The Laplacian coflow on almost-abelian Lie groups

Abstract

We find explicit solutions of the Laplacian coflow of $G_2$-structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existence.

1 Introduction

A $G_2$-structure on a seven-dimensional manifold M is given by a 3-form $\varphi$ on M with pointwise stabilizer isomorphic to the exceptional group $G_2 \subset SO(7)$. The 3-form $\varphi$ induces a Riemannian metric $g_{\varphi }$, an orientation and so a Hodge star operator $\star _{\varphi }$ on M. It is well-known [7] that $\varphi$ is parallel with respect to the Levi-Civita connection of $g_{\varphi }$ if and only if $\varphi$ is closed and coclosed and that when this happens the holonomy of $g_{\varphi }$ is contained in $G_2$.

The different classes of $G_2$-structures can be described in terms of the exterior derivatives $d \varphi$ and $d \star _{\varphi } \varphi$ [4, 7]. If $d \varphi$= 0, then the $G_2$-structure is called closed (or calibrated in the sense of Harvey and Lawson [13]) and if $\varphi$ is coclosed, then the $G_2$-structure is called coclosed (or cocalibrated [13]).

Flows of $G_2$-structures were first considered by Bryant in [4]. In particular, he considered the Laplacian flow of closed $G_2$-structures. Recently, Lotay and Wei investigated the properties of the Laplacian flow in the series of papers [19,20,21]. The Laplacian coflow has been originally proposed by Karigiannis, McKay and Tsui in [15], and, for an initial coclosed $G_2$-form $\varphi _0$ with $\star _{\varphi _0} \varphi _0 = \phi _0$, it is given by

$$\begin{aligned} \frac{\partial }{\partial t} \phi (t) = - \Delta _t \phi (t), \quad \mathrm{d}\phi (t) =0, \quad \phi (0) = \phi _0, \end{aligned}$$

(1)

where $\phi (t)$ is the Hodge dual 4-form of a $G_2$-structure $\varphi (t)$ with respect to the Riemannian metric $g_{\varphi (t)}.$ This flow preserves the condition of the $G_2$-structure being coclosed, and it was studied in [15] for warped products of an interval, or a circle, with a compact 6-manifold N which is taken to be either a nearly Kähler manifold or a Calabi–Yau manifold. No general result is known about the short-time existence of the coflow (1). In [2], the Laplacian coflow on the seven-dimensional Heisenberg group has been studied, showing that the solution is always ancient, that is it is defined in some interval $(-\, \infty , T)$, with $0< T < +\, \infty .$ Other examples of flows of $G_2$-structures are the modified Laplacian coflow [11, 12] and Weiss and Witt’s heat flow [24]. The first one is a flow of coclosed $G_2$-structures obtained by adding a fixing term to the Laplacian coflow in order to ensure weak parabolicity in the exact directions. The second one is the gradient flow associated with the functional which measures the full torsion tensor of a $G_2$-structure; generally it does not preserve any special class of $G_2$-structures, but it can be modified to fix the underlying metric (see [3]).

As for the Ricci flow (and other geometric flows), for the Laplacian coflow it is interesting to consider self-similar solutions which are evolving by diffeomorphisms and scalings. If $x_t$ is a 1-parameter family of diffeomorphisms generated by a vector field X on M with $x_0=\mathrm {Id}_M$ and $c_t$ is a positive real function on M with $c_0=1$, then a coclosed $G_2$-structure $\phi (t) = c_t (x_t)^* \phi _0$ is a solution of the coflow (1) if and only if $\phi _0$ satisfies

$$\begin{aligned} - \Delta _0 \phi _0 = L_X \phi _0 + c'_0 \phi _0 = d(X \lnot \phi _0) + c'_0 \phi _0, \end{aligned}$$

where by $L_X$ and $X\lnot$ we denote, respectively, the Lie derivative and the contraction with the vector field X. A coclosed $G_2$-structure satisfying the previous equation is called soliton. As in the case of the Ricci flow, the soliton is said to be expanding, steady, or shrinking if $c'_0$ is positive, zero, or negative, respectively. By Proposition 4.3 in [15], if M is compact, then there are no expanding or steady soliton solutions of (1), other than the trivial case of a torsion-free $G_2$-structure in the steady case. Examples of solitons for the Laplacian flow have been constructed in [5, 8, 16,17,18, 22].

In this paper, we study the coflow (1) on almost-abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. Coclosed and closed left-invariant $G_2$-structures on almost-abelian Lie groups have been studied by Freibert in [9, 10]. General obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra of dimension seven with non-trivial center are given in [1].

By [16], the Laplacian coflow on homogeneous spaces can be completely described as a flow of Lie brackets on the ordinary euclidean space, the so-called bracket flow. In particular, Lauret showed in [16] that any left-invariant closed Laplacian flow solution $\varphi (t)$ on an almost-abelian Lie group is immortal, i.e., defined in the interval $[0, +\,\infty )$. Moreover, he proved that the scalar curvature of $g_{\varphi (t)}$ is strictly increasing and converges to zero as t goes to $+\,\infty$.

In Sect. 3, we find an explicit description of the left-invariant solutions to the Laplacian coflow on almost-abelian Lie groups under suitable assumptions on the initial data, showing that the solution is ancient.

In Sect. 4 we show sufficient conditions for a left-invariant coclosed $G_2$-structure on an almost-abelian Lie group to be a soliton for the Laplacian coflow. In particular, we construct new examples of solitons which are not eigenforms of the Laplacian.

2 Preliminaries

A k-form on an n-dimensional real vector space is stable if it lies in an open orbit of the linear group $GL(n, {{\mathbb {R}}})$. In this section, we review the theory of stable forms in dimensions six and seven. We refer to [6, 14], and the references therein, for more details. Throughout the sections, we denote by $\vartheta$ and by $^*$ the actions of the endomorphism group and the general linear group, respectively.

2.1 Linear $G_2$-structures

A 3-form $\varphi$ on a seven-dimensional real vector space V is stable if the $\Lambda ^7(V^*)$-valued bilinear form $b_\varphi$, defined by

$$\begin{aligned} b_\varphi (x,y)=\frac{1}{6}(x\lnot \varphi )\wedge (y\lnot \varphi )\wedge \varphi ,\quad x,y\in V, \end{aligned}$$

is nondegenerate. In this case, $\varphi$ defines an orientation $vol_\varphi$ by $\root 9 \of {\mathrm {det}b_\varphi }$ and a bilinear form $g_\varphi$ by $b_\varphi =g_\varphi vol_\varphi$. A stable 3-form $\varphi$ is said to be positive, and we will write $\varphi \in \Lambda ^3_+V^*$, if, in addition, $g_\varphi$ is positive definite.

It is a well-known fact that the action of $\mathrm {GL}(V)$ on $\Lambda ^3_+V^*$ is transitive and the stabiliser of every $\varphi \in \Lambda ^3_+V^*$ is a subgroup of $\mathrm{SO}(g_\varphi )$ isomorphic to $G_2$. Therefore, if we assume that $||\varphi ||_{g_\varphi } =7$ we get a one-to-one correspondence between normalized positive 3-forms on V and presentations of $G_2$ inside $\mathrm {GL}(V)$.

We denote by $\star _\varphi$ the Hodge operator induced by $\varphi$, and we will always write $\phi$ to indicate the Hodge dual form $\star _\varphi \varphi$ of $\varphi$. Precisely, $\phi$ belongs to the $\mathrm {GL}(V)$-orbit, denoted by $\Lambda ^4_+V^*$, of positive 4-forms. It is another basic fact that the stabilisers of $\varphi$ and $\phi$ under $\mathrm {GL}^+(V)$ are equal, and therefore, the choice of $\phi$ and of an orientation vol is sufficient to define $\varphi$.

We will refer to a presentation of $G_2$ inside $\mathrm {GL}(V)$ as a linear $G_2$-structure on V, and we will call $\varphi$ and $\phi$ the fundamental forms associated with the linear $G_2$-structure.

