $h^*$. $\square$

As consequence of the above Lemma, we can relate $Q_\psi \in {\mathfrak {q}}_\psi$ to $Q_0\in {\mathfrak {q}}_{\psi _0}$:

$$\begin{aligned} \theta (Q_\psi )\psi =\,&\Delta _\psi \psi =\Delta _\psi ((h^{-1})^*\psi _0)=(h^{-1})^*(\Delta _0\psi _0)\\ =\,&(h^{-1})^*\theta (Q_0)\psi _0=(h^{-1})^*\theta (Q_0)h^*\psi \\ =\,&(h^{-1})^*\frac{d}{dt}\big (e^{tQ_0}\cdot (h^{-1}\cdot \psi )\big )|_{t=0}=\frac{d}{dt}\big ((he^{tQ_0}h^{-1})\cdot \psi )\big )|_{t=0}\\ =\,&\frac{d}{dt}\big ((e^{t{{\,\mathrm{Ad}\,}}(h)Q_0})\cdot \psi )\big )|_{t=0}=\theta ({{\,\mathrm{Ad}\,}}(h)Q_0)\psi , \end{aligned}$$

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

$$\begin{aligned} Q_\psi ={{\,\mathrm{Ad}\,}}(h)Q_0. \end{aligned}$$

(8)

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

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

(9)

Writing $\psi _t=:h_t^{-1}\cdot \psi _0$ for $h_t\in {{\,\mathrm{Gl}\,}}({\mathfrak {g}})$, we have

$$\begin{aligned} \frac{d}{dt}\psi _t=\,&\psi _0(h_t'\cdot ,h_t\cdot ,h_t\cdot ,h_t\cdot )+ \psi _0(h_t\cdot ,h_t'\cdot ,h_t\cdot ,h_t\cdot )\\&+\psi _0(h_t\cdot ,h_t \cdot ,h_t'\cdot ,h_t\cdot )+\psi _0(h_t\cdot ,h_t\cdot ,h_t\cdot ,h_t'\cdot )\\ {=}\,&\psi _t(h_t^{-1}h_t'\cdot ,\cdot ,\cdot ,\cdot ){+} \psi _t(\cdot ,h_t^{-1}h_t'\cdot ,\cdot ,\cdot ){+}\psi _t(\cdot ,\cdot ,h_t^{-1}h_t' \cdot ,\cdot ){+}\psi _t(\cdot ,\cdot ,\cdot ,h_t^{-1}h_t'\cdot )\\ =\,&-\theta (h_t^{-1}h_t')\psi _t, \end{aligned}$$

thus the evolution of $h_t$ under the flow (9) is given by

$$\begin{aligned} \frac{d}{dt}h_t=h_tQ_t. \end{aligned}$$

(10)

4 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 [15] as a tool for the study of regularity and long-time behaviour of solutions.

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

$$\begin{aligned} \mu =[\cdot , \cdot ]_h :=h\cdot [\cdot ,\cdot ] =h[h^{-1}\cdot ,h^{-1}\cdot ]. \end{aligned}$$

(11)

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

$$\begin{aligned} \eta : (G,\psi _\mu )\rightarrow (G_\mu ,\psi ), \end{aligned}$$

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 $\psi _\mu =\eta ^*\psi$. In particular, by Lemma 3.1, $\Delta _\mu \psi _\mu =\eta ^*\Delta _\psi \psi$, or, equivalently, $Q_\mu =hQ_\psi h^{-1}$, by Eq. (8).