On V there exists always a $g_\varphi$-orthonormal and positive oriented co-frame $\left( e^1,\dots ,e^7\right)$, called an adapted frame, such that

$$\begin{aligned} \varphi&=e^{127}+e^{347}+e^{567}+e^{135}-e^{146}-e^{236}-e^{245},\\ \phi&=e^{1234}+e^{3456}+e^{1256}-e^{2467}+e^{1367}+e^{1457}+e^{2357}. \end{aligned}$$

2.2 Linear $\mathrm{SU}(3)$-structures

Let U be a real vector space of dimension six. A 2-form $\omega$ on U is stable if it is nondegenerate, i.e., if $\omega ^3 \ne 0.$

Given a 3-form $\psi$ on U, the equivariant identification of $\Lambda ^5U^*$ with $U\otimes \Lambda ^6U^*$ allows us to define the operator

$$\begin{aligned} K_\psi :U\rightarrow U\otimes \Lambda ^6U^*,\quad x \mapsto (x\lnot \psi )\wedge \psi . \end{aligned}$$

We can consider the trace of its second iterate

$$\begin{aligned} \lambda (\psi )=\frac{1}{6}\mathrm {tr}(K^2_\psi )\in (\Lambda ^6U^*)\otimes (\Lambda ^6U^*), \end{aligned}$$

where

$$\begin{aligned} K^2_\psi :U\rightarrow U\otimes (\Lambda ^6U^*)\otimes (\Lambda ^6U^*). \end{aligned}$$

Then $\psi$ is stable if and only if $\lambda (\psi )\ne 0$. If $\lambda (\psi )<0$, the 3-form $\psi$ is called negative. In this case, we will write $\psi \in \Lambda ^3_-U$. Here, the basic fact is that the action of $\mathrm {GL}^+(U)$ is transitive on $\Lambda ^3_-U$ with stabiliser of $\psi$ isomorphic to $\mathrm {SL}(3,{{\mathbb {C}}})$, where the associated complex structure $J_\psi$ and complex volume form $\Psi$ on U are given, respectively, by

$$\begin{aligned} J_\psi =\frac{1}{\sqrt{-\lambda }}\,K_\psi ,\quad \Psi =-J^*_\psi \psi +i\psi . \end{aligned}$$

It is important to note that the 3-form $J_\psi ^*\psi$ is still negative and that it defines the same complex structure of $\psi$.

If $\psi$ is a negative 3-form and $\omega$ a stable 2-form, then $\omega$ is of type (1, 1) with respect to $J_\psi$, meaning that $J_\psi ^*\omega =\omega$, if and only if $\psi \wedge \omega =0$. In this case, we can define a symmetric bilinear form h on U by

$$\begin{aligned} h(x,y)=\omega (x,J_\psi y),\quad x,y\in U. \end{aligned}$$

When h is positive definite, the couple $(\omega ,\psi )$ is said to be a positive couple and it defines a linear $\mathrm{SU}(3)$-structure, meaning that its stabiliser in $\mathrm {GL}(U)$ is isomorphic to $\mathrm{SU}(3)$. In this case, h is hermitian with respect to $J_\psi$ and $\Psi =-J^*_\psi \psi +i\psi$ is a complex volume form. A positive couple is said to be normalized if

$$\begin{aligned} 2\,\omega ^3=3\,\psi \wedge J^*_\psi \psi . \end{aligned}$$

If a positive couple $(\omega ,\psi )$ is normalized, then there exists an h-orthonormal and positive oriented co-frame of U, called an adapted frame, $\left( f^1,J^*f^1,f^2,J^*f^2,f^3,J^*f^3\right)$ such that

$$\begin{aligned} \omega&=f^1\wedge J^*f^1+f^2\wedge J^*f^2+f^3 \wedge J^*f^3,\\ \psi&=-f^2\wedge f^4\wedge f^6+f^1\wedge J^*f^3\wedge J^*f^6+J^*f^1\wedge f^4\wedge J^*f^5+J^*f^2\wedge J^*f^3\wedge f^5. \end{aligned}$$

Therefore, if we denote by $*_h$ the Hodge operator on U associated with h, it follows that

$$\begin{aligned} *_h \omega =\frac{1}{2}\omega ^2,\quad *_h \psi =J^*_\psi \psi . \end{aligned}$$

2.3 From $G_2$ to $\mathrm{SU}(3)$

Given a linear $G_2$-structure $\varphi$ on V, with fundamental forms $\varphi$ and $\phi$, the six-dimensional sphere

$$\begin{aligned} S^6 =\{ x \in V \mid g_{\varphi } (x, x) = 1 \} \subset V \end{aligned}$$

is $G_2$-homogeneous and, for any nonzero vector $v\in S^6$, there is an induced linear $\mathrm{SU}(3)$-structure on the $g_{\varphi }$-orthogonal complement $U = ({\text{ span }} <v>)^{\perp }$. This structure is constructed as follows. Let

$$\begin{aligned} \omega =v\lnot \varphi ,\quad \psi =-v\lnot \phi . \end{aligned}$$

Then, $(\omega ,\psi )$ is a positive couple on U defining the linear $\mathrm{SU}(3)$-structure. It is then clear that the restriction of an adapted co-frame of $(V, \varphi )$, with $v=e_7$, to U gives an adapted frame of $(U, \omega , \psi )$ and it follows that

$$\begin{aligned} \varphi =\omega \wedge e^7-J^*_\psi \psi ,\quad \phi =\frac{1}{2}\omega ^2+ \psi \wedge e^7. \end{aligned}$$

3 Explicit solutions to the Laplacian coflow on almost-abelian Lie groups

We recall that a Lie group G is said to be almost-abelian if its Lie algebra $\mathfrak {g}$ has a codimension-one abelian ideal $\mathfrak {h}$. Such a Lie algebra will be called almost-abelian, and it can be written as a semidirect product $\mathfrak {g} = {{\mathbb {R}}}x \ltimes _A \mathfrak {h}$. We point out that an almost-abelian Lie algebra is nilpotent if and only if the operator $ad_x \vert _{\mathfrak {h}}$ is nilpotent.

Freibert showed in [9] that if $\mathfrak {g}$ is a 7-dimensional almost-abelian Lie algebra, then, the following are equivalent:

• 1. $\mathfrak {g}$ admits a coclosed $G_2$-structure $\varphi$.

• 2. For any $x \in \mathfrak {g} \setminus \mathfrak {h}$, $ad(x) \vert _\mathfrak {h} \in \mathfrak {gl}(\mathfrak {h})$ belongs to $\mathfrak {sp}( {\mathfrak {h}}, \omega ),$ where $\omega$ is a nondegenerate 2-form $\omega$ on $\mathfrak {h}.$

• 3. For any $x \in \mathfrak {g} \setminus \mathfrak {h}$, the complex Jordan normal form of $ad(x) \vert _\mathfrak {h}$ has the property that for all $m \in {\mathbb {N}}$ and all $\lambda \ne 0$ the number of Jordan blocks of size m with $\lambda$ on the diagonal is the same as the number of Jordan blocks of size m with $- \lambda$ on the diagonal and the number of Jordan blocks of size $2m - 1$ with 0 on the diagonal is even.

In this section, we obtain an explicit description of the solutions to the Laplacian coflow on almost-abelian Lie groups under suitable assumptions on the initial data.

Let G be a seven-dimensional, simply connected, almost-abelian Lie group equipped with an invariant coclosed $G_2$-structure $\varphi _0$ with 4-form $\phi _0$ and let $\mathfrak {h}$ be a codimension-one abelian ideal of the Lie algebra $\mathfrak {g}$ of G. By Proposition 4.5 in [23], if we choose a vector $e_7$ in the orthogonal complement of $\mathfrak {h}$ with respect to $g_{\varphi _0}$ such that $g_{\varphi _0}(e_7,e_7) = 1,$ the forms

$$\begin{aligned} \omega _0 = {e_7} \lnot \varphi _0, \quad \psi _0 =- e_7\lnot \phi _0, \end{aligned}$$

(2)

define an $\mathrm{SU}(3)$-structure $(\omega _0, \psi _0)$ on $\mathfrak {h}$. Let $\eta =e_7\lnot g_{\varphi _0}$. Then, we can identify $\mathfrak {g}^*$ with $\mathfrak {h}^*\oplus {{\mathbb {R}}}\eta$ and we have $d \eta =0$, since $\eta$ vanishes on the commutator $[\mathfrak {g}, \mathfrak {g}] \subseteq \mathfrak {h}$. Moreover

$$\begin{aligned} d\alpha =\eta \wedge \vartheta (A)\alpha , \end{aligned}$$

for every $\alpha \in \Lambda ^p\mathfrak {h}^*$, where $A = ad_{e_7} \vert _{\mathfrak {h}}$. In particular, if $\phi$ is any 4-form on $\mathfrak {g}$, we can consider the decomposition

$$\begin{aligned} \phi =\phi ^{(4)}+\phi ^{(3)}\wedge \eta ,\quad \phi ^{(i)}\in \Lambda ^i\mathfrak {h}^*,\,i=3,4. \end{aligned}$$

(3)

So $(\phi _0)^{(4)}=1/2\omega _0^2$ and $(\phi _0)^{(3)}=\psi _0$. Finally let us observe that $d\phi _0=0$ if and only if $\vartheta (A)(\omega ^2_0)=2\vartheta (A)(\omega _0)\wedge \omega _0=0$, which means that $A\in \mathfrak {sp}(\mathfrak {h}, \omega _0)$, since $\omega _0$ is a nondegenerate 2-form.

Lemma 3.1

Let U be a real vector space of dimension 6 endowed with a linear $\mathrm{SU}(3)$-structure $(\omega , \psi )$ and $A\in \mathfrak {sp}(\omega )$ be normal with respect to the inner product h defined by $(\omega , \psi )$. Denote by J the complex structure induced by $\psi$ and by S and L the symmetric and skew-symmetric part of A, respectively. Then, there exist $\theta \in [0,2\pi ]$ and a basis $\left( e_1, e_2, e_3, J e_1,J e_2, Je_3\right)$ of U such that

$$\begin{aligned} \begin{array}{l} \omega = e^1\wedge J^*e^1 +e^2\wedge J^*e^2+e^3\wedge J^*e^3, \\ \Psi = (e^1 + i J^* e^1) \wedge (e^2 + i J^* e^2) \wedge (e^3+ i J^* e^3) \end{array} \end{aligned}$$

(4)

and

$$\begin{aligned} S (e_i) = s_i (\cos (\theta ) e_i + \sin (\theta ) J e_i), \quad S (J e_i) = - s_i (- \sin (\theta ) e_i + \cos (\theta ) Je_i), \quad i =1,2,3,\nonumber \\ \end{aligned}$$

(5)

where the real numbers $\left\{ \pm s_i, i=1,2,3\right\}$ are the eigenvalues of S (counted with their multiplicities), and $J V_{s_i} = V_{- s_i}$, where $V_{s_i}$ denotes the eigenspace of S associated with the eigenvalue $s_i$. Moreover,

• 1. if $s_j=0$, then $Le_j=l_j Je_j$ and $LJe_j=-l_j e_j$, for $l_j\in {{\mathbb {R}}}$;

• 2. if $s_j\ne 0$ with multiplicity $m_j$, then $L |_{V_{s_j} \oplus V_{-s_j}}$ is given by the block matrix

  $$\begin{aligned} L= \left[ \begin{array}{ll} L' &{}\quad 0\\ 0&{}\quad L'\end{array} \right] , \quad L'\in \mathfrak {so}(m_j), \end{aligned}$$

  with respect to the basis $(e_{i_1},\dots ,e_{i_{m_j}},Je_{i_1},\dots ,Je_{i_{m_j}})$ of $V_{s_j} \oplus V_{-s_j}$,

Proof

Clearly S and L belong to $\mathfrak {sp}(\omega )$ since A does. Therefore we have

$$\begin{aligned} h(x,SJy)=h(Sx,Jy)=-\omega (Sx,y)=\omega (x,Sy)=-h(x,JSy),\quad x,y\in V. \end{aligned}$$

Thus $SJ=-JS$ and, similarly, $LJ=JL$.

The spectrum of S must be real and centrally symmetric, since S is symmetric and anti-commutes with J. Let $\left\{ \pm \, s_i, i = 1,2,3 \right\}$ be the spectrum of S. Denote by $V_{s_i}$ the eigenspace of S associated with the eigenvalue $s_i$, and by $m(s_i)$ its multiplicity. It is then clear that, since $[S,L]=0$ and $SJ=-JS$, L preserves each eigenspace $V_{s_i}$ and $JV_{s_i}=V_{-s_i}$.

Now we show that on each J-invariant subspace $W_{s_i}=V_{s_i}+ V_{-s_i}$, both S and L are given as in the statement with respect to some orthonormal basis. First, let us consider the case when $s_i =0$ is an eigenvalue of S. Clearly, its multiplicity $m_0=m(0)$ is even and the restriction $L|_{V_0}$ of L to the eigenspace $V_0$ belongs to $\mathfrak {sp}(m_0,{{\mathbb {R}}})\cap \mathfrak {so}(m_0)=\mathfrak {u}(m_0/2)$. Therefore, we can diagonalize L over ${{\mathbb {C}}}$ as a complex matrix finding the desired expression; indeed, its eigenvalues are all imaginary numbers.

Now let $s_i\ne 0$ and $m(s_i)=m_i$. Then, $W_{s_i}$ has real dimension $2m_i$ and there exists an orthonormal basis $\left( e_{r_1},\dots ,e_{r_{m_i}}\right)$ of $V_{s_i}$ such that, $L \vert _{W_i}$ has the following expression with respect to the orthonormal basis $\left( e_{r_1},\dots ,e_{r_{m_i}}, Je_{r_1},\dots ,Je_{r_{m_i}}\right)$

$$\begin{aligned} L=\left[ \begin{array}{ll} L_1 &{}\quad L_{3}\\ -\,L_{3}^\dagger &{}\quad L_2\end{array} \right] , \, L_1,L_2\in \mathfrak {so}(m_i),\; \, L_{3}\in \mathfrak {gl}(m_i,{{\mathbb {R}}}), \end{aligned}$$

where by $\dagger$ we denote the transpose. So, by $LJ=JL$ and $LS=SL$ we get $L_{3}=0$ and $L_1=L_2$.

Putting all together the basis of $W_i$ we get an orthonormal basis $(e_1, e_2, e_3, Je_1, J e_2, J e_3)$ of U but, generally, the basis is not an adapted frame with respect to the linear $\mathrm{SU}(3)$-structure $(\omega , \psi )$. Indeed $\Psi _0=(e^1+iJ^*e^1)\wedge (e^2+iJ^*e^2)\wedge (e^3+iJ^*e^3)$ does not necessarily coincide with $\Psi$. However, there exists a complex number z of modulus 1 such that $z^{-1}\Psi _0=\Psi$. If we take a cubic root w of z and we consider the linear map Q defined by $Q=\mathrm {Re}(w)\mathrm {id}+\mathrm {Im}(w)J$ we get that $Q^*\Psi _0=\Psi$. The transformation Q commutes with J and preserves each vector subspace $W_{s_i}$. Moreover,

$$\begin{aligned} Q^*S=QSQ^{-1}&=(\mathrm {Re}(w)\mathrm {id}+\mathrm {Im}(w)J)S(\mathrm {Re}(w)\mathrm {id}-\mathrm {Im}(w)J)\\&=S(\mathrm {Re}(w)\mathrm {id}-\mathrm {Im}(w)J)(\mathrm {Re}(w)\mathrm {id}-\mathrm {Im}(w)J)\\&=S\left\{ \left( \mathrm {Re}(w)^2-\mathrm {Im}(w)^2\right) \mathrm {id}-2\left( \mathrm {Re}(w)\mathrm {Im}(w)\right) J\right\} \\&=\cos (\theta )S+\sin (\theta )JS,\\ Q^*L= QLQ^{-1}&=(\mathrm {Re}(w)\mathrm {id}+\mathrm {Im}(w)J)L(\mathrm {Re}(w)\mathrm {id}-\mathrm {Im}(w)J)\\&=L(\mathrm {Re}(w)\mathrm {id}+\mathrm {Im}(w)J)(\mathrm {Re}(w)\mathrm {id}-\mathrm {Im}(w)J)\\&=L\left( \mathrm {Re}(w)^2+\mathrm {Im}(w)^2\right) \mathrm {id}\\&=L. \end{aligned}$$

Therefore, the new basis $\left( Qe_1, Q e_2, Qe_3,J Q e_1, J Q e_2,JQe_3\right)$ satisfies all the requested properties. $\square$

Lemma 3.2

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _{A} \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. Let $(\omega _0, \psi _0)$ be the induced $\mathrm{SU}(3)$-structure on $\mathfrak {h}$ defined by (2), with $\eta (e_7) \ne 0$, $\eta \vert _{\mathfrak {h}} =0$ and $\Vert \eta \Vert _{g_{\varphi _0}} = 1$. The solution $\phi _t$ of the Laplacian coflow on $\mathfrak {g}$

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\phi }_t=-\Delta _t\phi _t,\\ d\phi _t=0,\\ \phi _0=\star _0\varphi _0, \end{array}\right. } \end{aligned}$$

(6)

is given by

$$\begin{aligned} \phi _t=\frac{1}{2}\omega _0^2+p_t\wedge \eta , \end{aligned}$$

where $p_t$ is a time-dependent negative 3-form on $\mathfrak {h}$ solving

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{p}_t=-\varepsilon (p_t)^2 \,\vartheta (A)\vartheta (B_t)p_t,\\ p_0= \psi _0, \end{array}\right. } \end{aligned}$$

(7)

where $\varepsilon (p_t)$ is a function such that $(\omega _0,\varepsilon (p_t)p_t)$ defines an $\mathrm{SU}(3)$-structure on $\mathfrak {h}$ and $B_t$ is the adjoint of $A = ad_{e_7} \vert _{\mathfrak {h}}$ with respect to the scalar product $h_t$ induced by the $\mathrm{SU}(3)$-structure $(\omega _0,\varepsilon (p_t)p_t)$.