Lemma 4.1

[15, §4.1] Let $\{h_t\}\subset {{\,\mathrm{Gl}\,}}({\mathfrak {g}})$ be a solution of (10), then the bracket $\mu _t:=[\cdot ,\cdot ]_{h_t}$ evolves under the flow

$$\begin{aligned} \frac{d}{dt}\mu _t=-\delta _{\mu _t}(Q_{\mu _t}), \end{aligned}$$

(12)

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

$$\begin{aligned} \delta _\mu (A):=-A\mu (\cdot ,\cdot )+\mu (A\cdot ,\cdot )+\mu (\cdot ,A\cdot ). \end{aligned}$$

Proof

Setting $Q_{\mu _t}:=h_tQ_th_t^{-1}$, we compute:

$$\begin{aligned} \frac{d}{dt}\mu _t=\,&h_t'[h_t^{-1}\cdot ,h_t^{-1}\cdot ]+h_t[(h_t^{-1})' \cdot ,h_t^{-1}\cdot ]+h_t[h_t^{-1}\cdot ,(h_t^{-1})'\cdot ]\\ =\,&h_t'h_t^{-1}\mu _t(\cdot ,\cdot )-\mu _t(h_t'h_t^{-1}\cdot , \cdot )-\mu _t(\cdot ,h_t'h_t^{-1}\cdot )\\ =\,&-\delta _{\mu _t}(h_t'h_t^{-1})=-\delta _{\mu _t}(h_tQ_th_t^{-1})=- \delta _{\mu _t}(Q_{\mu _t}), \end{aligned}$$

since $(h_t^{-1})'=-h_t^{-1}h_t'h_t^{-1}$. $\square$

Remark

Notice that, if $\{h_t\}\subset {{\,\mathrm{Gl}\,}}({\mathfrak {g}})$ solves

$$\begin{aligned} \frac{d}{dt}h_t=Q_{\mu _t}h_t, \end{aligned}$$

then $\mu _t$ solves the bracket flow (12).

5 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 $\psi _t=b_tf_t^*\psi$, 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

$$\begin{aligned} -\Delta \psi =\lambda \psi +{\mathcal {L}}_X\psi , \end{aligned}$$

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

$$\begin{aligned} Q_\psi =cI+D \quad \text {for} \quad c\in {\mathbb {R}} \quad \text {and} \quad D\in {{\,\mathrm{Der}\,}}({\mathfrak {g}}). \end{aligned}$$

(13)

Then we have

$$\begin{aligned} \theta (Q_\psi )\psi =&-4c\psi +\theta (D)\psi =-4c\psi - \frac{d}{dt}\Big ((e^{tD})^*\psi \Big )|_{t=0}\\ =&-4c\psi -{\mathcal {L}}_{X_D}\psi , \end{aligned}$$

where $X_D$ is a vector field on ${\mathfrak {g}}$ defined by the 1-parameter group of automorphisms $e^{tD}\in {{\,\mathrm{Aut}\,}}({\mathfrak {g}})$.

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

$$\begin{aligned} -\Delta _\psi \psi =4c\psi +{\mathcal {L}}_{X_D}\psi , \end{aligned}$$

where $X_D$ also denotes the invariant vector field on G defined by the 1-parameter subgroup $\beta _t$ in ${{\,\mathrm{Aut}\,}}(G)$ such that $d(\beta _t)_1=e^{tD}\in {{\,\mathrm{Aut}\,}}({\mathfrak {g}})$.

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 5.1

Given $\psi _2=c\psi _1$ with $c\in {\mathbb {R}}^*$, the Laplacian operator satisfies the scaling property

$$\begin{aligned} \Delta _2\psi _2=c^{1/4}\Delta _1\psi _1 \end{aligned}$$

(14)

Proof

Notice that $c\psi _1=(c^{1/4}I)^*\psi _1$ and apply Lemma 3.1. $\square$

Lemma 5.2

If $\psi$ is an algebraic soliton with $Q_\psi =cI+D$, then $\psi _t=b_th_t^*\psi$ is a self-similar solution for the Laplacian co-flow (9), with

$$\begin{aligned} b_t=(3ct+1)^{4/3} \quad \text {and} \quad \quad h_t=e^{s_tD}, \quad \text {for} \quad s_t=-\frac{1}{3c}\log (3ct+1). \end{aligned}$$

(15)