Proof

By Cauchy theorem the system of ODEs (6) admits a unique solution. Let $\phi _t$ be the solution of (6) and $\varepsilon _t$ be the norm $||\eta ||_t$ with respect to the scalar product $g_t$ induced by $\phi _t$. Then, we can write

$$\begin{aligned} \phi _t=\frac{1}{2}\omega ^2_t+\psi _t\wedge \frac{1}{\varepsilon _t}\eta , \end{aligned}$$

where the couple $(\omega _t,\psi _t)$ defines an $\mathrm{SU}(3)$-structure on $\mathfrak {h} = \mathrm {Ker}(\eta )$. To see this observe that if we define $x_t$ by $g_t(x_t,y)=\eta (y)$, for any $y\in \mathfrak {g}$, then $\mathfrak {h}=\mathrm {Ker}(\eta )=\left\{ y\in \mathfrak {g}\,|\,g_t(x_t,y)=0\right\}$. Therefore, for every t, the 4-form $\phi _t$ defines an $\mathrm{SU}(3)$-structure $(\omega _t, \psi _t)$ on $\mathfrak {h}$.

With respect to the decomposition (3), we can write $\phi _t$ as $\phi _t = \phi _t^{(4)} + \phi _t^{(3)} \wedge \eta$ with

$$\begin{aligned} \phi ^{(4)}_t=\frac{1}{2}\omega _t^2,\quad \phi ^{(3)}_t=\frac{1}{\varepsilon _t}\psi _t. \end{aligned}$$

Since the cohomology class of $\phi _t$ is fixed by the flow, i.e., $\phi _t = \phi _0 + \mathrm{d} \alpha _t$ it turns out that

$$\begin{aligned} \dot{\phi }_t=\dot{\phi }^{(4)}_t+ \dot{\phi }^{(3)}_t\wedge \eta = \mathrm{d} \dot{\alpha _t} \in d\Lambda ^3\mathfrak {g}^*\subseteq \Lambda ^3\mathfrak {h}^*\wedge {{\mathbb {R}}}\eta . \end{aligned}$$

Therefore, $\dot{\phi }^{(4)}_t=0$, i.e., $\omega _t \equiv \omega _0$, and consequently,

$$\begin{aligned} \phi _t=\frac{1}{2}\omega _0^2+\psi _t\wedge \frac{1}{\varepsilon _t}\eta . \end{aligned}$$

Now define $\eta _t=\frac{1}{\varepsilon _t}\,\eta$ and denote by $\star _{g_t}$ and $\star _{h_t}$ the star Hodge operators on $\mathfrak {g}$ and $\mathfrak {h}$ with respect to $g_t$ and $h_t$, respectively. Note that

$$\begin{aligned} \star _{g_t} \phi _t = \omega _0\wedge \eta _t - \star _{h_t} \psi _t, \end{aligned}$$

since

$$\begin{aligned} \star _{g_t}\beta =\star _{h_t}\beta \wedge \eta _t, \quad \star _{g_t}(\beta \wedge \eta _t)=(-1)^k\star _{h_t}\beta , \end{aligned}$$

for every k-form $\beta$ on $\mathfrak {h}$.

Then,

$$\begin{aligned} \begin{array}{lcl} \Delta _t\phi _t&{}=&{}d\star _{g_t} d\star _{g_t} \phi _t=d\star _td\left( \omega \wedge \eta _t -*_t\psi _t\right) \\ &{}=&{} - d\star _t\left( \eta \wedge \vartheta (A)*_t\psi _t\right) =-d\star _t\left( \varepsilon _t\eta _t\wedge \vartheta (A)*_t\psi _t\right) \\ &{}=&{}-\varepsilon _td*_t\left( \vartheta (A)*_t\psi _t\right) \\ &{}=&{}\varepsilon _t\,\left( \vartheta (A)*_t\vartheta (A)*_t\psi _t\right) \wedge \eta \\ &{}=&{}\varepsilon ^2_t\,\left( \vartheta (A)*_t\vartheta (A)*_t\psi _t\right) \wedge \eta _t. \end{array} \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \dot{\phi }_t=\dot{\psi }_t\wedge \frac{1}{\varepsilon _t}\,\eta -\psi _t\wedge \frac{\dot{\varepsilon }_t}{\varepsilon _t^2}\,\eta =\dot{\psi }_t\wedge \eta _t-\frac{\dot{\varepsilon }_t}{\varepsilon _t}\,\psi _t\wedge \eta _t. \end{aligned}$$

Imposing $\dot{\phi }_t = - \Delta _t\phi _t$, we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\psi _t-\frac{d}{dt}(\varepsilon _t)\varepsilon _t^{-1}\psi _t=-\varepsilon ^2_t\,\left( \vartheta (A)*_t\vartheta (A)*_t\psi _t\right) . \end{aligned}$$

(8)

Consider the 3-form $p_t = \varepsilon _t ^{-1}\, \psi _t$. It is clear that $p_t$ is a negative 3-form, compatible with $\omega _0$ and defining the same complex structure $J_t$ induced by $\psi _t$. Moreover, it satisfies the condition

$$\begin{aligned} -6\, p_t\wedge J_t^*p_t=4\, \varepsilon _t^{-2} {\omega _0}^3. \end{aligned}$$

Then, by (8) we obtain

$$\begin{aligned} \varepsilon _t\dot{p}_t+\dot{\varepsilon }_t p_t-\dot{\varepsilon }_tp_t = -\varepsilon _t^3\left( \vartheta (A)*_t\vartheta (A)*_t p_t\right) , \end{aligned}$$

and thus, the following equation in terms of the 3-form $p_t$

$$\begin{aligned} \dot{p}_t=-\varepsilon (p_t)^2\left( \vartheta (A)*_t\vartheta (A)*_t p_t\right) ,\quad p_0= \psi _0, \end{aligned}$$

(9)

where the function $\varepsilon (p_t) = \varepsilon _t = \Vert \eta \Vert _t$ is defined in terms of the 3-form $p_t$ by

$$\begin{aligned} 6\, p_t\wedge J_t^*p_t=4\, \varepsilon (p_t)^{-2} {\omega _0}^3. \end{aligned}$$

It is easy to see that $*_t\vartheta (A)*_t$ is the $h_t$-adjoint operator of $\vartheta (A)$ on $\Lambda ^3\mathfrak {h}^*$. Indeed, if $\alpha ,\beta \in \Lambda ^3\mathfrak {h}^*$, then

$$\begin{aligned} \langle ( (*_t \vartheta (A) *_t)\alpha ,\beta \rangle _t \, \omega _0^3/6=-\beta \wedge \vartheta (A) (*_t \alpha )=\vartheta (A) (\beta )\wedge *_t(\alpha )= \langle \alpha ,\vartheta (A)(\beta ) \rangle _t \,\omega _0^3/6, \end{aligned}$$

where in the second equality we have used that A is traceless and consequently that $\vartheta (A)$ acts trivially on 6-forms.

Now let $B_t$ be the $h_t$-adjoint of A on $\mathfrak {h}$. We claim that $(*_t \vartheta (A) *_t)\alpha =\vartheta (B_t)\alpha$ for any 3-form $\alpha$ on $\mathfrak {h}$. To see this let $\left( e_1\dots ,e_6\right)$ be an $h_t$-orthonormal basis of $\mathfrak {h}$, so¹

$$\begin{aligned} (B_t)^i_{\;j}=\sum _{a,b}(A)^{a}_{\phantom {a}b}\,(h_t)_{aj}(h_t)^{bi}=(A)^j_{\phantom {j}i},\quad i,j=1,\dots ,6. \end{aligned}$$

On the other hand, for any choice of ordered triples (i, j, k) and (a, b, c), we get

$$\begin{aligned} h_t\left( \vartheta (A)e^{ijk},e^{abc}\right)&=-h_t\sum _{l,m,n}\left( A^i_{\;l}e^{ljk}+A^j_{\;m}e^{imk}+A^k_{\;n}e^{ijn},e^{abc}\right) \\&=-\,\sum _{l,m,n}h_t\left( A^i_{\;i'}e^{i'j'k'}+A^j_{\;j'}e^{i'j'k'}+A^k_{\;k'}e^{i'j'k'},e^{abc}\right) \\&=-\,(A^i_{\;a}+A^j_{\;b}+A^k_{\;c}) \end{aligned}$$

and

$$\begin{aligned} h_t\left( e^{ijk},\vartheta ({B_t})e^{abc}\right)&=-\sum _{l,m,n}h_t\left( e^{ijk},B^a_{\;l}e^{lbc}+B^b_{\;m}e^{amc}+B^c_{\;n}e^{abn}\right) \\&=-\,\sum _{l,m,n}h_t\left( e^{ijk},B^{a}_{\;a'}e^{a'b'c'}+B^{b}_{\;b'}e^{a'b'c'}+B^{c}_{\;c'}e^{a'b'c'}\right) \\&=-\,(B^{a}_{\;i}+B^{b}_{\;j}+B^{c}_{\;k})\\&=-\,(A^i_{\;a}+A^j_{\;b}+A^k_{\;c}), \end{aligned}$$

since $h_t\left( e^{ijk},e^{abc}\right) =\delta ^{ia}\delta ^{bj}\delta ^{kc}$. Therefore,

$$\begin{aligned} h_t\left( \vartheta (A)\alpha ,\beta \right) =h_t\left( \alpha ,\vartheta (B_t)\beta \right) ,\quad \alpha ,\beta \in \Lambda ^3\mathfrak {h}^*, \end{aligned}$$

as we claimed and (7) holds. $\square$

Theorem 3.3

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _{A} \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. Let $(\omega _0, \psi _0)$ be the induced $\mathrm{SU}(3)$-structure on $\mathfrak {h}$ defined by (2), with $\eta (e_7) \ne 0$, $\eta \vert _{\mathfrak {h}} =0$ and $\Vert \eta \Vert _{g_{\varphi _0}} = 1$, and let $J_0 = J_{\psi _0}$. Suppose that $A = ad_{e_7} \vert _{\mathfrak {h}}$ is symmetric with respect to the inner product $h_0 = g_0 \vert _{\mathfrak {h}}$ and fix an adapted frame $(e_1, J_0 e_1, e_2, J_0 e_2, e_3, J_0 e_3)$ of $(\mathfrak {h}, \omega _0, \psi _0)$ such that $\omega _0$ and $\psi _0$ are given by (4) and A has the normal form (5). Furthermore, assume that A satisfies $\theta =0$. Then, the solution $p_t$ of (7) is ancient and it is given by

$$\begin{aligned} p_t=-b_1(t)e^{246}+b_{2}(t)e^{136}+b_3(t)e^{145}+b_{4}(t)e^{235}, \quad t \in \left( -\infty , \frac{1}{8 \left( s_{1}^2+s_2^2+s_3^2\right) } \right) , \end{aligned}$$

where $b_i(t)=e^{-\sigma _i\epsilon (t)}$ for suitable constants $\sigma _i$ and

$$\begin{aligned} \epsilon (t) = \int _0^t \frac{1}{1-8\left( s_{1}^2+s_2^2+s_3^2\right) u}\mathrm{d}u. \end{aligned}$$

Proof

Consider the following system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\chi }_t=-f(t)^2\vartheta (A)\vartheta (A)\chi _t,\\ \chi _0= \psi _0, \end{array} \right. \end{aligned}$$

(10)

where f(t) is a positive function which will be defined later. Moreover, let

$$\begin{aligned} \left( f_1,f_2,f_3,f_4,f_5,f_6\right) = (e_1, J_0 e_1, e_2, J_0 e_2, e_3, J_0 e_3) \end{aligned}$$

be an adapted frame of $\mathfrak {h}$ such that $\omega _0$ and $\psi _0$ are given by (4) and A has the normal form (5). It is clear that

$$\begin{aligned} \vartheta (A)\vartheta (A){\psi }_0&=-(s_1+s_2+s_3)^2f^{246}\\&\quad +\,(s_1+s_2-s_3)^2f^{136}\\&\quad +\,(s_1-s_2+s_3)^2f^{145}\\&\quad +\,(-s_1+s_2+s_3)^2f^{235}.\\ \end{aligned}$$

So

$$\begin{aligned} \vartheta (A)\vartheta (A){\psi }_0=-\sigma _1f^{246}+\sigma _2f^{136}+\sigma _3f^{145}+\sigma _4f^{235}, \end{aligned}$$

for suitable constants $\sigma _1,\sigma _2,\sigma _3$ and $\sigma _4$. The solution of (10) is then given by

$$\begin{aligned} \chi _t=-b_1(t)f^{246}+b_{2}(t)f^{136}+b_3(t)f^{145}+b_{4}(t)f^{235}, \end{aligned}$$

(11)

where $b_i(t)=e^{-\sigma _i\epsilon (t)}$ for a function $\epsilon (t)$ satisfying $\dot{\epsilon }(t) = f(t)^2$. In order to determine the function f(t), note that, for every t where it is defined, the 3-form $\chi _t$ is negative, compatible with $\omega _0$ and it defines a complex structure $J_t$, given by

$$\begin{aligned} J_t=\frac{2}{\sqrt{-\nu _t}} \left[ \begin{array}{llllll} 0&{}\quad -b_{{4}}b_{{1}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ b_{{2}}b_{{3}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -b_{{3}}b_{{1}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad b_{{2}}b_{{4}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad -b_{{2}}b_{{1}}\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad b_{{3}}b_{{4}}&{}\quad 0\end{array} \right] ,\quad \nu _t=-4b_1^2b_2^2b_3^2b_4^2,\nonumber \\ \end{aligned}$$

(12)

with respect to the adapted frame $\left( f_1,f_2,f_3,f_4,f_5,f_6\right)$. Moreover,

$$\begin{aligned} 6\,\chi _t\wedge J_t^*\chi _t=4b_1^2b_2^2b_3^2b_4^2\, {\omega _0^3}. \end{aligned}$$

The previous condition is satisfied if we choose f(t) such that

$$\begin{aligned} f(t)^{-2}=b_1^2b_2^2b_3^2b_4^2=e^{-2\left( \sigma _{1}+\sigma _2+\sigma _3+\sigma _4\right) \int _0^tf(u)^2 \mathrm{d}u}=e^{-8(s_1^2+s_2^2+s_3^2)\int _0^tf(u)^2\mathrm{d}u}. \end{aligned}$$

The above identity is satisfied if and only if the function $F_t=\int _0^tf(u)^2\mathrm{d}u$ solves the following Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} \dot{F}_t=e^{8 \delta F_t},\\ F_0=0,\\ \end{array} \right. \end{aligned}$$

where $\delta =s_{1}^2+s_2^2+s_3^2$. Integrating we get

$$\begin{aligned} t=\frac{1-e^{-8\delta F_t}}{8\delta }. \end{aligned}$$

Therefore,

$$\begin{aligned} F_t=\frac{\mathrm {ln}(1-8\delta t)}{-8\delta }, \end{aligned}$$

and consequently, $f(t)=\frac{1}{\sqrt{1-8\delta t}}.$ Finally, we observe that the metric $h_t$ defined by $(\omega _0, J_t)$ is positive definite. Moreover, the endomorphism $H_t$, defined by $g_0(x,H_ty)=h_t(x,y)$ for any $x,y\in \mathfrak {h}$, has the following matrix representation

$$\begin{aligned} H_t=\frac{2}{\sqrt{-\nu _t}} \left[ \begin{array}{llllll} b_{{2}}b_{{3}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad b_{{1}}b_{{4}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad b_{{2}}b_{{4}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad b_{{1}}b_{{3}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad b_{{3}}b_{{4}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad b_{{1}}b_{{2}} \end{array} \right] , \end{aligned}$$

(13)

with respect to the adapted frame $\left( f_1,f_2,f_3,f_4,f_5,f_6\right)$. Now we claim that $\chi _t$, given by (11), with

$$\begin{aligned} \epsilon (t) = \int _0^t f(u)^2 \, \mathrm{d}u = \int _0^t \frac{1}{1-8\left( s_{1}^2+s_2^2+s_3^2\right) u}\mathrm{d}u, \end{aligned}$$

is the solution of (7). To see this, first observe that the choice of f(t) ensures that $\varepsilon (\chi _t)^2=f(t)^2$. The only thing we have to prove is that the adjoint $C_t$ of A with respect to $h_t$ is constant. It is clear that $C_t=H^{-1}_tB_0H_t$ and then the claim is equivalent to show that $[H_t,B_0]=0$. With respect to the adapted frame $\left( f_1,f_2,f_3,f_4,f_5,f_6\right)$, the endomorphism $H_t$ is diagonal as well as $B_0=A$, and then the claim follows. Thus, the solution $p_t$ of (7) is given by $\chi _t$, and in particular $B_t\equiv B_0$. $\square$

Remark 3.4

The previous proof can be adapted to the case $\theta =\pi$. Indeed, if $\theta =\pi$ then $\vartheta (A)\vartheta (A)\psi _0$ is again a linear combination of elements of the form $e^a\wedge e^b\wedge J_t^*e^c$ with coefficients given by a suitable choice of $\pm (s_a + s_b- s_c)$. On the other hand, when $\theta$ is different from 0 and $\pi$, it turns out that $\vartheta (A)\vartheta (A)\psi _0$ is a linear combination of elements $e^a\wedge e^b\wedge J_t^* e^c$ and $e^a \wedge J_t^* e^b\wedge J_t^* e^c$. Therefore, the derivative of $J_t^*$ at $t =0$ is much more complicated than in the other cases (see Remark 3.6).