Moreover,

$$\begin{aligned} Q_t=b_t^{-3/4}Q_\psi . \end{aligned}$$

Proof

Applying Lemmas 3.1 and 5.1 , we have

$$\begin{aligned} \Delta _t\psi _t&=b_t^{1/4}h_t^*\Delta \psi =b_t^{1/4}h_t^*\theta (Q_\psi )\psi \\&=b_t^{1/4}h_t^*\Big (-4c\psi +\theta (D)\psi \Big )\\&=-4cb_t^{1/4}h_t^*\psi +\theta (b_t^{1/4}h_t^{-1}Dh_t)h_t^*\psi . \end{aligned}$$

On the other hand,

$$\begin{aligned} \frac{d}{dt}\psi _t=\,&b_t'h_t^*\psi +b_t(h_t^*\psi )'\\ =\,&b_t'h_t^*\psi +b_t\theta (h_t^{-1}h_t')h_t^*\psi . \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} b_t'=4cb_t^{1/4}, &{} b(0)=1 \\ b_th_t'=-b_t^{1/4}Dh_t, &{} h(0)=I \\ \end{array}\right. }, \end{aligned}$$

the solutions of which are as claimed.

Finally, we have

$$\begin{aligned} \theta (Q_t)\psi _t&=\Delta _t\psi _t=b_t^{1/4}h_t^*\Delta \psi =b_t^{1/4}h_t^*\theta (Q_\psi )\psi \\&=b_t^{1/4}\theta (h_t^{-1}Q_\psi h_t)h_t^*\psi =\theta (b_t^{-3/4}h_t^{-1}Q_\psi h_t)\psi _t, \end{aligned}$$

so $Q_t=b_t^{-3/4}h_t^{-1}Q_\psi h_t$, which yields the second claim, since $Q_\psi h_t=h_tQ_\psi$. $\square$

In terms of the bracket flow, we have $Q_{\mu _t}=h_tQ_th_t^{-1}=b_t^{-3/4}Q_\psi$. Then, replacing in (12) the Ansatz

$$\begin{aligned} \mu _t=\left( \frac{1}{c(t)}I\right) \cdot [\cdot ,\cdot ]=c(t)[\cdot ,\cdot ] \quad \text {for} \quad c(t)\ne 0 \quad \text {and} \quad c(0)=1, \end{aligned}$$

(16)

we obtain $c_t'=cb_t^{-3/4}c_t$, 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 5.2:

Theorem 5.3

[15, Theorem 6] Let $(G,\varphi )$ be a 1-connected Lie group with an invariant $\mathrm {G}_2$-structure. The following conditions are equivalent:

• (i) The bracket flow solution starting at $[\cdot ,\cdot ]$ is given by

  $$\begin{aligned} \mu _t=\left( \frac{1}{c(t)}I\right) \cdot [\cdot ,\cdot ] \quad \text {for} \quad c(t)>0, c(0)=1. \end{aligned}$$

• (ii) The operator $Q_t\in {\mathfrak {q}}_\psi \subset {{\,\mathrm{End}\,}}({\mathfrak {g}})$, such that $\Delta _\psi \psi =\theta (Q_\psi )\psi$, satisfies

  $$\begin{aligned} Q_\psi =cI+D, \quad \text {for} \quad c\in {\mathbb {R}} \quad \text {and} \quad D\in {{\,\mathrm{Der}\,}}({\mathfrak {g}}). \end{aligned}$$

6 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 $\rho (t)={{\,\mathrm{exp}\,}}(tA)\in {{\,\mathrm{Aut}\,}}({\mathfrak {g}})$, with

$$\begin{aligned} A=\left( \begin{array}{ccc|ccc} &{} &{} &{} &{} &{}\, 1 \\ &{} &{} &{} &{} \,0&{} \\ &{} &{} &{}\, 1&{} &{} \\ \hline &{} &{}\, 1&{} &{} &{} \\ &{}\, 0&{} &{} &{} &{} \\ \,1&{} &{} &{} &{} &{} \ \end{array} \right) . \end{aligned}$$