Theorem 3.5

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _{A} \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. Let $(\omega _0, \psi )$ be the induced $\mathrm{SU}(3)$-structure on $\mathfrak {h}$ defined by (2), with $\eta (e_7) \ne 0$, $\eta \vert _{\mathfrak {h}} =0$ and $\Vert \eta \Vert _{g_{\varphi _0}} = 1$. Suppose that $A = ad_{e_7} \vert _{\mathfrak {h}}$ is skew-symmetric with respect to the inner product $h_0 = g_0 \vert _{\mathfrak {h}}$ and define $l=l_1+l_2+l_3$, where $l_1,l_2$ and $l_3$ are as in Lemma 3.1. Then, the solution $p_t$ of (7) is given by

$$\begin{aligned} p_t=b(t) \psi _0, \end{aligned}$$

where $b(t)=e^{-l^2\int _0^t \varepsilon ^2_u\mathrm{d}u}$ and $\varepsilon _t$ is a positive function given by

$$\begin{aligned} \varepsilon _t=\frac{1}{\sqrt{1-2l^2t}}. \end{aligned}$$

In particular, $p_t$ is an ancient solution, defined for every t in $\left( -\,\infty ,\frac{1}{2l^2} \right) .$

Proof

Let $f_t$ be a positive function which will be fixed later and let us consider the following system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\chi }_t=-f^2_t\vartheta (A)\vartheta (-A)\chi _t,\\ \chi _0= \psi _0. \end{array} \right. \end{aligned}$$

(14)

Moreover, let $\left( f_1,\dots ,f_6\right) = (e_1, J_0 e_1, e_2, J_0 e_2, e_3, J_0 e_3)$ be an adapted frame such that $\omega _0$ and $\psi _0$ are given by (4) and A has the normal form (5). It is clear that

$$\begin{aligned} \begin{array}{ccc} \vartheta (A)\vartheta (A){\psi }_0=-l^2{\psi _0}. \end{array} \end{aligned}$$

Therefore, the solution of (14) is given by

$$\begin{aligned} \chi _t=b(t) \psi _0, \end{aligned}$$

where $b(t)=e^{-l^2\int _0^tf_u^2\mathrm{d}u}$.

The 3-form $\chi _t$ is negative, compatible with $\omega _0$ and it defines a constant complex structure $J_t\equiv J_0$. Moreover, it satisfies

$$\begin{aligned} 6\,\chi _t\wedge I_t\chi _t=4b^2(t)\,\omega ^3. \end{aligned}$$

Now we choose $f_t$ so that

$$\begin{aligned} f^{-2}_t=b(t)^2=e^{-2l^2\int _0^tf_u^2\mathrm{d}u}. \end{aligned}$$

To do this, we solve the system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{F}_t=e^{2l^2F_t\mathrm{d}u},\\ F_0=0, \end{array} \right. \end{aligned}$$

and then we put $f_t=\sqrt{\dot{F}_t}$. Integrating by t we get

$$\begin{aligned} t=\frac{1-e^{2l^2 F_t}}{2l^2}. \end{aligned}$$

Thus,

$$\begin{aligned} F_t=\frac{\mathrm {ln}(1-2l^2 t)}{-2l^2}, \end{aligned}$$

and consequently, $\varepsilon _t=\frac{1}{\sqrt{1-2l^2 t}}.$

Now it is easy to show that $\chi _t$ is a solution of (7). Indeed, the choice of $f_t$ ensures that $\varepsilon (\chi _t)^2=f_t^2$, and moreover, that the metric $h_t$ induced by $\omega _0$ and $\chi _t$ is constant. Therefore, the adjoint of A is constantly equal to $-A$ and, as a consequence, the solution $p_t$ of (7) is given by $\chi _t$. $\square$

Remark 3.6

It is not hard to prove that if A is normal with respect to $h_0$, then the solution $p_t$ of (7) is given by

$$\begin{aligned} p_t= & {} -\,b_1(t)f^{246}+b_2(t)f^{136}+b_3(t)f^{145}+b_4(t)f^{235}+c_1(t)f^{135}\\&-\,c_2(t)f^{245}-c_3(t)f^{236}-c_4(t)f^{145}, \end{aligned}$$

where $(f_1,\dots ,f_6)=(e_1,J_0e_1, e_2, J_0 e_2, ,e_3,J_0 e_3)$ is an adapted frame of $\mathfrak {h}$ and A is given by (5). Unfortunately in this case, we cannot find an explicit solution of (7).

However, note that if we write $p_t=(x_t)^* \psi _0$, for $[x_t]\in \mathrm {GL}(\mathfrak {h})/\mathrm {SL}(3,{{\mathbb {C}}})$, then $x_t$ belongs to $\mathrm {GL}^+(2,{{\mathbb {R}}})^3$ acting on $<e_1,J_0 e_1>\oplus<e_2,J_0 e_2>\oplus <e_3,J_0 e_3>$. Therefore, $[x_t]=([x^{(1)}_t],[x^{(2)}_t],[x^{(3)}_t])\in \left( (\mathrm {GL}^+(2,{{\mathbb {R}}})/\mathrm{SO}(2)\right) ^3$.

4 Solitons for the Laplacian coflow on almost-abelian Lie groups

In this section, we find sufficient conditions for a left-invariant coclosed $G_2$-structure on an almost-abelian Lie group G to be a soliton for the Laplacian coflow.

Let $\mathfrak {g}$ be a Lie algebra. We recall the following

Definition 4.1

Let $\mathfrak {g}$ be a seven-dimensional Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. A solution $\phi _t$ to the Laplacian coflow (6) on $\mathfrak {g}$ is self-similar if

$$\begin{aligned} \phi _t=c_t(x_t)^*\phi _0, \end{aligned}$$

for a real-valued function $c_t$ and a $\mathrm {GL}(\mathfrak {g})$-valued function $x_t$.

It is well-known that a solution $\phi _t$ of (6) is self-similar if and only if the Cauchy datum $\phi _0$ at $t=0$ is a soliton, namely if it satisfies

$$\begin{aligned} -\Delta _0\phi _0=-4c\phi _0+\vartheta (D)\phi _0, \end{aligned}$$

for some real number c and some derivation D of the Lie algebra $\mathfrak {g}$ (see [16]). A soliton is said to be expanding if $c <0$, shrinking if $c>0$ and steady if $c=0$.

Let $K_t$ be the stabiliser of $\phi _t$ and fix a $K_t$-invariant decomposition of $\mathrm {End}(\mathfrak {g})$. Since $\phi _t$ is stable at any time t, there exists a time-dependent endomorphism $X_t$ of $\mathfrak {g}$, transversal to the Lie algebra of $K_t$ (in the sense that, for every t, $X_t$ takes values in an ad-invariant complement of the Lie algebra of $K_t$), such that

$$\begin{aligned} -\Delta _t\phi _t=\vartheta (X_t)\phi _t. \end{aligned}$$

Therefore, $\phi _0$ is a soliton on $\mathfrak {g}$ if and only if

$$\begin{aligned} X_0=c \, \mathrm {Id}+D. \end{aligned}$$

Suppose now that $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}, \varphi _0)$ is an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. In Lemma 3.2, we have seen that, with no further assumptions on $A = ad_{e_7} \vert _{\mathfrak {h}}$, the Laplacian coflow reads as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\mathrm{d}}{\mathrm{d}t}\phi _t=-\varepsilon _t \vartheta (A)\vartheta (B_t)\psi _t\wedge \eta ,\\ \phi _0=\star _0\varphi _0. \end{array}\right. } \end{aligned}$$

We can show that the term $-\varepsilon _t \vartheta (A)\vartheta (B_t)\psi _t\wedge \eta$ can be rewritten as

$$\begin{aligned} -\varepsilon _t\left( \vartheta (A)\vartheta (B_t)\psi _t\right) \wedge \eta =\vartheta (X_t)\phi _t, \end{aligned}$$

(15)

for a time-dependent endomorphism $X_t$ of $\mathfrak {g}$ in the following way.

Let $(\omega _t, \psi _t)$ be the $\mathrm{SU}(3)$-structure on $\mathfrak {h}$ induced by $\phi _t$. By Lemma 3.1 there exist $\theta (t) \in [0, 2 \pi ]$ and an adapted frame of $\mathfrak {h}$ such that $\eta =\varepsilon _te^7$ and the symmetric part S(t) of A has the normal form (5). More precisely, let

$$\begin{aligned} a(t)=\cos (\theta (t)), \quad b (t)=\sin (\theta (t)). \end{aligned}$$

With respect to the adapted frame at time t, S(t) has the form (5), so it is given by

$$\begin{aligned} S(t) =\left[ \begin{array}{llll} S_1(t)&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad S_2(t)&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad S_3(t)&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} S_i (t)=\left[ \begin{array}{ll} a(t) s_i(t)&{}\quad b(t) s_i(t)\\ b(t) s_i(t) &{}\quad -a(t) s_i(t)\\ \end{array} \right] . \end{aligned}$$

Define l(t) to be the imaginary part of the complex trace of the skew-symmetric part L(t) of A at time t, and let

$$\begin{aligned} \Sigma (t)=\left[ \begin{array}{llll} \Sigma _1(t)&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad \Sigma _2(t)&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad \Sigma _3(t)&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -s^2(t)\\ \end{array} \right] ,\quad \Lambda (t)= \left[ \begin{array}{llll} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -l(t)^2\\ \end{array}\right] \end{aligned}$$

(16)

where

$$\begin{aligned} \Sigma _1 (t)=\left[ \begin{array}{ll} -2s_2(t)s_3(t)+4a(t)^2 s_2(t)s_3(t)&{}\quad -4a(t)b(t)s_2(t)s_3(t)\\ -4a(t)b(t)s_2(t)s_3(t)&{}\quad 2s_2(t)s_3(t)-4a(t)^2s_2(t)s_3(t)\\ \end{array} \right] ,\\ \Sigma _2 (t)=\left[ \begin{array}{ll} -2s_1(t)s_3(t)+4a(t)^2s_1(t)s_3(t)&{}\quad -4a(t)b(t)s_1(t)s_3(t)\\ -4a(t)b(t)s_1(t)s_3(t)&{}\quad 2s_1(t)s_3(t)-4a(t)^2s_1(t)s_3(t)\\ \end{array} \right] ,\\ \Sigma _3(t)=\left[ \begin{array}{ll} -2s_1(t)s_2(t)+4a^2(t)s_1(t)s_2(t)&{}\quad -4a(t)b(t)s_1(t)s_2(t)\\ -4a(t)b(t)s_1(t)s_2(t)&{}\quad 2s_1(t)s_2(t)-4a(t)^2s_1(t)s_2(t)\\ \end{array} \right] , \end{aligned}$$

and $s(t)^2=s_1(t)^2+s_2(t)^2+s_3(t)^2.$ We claim that

$$\begin{aligned} X_t=- \varepsilon _t \left( \Sigma (t) +\Lambda (t) -[S(t),L(t)]\right) . \end{aligned}$$

(17)

To prove this first observe that

$$\begin{aligned} \begin{array}{lcl} \vartheta (A)\vartheta (B_t)\psi _t&{}=&{}\vartheta (S(t)+L(t))\vartheta (S(t)-L(t))\psi _t\\ &{}=&{}\left( \vartheta (S(t))\vartheta (S(t))-\vartheta (L(t))\vartheta (L(t))-\vartheta ([S(t),L(t)])\right) \psi _t. \end{array} \end{aligned}$$

Then, a direct computation shows that

$$\begin{aligned} \vartheta (\Sigma (t))\phi _t=\vartheta (\Sigma (t))(\psi _t\wedge \eta )=(\vartheta (S(t))\vartheta (S(t))\psi _t)\wedge \eta . \end{aligned}$$

On the other hand, if we change the adapted frame so that $L_t$ has the form (5) we see that, as already seen in the proof of Theorem 3.5, $l(t)=l_1(t)+l_2(t)+l_3(t)$ and

$$\begin{aligned} -\vartheta (L(t))\vartheta (L(t))\psi _t=l^2(t)\psi _t, \end{aligned}$$

which is an expression independent on the choice of the adapted frame. Thus,

$$\begin{aligned} \vartheta (\Lambda (t))\phi _t=\vartheta (\Lambda (t))(\psi _t\wedge \eta )=l(t)^2\psi _t\wedge \eta =-(\vartheta (L(t))\vartheta (L(t))\psi _t)\wedge \eta , \end{aligned}$$

proving the claim.

Theorem 4.2

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. The 4-form $\phi _0=\star _0\varphi _0$ is a soliton for the Laplacian coflow if and only if A satisfies

$$\begin{aligned}{}[-\Sigma (0)+1/2[A,A^\dagger ],A]=\delta A, \end{aligned}$$

(18)

where $A^\dagger$ denotes the transpose of A with respect to the underlying metric on $\mathfrak {h}$, $\delta = l^2_0+s^2_0 - c$ for a constant $c \in {{\mathbb {R}}}$ and $\Sigma (0)$ the endomorphism (16). If $\phi _0$ is a soliton, then the solution $\phi _t$ to the Laplacian coflow is given by

$$\begin{aligned} \phi _t=c(t)e^{f(t)D}\phi _0,\quad c(t)=\left( 1-2ct\right) ^{2},\quad f(t)=-\frac{1}{2c}\mathrm {ln}\left( 1-2ct\right) ,\quad t < \frac{1}{2c}, \end{aligned}$$

(19)

where the derivation D of $\mathfrak {g}$ is given by $X_0-c \, \mathrm {Id}$, with $X_0$ as in (17).

Proof

In the light of the previous results, we can write down the soliton equation for the Laplacian coflow as follows. Suppose that $\phi _0$ is a soliton, that is, $X_0=c\mathrm {Id}+D$ for some $c\in {{\mathbb {R}}}$ and a derivation D of $\mathfrak {h}$. Then, by [16, Theorem 4.10], (19) holds. Therefore, A corresponds to a soliton if and only if there exists $c\in {{\mathbb {R}}}$ such that $D=X_0-c \, \mathrm {Id}$ is a derivation of $\mathfrak {g}$.

This condition can be read as a system of algebraic equations for c and the elements of the matrix associated with A. Note that $De_7=\delta e_7,$ with $\delta = l^2_0+s^2_0 - c$. Hence, denoting by $\mu _A$ the Lie bracket structure defined by A,

$$\begin{aligned}{}[D,A]v=DAv-ADv=D\mu _A(e_7,v)-\mu _A(e_7,Dv)=\mu _A(De_7,v)=\delta Av, \quad v\in \mathfrak {h}. \end{aligned}$$

This reads as

$$\begin{aligned}{}[D,A]=\delta A. \end{aligned}$$

(20)

Finally, writing $D=X_0-c \, \mathrm {Id}$ for $X_0$ as in (17) and recalling that $[A,A^\dagger ]=-2[S(0),L(0)]$, we derive (18) from (20). $\square$

We call (18) the soliton equation of the almost-abelian Laplacian coflow.

Notice that we can split the soliton equation into two coupled equations, for the symmetric and skew-symmetric parts of A, in the following way. Since the commutator of two symmetric matrices is skew-symmetric and the commutator of a symmetric matrix and a skew-symmetric one is symmetric, we find

$$\begin{aligned}{}[-\Sigma (0)+[S(0),L(0)],L(0)]=\delta S(0),\quad [-\Sigma (0)+[S(0),L(0)],S(0)]=\delta L(0). \end{aligned}$$

(21)

We have just proved the following result.

Corollary 4.3

If a soliton $\phi _0$ of the Laplacian coflow on the almost-abelian Lie algebra $\mathfrak {g}= {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}$ is an eigenform of the Laplacian then it is harmonic, namely $A\in \mathrm {su}(6)$.

Proof

Clearly, if $\phi _0$ is an eigenform of the Laplacian, then $D=0$, hence $X_0=c \, \mathrm {Id}$. Taking the trace of $X_0|_{\mathfrak {h}}=c\mathrm {Id}|_{\mathfrak {h}}$ we find $c=0$. $\square$

Corollary 4.4

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$. Assume that A is normal with respect to the underlying metric. Then, $\phi _0 = \star _0 \varphi _0$ is soliton on $\mathfrak {g}$ if and only if $4b(1-4a^2)s_1s_2s_3=0$ and $[\Sigma (0),L(0)]=0$. In such a case $\delta =0$ and hence $c\ge 0$.

Proof

By hypotheses, Eq. (21) reduce to

$$\begin{aligned} -[\Sigma (0),S(0)]=\delta L(0),\quad -[\Sigma (0),L(0)]=\delta S(0). \end{aligned}$$

Using the normal form (5), a direct computation shows that

$$\begin{aligned}{}[\Sigma (0),S(0)]=4b(1-4a^2)s_1s_2s_3J_0. \end{aligned}$$

If $\delta$ was different from zero, then S(0) would be invertible (each $s_i$ should be nonzero), and therefore, L(0) would be a nonzero multiple of $J_0$, contradicting Lemma 3.1. Thus, $\delta =0$, that is $4b(1-4a^2)s_1s_2s_3=0$. Clearly, if $L(0)\ne 0$, equation $[\Sigma (0),L(0)]=0$ is not generically satisfied. $\square$

Corollary 4.5

Let $(\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}, \varphi _0)$ be an almost-abelian Lie algebra endowed with a coclosed $G_2$-structure $\varphi _0$ and suppose that $A = ad_{e_7} \vert _{\mathfrak {h}}$ is skew-symmetric with respect to the underlying metric. Then, the solution to the Laplacian coflow obtained in Theorem  3.5 is a soliton.