The canonical ${{\,\mathrm{SU}\,}}(3)$-structure on ${\mathbb {R}}^6$ with respect to the orthonormal basis $\{e_1,e_6,e_2,e_5,e_3,e_4\}$ is

$$\begin{aligned} \omega =e^{16}+e^{25}+e^{34}, \quad \rho _+=e^{123}+e^{145}+e^{356}-e^{246} \end{aligned}$$

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

$$\begin{aligned} J=\begin{pmatrix} 0 &{}\quad -I_{3}\\ I_{3} &{} \quad 0 \end{pmatrix}. \end{aligned}$$

We also have the natural 3-form

$$\begin{aligned} \rho _-:=J\cdot \rho _+=-e^{135}+e^{124}+e^{236}+e^{456}. \end{aligned}$$

The structure equations of ${\mathfrak {g}}^*$ with respect to the dual basis of $\{e_1,e_6,e_2,e_5,e_3,e_4,e_7\}$ are

$$\begin{aligned} de^1=e^{67}, \quad de^6=e^{17}, \quad de^3=e^{47}, \quad de^4=e^{37}, \quad de^j=0 \quad \text {for} \quad j=2,5. \end{aligned}$$

From the above, we have

$$\begin{aligned} d\omega =0, \quad d\rho _+=2(e^{1357}+e^{4567}), \quad \text {and} \quad d\rho _-=2(e^{2467}+e^{1237}). \end{aligned}$$

There is a natural co-closed $\mathrm {G}_2$-structure on ${\mathfrak {g}}$, given by

$$\begin{aligned} \varphi :=\omega \wedge e^7 -\rho _-=e^{167}+e^{257}+e^{347}+e^{135}-e^{124}-e^{236}-e^{456}, \end{aligned}$$

(17)

with dual 4-form

$$\begin{aligned} \psi =*\varphi =\frac{\omega ^2}{2}+\rho _+\wedge e^7=e^{1256}+e^{1346}+e^{2345}+e^{1237}+e^{1457}+e^{3567}-e^{2467}. \end{aligned}$$

(18)

Clearly $\tau _1=0$ and $\tau _2=0$, and

$$\begin{aligned} d\varphi =-d\rho _-=-2(e^{2467}+e^{1237})=*\tau _3, \end{aligned}$$

since $d\varphi \wedge \varphi =0$, i.e. $\tau _0=0$. Therefore, using (4),

$$\begin{aligned} \tau _3=2(e^{135}+e^{456}) \quad \text {or, alternatively,}\quad \tau _{27}=(e^1)^2+(e^3)^{2}-((e^4)^2+(e^6)^2). \end{aligned}$$

The Laplacian of $\psi$ is

$$\begin{aligned} \Delta \psi =\,&d*d*\psi +*d*d\psi =d*d\varphi \\ =\,&d\tau _ 3=4(e^{1457}+e^{3567}). \end{aligned}$$

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

$$\begin{aligned} X_D(x)=\frac{d}{dt}({{\,\mathrm{exp}\,}}(tD)(x)), \quad \text {for} \quad x\in {\mathfrak {g}}. \end{aligned}$$

Then we have

$$\begin{aligned} {\mathcal {L}}_{X_D}\psi =&\;\frac{d}{dt}({{\,\mathrm{exp}\,}}(-tD)^*\psi )|_{t=0}=-\theta (D)\psi \\ {=}&\;(2a+b+d)e^{1256}{+}(2a+2c)e^{1346}{+}(b+2c+d)e^{2345}{+}(a{+}b{+}c)e^{1237}\\&+(a+c+d)e^{1457}+(a+c+d)e^{3567}-(a+b+c)e^{2467}. \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{rcc} 2a+b+d+\lambda &{}=&{} 0 \\ 2a+2c+\lambda &{}=&{} 0 \\ a+b+c+\lambda &{}=&{} 0 \\ a+c+d+\lambda &{}=&{}-4 \end{array} \right. , \end{aligned}$$

which has solution $D={{\,\mathrm{diag}\,}}(2,4,2,2,0,2,0)$ 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 5.2, the functions

$$\begin{aligned} c(t)=(1-6t)^{4/3} \quad \text {and} \quad s(t)=\frac{1}{6}\log (1-6t) \quad \text {for} \quad \frac{1}{6}>t, \end{aligned}$$