Remark 4.6

Differently from the Laplacian flow studied in [16] there exist almost-abelian Lie algebras $\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}$ with A symmetric and admitting coclosed $\mathrm {G}_2$-structures that are no solitons for the Laplacian coflow. Indeed, in the light of Corollary 4.4 it is enough to choose a symmetric matrix A and a suitable $\mathrm {G}_2$-structure for which the constant $4b(1-4a^2)s_1s_2s_3$ is nonzero. For instance, we can consider the $\mathrm {G}_2$-structure

$$\begin{aligned} \varphi _0 = e^{127}+e^{347}+e^{567}+e^{135}-e^{146}-e^{236}-e^{245} \end{aligned}$$

on the Lie algebra $\mathfrak {g} = {{\mathbb {R}}}e_7 \ltimes _A \mathfrak {h}$, where $\mathfrak {h} = {{\mathbb {R}}}< e_1, \ldots , e_6>$ and

$$\begin{aligned} A = ad_{e_7} \vert _{\mathfrak {h}} = \left[ \begin{array}{llllll} 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0\\ \end{array} \right] . \end{aligned}$$

Moreover, we are able to prove that the interval of existence of the corresponding solution is bounded. To this aim, and in analogy with the proof of Theorem 3.3, observe that the solution to (10) has the following expression:

$$\begin{aligned} \chi _t=-b_1(t)e^{246}+b_2(t)e^{136}+b_3(t)e^{145}+b_4(t)e^{235}. \end{aligned}$$

Indeed,

$$\begin{aligned} \vartheta (A)\vartheta (A)\chi _t&=-(3b_1(t)-2b_2(t)-2b_3(t)-2b_4(t))e^{246}\\&\quad +\,(-2b_1(t)+3b_2(t)+2b_3(t)+2b_4(t))e^{136}\\&\quad +\,(-2b_1(t)+2b_2(t)+3b_3(t)+2b_4(t))e^{145}\\&\quad +\,(-2b_1(t)+2b_2(t)+2b_3(t)+3b_4(t))e^{235}, \end{aligned}$$

and therefore, the vector-valued function $(b_1(t),b_2(t),b_3(t),b_4(t))$ satisfies a linear ODE whose matrix is

$$\begin{aligned} -f_t^2\left( \begin{matrix} 3&{}\quad -\,2&{}\quad -\,2&{}\quad -\,2\\ -\,2&{}\quad 3&{}\quad 2&{}\quad 2\\ -\,2&{}\quad 2&{}\quad 3&{}\quad 2\\ -\,2&{}\quad 2&{}\quad 2&{}\quad 3\\ \end{matrix}\right) . \end{aligned}$$

Taking into account that this matrix is symmetric, with eigenvalues $-9f_t^2$, $-f_t^2$, $-f_t^2$, $-f_t^2$ and eigenvectors $(-\,1,1,1,1)$, (1, 1, 0, 0), (1, 0, 1, 0) and (1, 0, 0, 1), it follows that

$$\begin{aligned} 2b_1(t)=-e^{-9\int _0^tf_u^2\mathrm{d}u}+3e^{-\int _0^tf_u^2\mathrm{d}u},\quad 2b_2(t)=2b_3(t)=2b_4(t)=e^{-9\int _0^tf_u^2\mathrm{d}u}+e^{-\int _0^tf_u^2\mathrm{d}u}. \end{aligned}$$

The function $F_t=\int _0^tf_u^2\mathrm{d}u$ can be fixed, as we did in Theorem 3.3, by imposing

$$\begin{aligned} 1=\dot{F}_t b_1^2(t)b_2^2(t)b_3^2(t)b_4^2(t)=\dot{F}_t \left( -e^{-9F_t}+3e^{-F_t}\right) ^2\left( e^{-9F_t}+e^{-F_t}\right) ^6\frac{1}{32}. \end{aligned}$$

This guarantees that $\chi _t$ actually solves (7) (note also that A is symmetric for any time).

Notice that the previous equation, after integration, ensures that, since $F_t\ge 0$ if and only if $t\ge 0$, the solution extinguishes in finite time. With an analogous argument, we see that $F_t$ cannot exist for any negative time. To be more precise, let I be the maximal interval of existence of F, then

$$\begin{aligned} 32t=\int _0^{F_t}(-e^{-9x}+3e^{-x})^2(e^{-9x}+e^{-x})^6 \mathrm{d}x,\quad t\in I. \end{aligned}$$

We immediately see that $\mathrm {sup}_I<+\infty$. On the other hand, if $\mathrm {inf}_I=-\infty$ then $F_t$ should be unbounded near $-\infty$: indeed when $M<F_t<0$ it turns out that

$$\begin{aligned} 32 t=\int _0^{F_t}(-e^{-9x}+3e^{-x})^2(e^{-9x}+e^{-x})^6 \mathrm{d}x>\int _0^{M}(-e^{-9x}+3e^{-x})^2(e^{-9x}+e^{-x})^6 \mathrm{d}x. \end{aligned}$$

Therefore, it would exist a sufficiently large negative time t such that $0=-e^{-9F_t}+3e^{-F_t}=2b_1(t)$. Clearly, this cannot happen because $\chi _t$ must be a stable and negative form. By these considerations, we also deduce that the only negative singular time $\tau$ for the monotone function $F_t$ satisfies $b_1(\tau )=0$, that is $F_\tau =-1/8\mathrm {ln}(3)$.

We will now construct an explicit example of soliton on a nilpotent almost-abelian Lie group.

Example 4.7

Let $\mathfrak {g}$ be the nilpotent almost-abelian Lie algebra with structure equations

$$\begin{aligned} \begin{array}{l} d e^1 = e^{27}, \\ d e^j =0, \, \, j = 2,4, 6, 7,\\ d e^3 = e^{47},\\ d e^5 = e^{67}. \end{array} \end{aligned}$$

Then in this case, we have $\mathfrak {h} = {{\mathbb {R}}}< e_1, \ldots , e_6>$ and

$$\begin{aligned} A = ad_{e_7} \vert _{\mathfrak {h}} = \left( \begin{array}{llllll} 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array} \right) . \end{aligned}$$

Consider the $G_2$-structure $\varphi _0 = e^{127} + e^{347} + e^{567} + e^{135} - e^{146} - e^{236} - e^{245}.$ The 4-form

$$\begin{aligned} \phi _0 = \star _{\varphi _0} \, \varphi _0 = e^{1234} + e^{3456} + e^{1256} - e^{2467} + e^{1367} + e^{1457} - e^{2357} \end{aligned}$$

is closed, and thus, $\varphi$ defines a coclosed $G_2$-structure. The basis $(e_1, \dots , e_7)$ is orthonormal with respect to $g_{\varphi _0}$, and one can check that A is not normal. We will apply Theorem 4.2 to show that $\phi _0$ is a soliton for the Laplacian coflow. First observe that S(0) and L(0), on $\mathfrak {h}$, restrict to

$$\begin{aligned} \left( \begin{array}{llllll} 0&{}\quad 1/2&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 1/2&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1/2&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1/2&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1/2\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1/2&{}\quad 0 \end{array} \right) ,\quad \left( \begin{array}{llllll} 0&{}\quad 1/2&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ -\,1/2&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1/2&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad -\,1/2&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1/2\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad -\,1/2&{}\quad 0 \end{array} \right) . \end{aligned}$$

So S(0) is in normal form (5), and therefore, the matrix $\Sigma (0)$, restricted to $\mathfrak {h}$, turns out to be

$$\begin{aligned} \left( \begin{array}{llllll} -\,1/2&{}\quad 0&{}\quad 0 &{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1/2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -\,1/2 &{}\quad 0&{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1/2 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\,1/2 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1/2 \end{array} \right) . \end{aligned}$$

A direct computation then shows that $[S(0),L(0)]=\Sigma (0)$ on $\mathfrak {h}$, which leads to $[-\Sigma (0)+[S(0),L(0)],A]=0$, so A solves the soliton equation for $\delta =0$. In particular, we have $s_1(0)=s_2(0)=s_3(0)=1/2$ and $l(0)=l_1(0)+l_2(0)+l_3(0) = 3/2$. Thus,

$$\begin{aligned} s^2(0)= 3/4, \quad l^2(0)= 9/4 \end{aligned}$$

and $c=3$. Then, the associated derivation D is given by $D=X-3 \, {{\mathrm{Id}}}$, with

$$\begin{aligned} X = \left( \begin{array}{lllllll} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 3 \end{array} \right) , \end{aligned}$$

and the existence interval is $(-\,\infty ,1/6)$. Note that $\phi _0$ is not an eigenform of the Laplacian since

$$\begin{aligned} \vartheta (X)\phi _0=-3 (- e^{2467} + e^{1367} + e^{1457} - e^{2357}). \end{aligned}$$