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

$$\begin{aligned} f(t)^{-1}&=\exp (-s(t)D)\\&{=}{{\,\mathrm{diag}\,}}((1{-}6t)^{-1/3},(1{-}6t)^{-2/3},(1{-}6t)^{-1/3},(1{-}6t)^{-1/3},1,(1{-}6t)^{-1/3},1). \end{aligned}$$

Hence,

$$\begin{aligned} \psi _t=e^{1256}+e^{1346}+e^{2345}+e^{1237}+ (1-6t)^{2/3}(e^{1457}+e^{3567})-e^{2467} \end{aligned}$$

(19)

defines a soliton of the Laplacian co-flow:

$$\begin{aligned} \frac{d\psi _t}{dt}=-4(1-6t)^{-1/3}(e^{1457}+e^{3567})= -c(t)^{1/4}(f(t)^{-1})^*\Delta \psi =-\Delta _t\psi _t. \end{aligned}$$

Corollary 6.1

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

• (i) the $\mathrm {G}_2$-structure

  $$\begin{aligned} \varphi _t=c(t)^{1/4}(e^{167}+e^{257}+e^{347}+e^{135}-e^{456}) -c(t)^{-1/4}(e^{124}+e^{236}); \end{aligned}$$

• (ii) the $\mathrm {G}_2$-metric

  $$\begin{aligned} g_t=(e^1)^2+(e^3)^2+(e^4)^2+(e^6)^2+c(t)^{-1/2}(e^2)^2+c(t)^{1/2}((e^5)^2+(e^7)^2); \end{aligned}$$

• (iii) the volume form

  $$\begin{aligned} {{\,\mathrm{vol}\,}}_t=c(t)^{1/4}{{\,\mathrm{vol}\,}}_\psi ; \end{aligned}$$

• (iv) the torsion form and the full torsion tensor

  $$\begin{aligned} \tau _3(t)=2(e^{135}+e^{456}) \quad \text {and} \quad T(t)=c(t)^{-1/4}\Big (-(e^1)^2-(e^3)^2+(e^4)^2+(e^6)^2\Big ); \end{aligned}$$

• (v) the Ricci tensor and the scalar curvature

  $$\begin{aligned} {{\,\mathrm{Ric}\,}}(g_t)=-4c(t)^{-1/2}(e^7)^2 \quad \text {and} \quad R_t=-\frac{1}{2}|\tau _3(t)|^2=-4c(t)^{-1/2}; \end{aligned}$$

• (vi) the bracket flow solution

  $$\begin{aligned} \mu _t=c(t)^{-1/4}[\cdot ,\cdot ]. \end{aligned}$$

Remark 6.2

   

• (1) From Corollary 6.1$\mathrm {(iv)}$ and $\mathrm {(v)}$, if $t\rightarrow -\infty$ then ${{\,\mathrm{Ric}\,}}(g_t)\rightarrow 0$, $T(t)\rightarrow 0$ and $\mu _t\rightarrow 0$. Since $G_{\mu _t}$ is solvable for each t [15, Proposition 6], $(G_{\mu _t},\psi )$ smoothly converges to the flat $\mathrm {G}_2$-structure $({\mathbb {R}}^7,\varphi _0)$.

• (2) Since ${{\,\mathrm{Ric}\,}}(g_\psi )={{\,\mathrm{diag}\,}}(0,0,0,0,0,0,-4)+4I_7 \in {{\,\mathrm{Der}\,}}({\mathfrak {g}})$, the metric $g_\psi$ is a shrinking Ricci soliton (cf. [1]).

7 Afterword

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

• (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) 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. [5, (4.28)] or [13, (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 [5, 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 $g_\varphi =g$ [7]. 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 6.1 that

  $$\begin{aligned} F(t)=\frac{R_t^2}{|{{\,\mathrm{Ric}\,}}(g_t)|^2}=1. \end{aligned}$$