Laplacian coflow for warped $\mathrm{G}_2$-structures

TL;DR

This paper studies the Laplacian coflow of G2-structures on warped products, providing explicit evolution equations, existence conditions, and new long-term solutions for specific types of SU(3)-structures.

Contribution

It offers a new reinterpretation of the Laplacian coflow as evolution equations on the base manifold and identifies conditions for solutions, including long-term solutions for special SU(3)-structures.

Findings

Derived explicit evolution equations for the flow.

Established necessary and sufficient conditions for solution existence.

Constructed new long-time solutions for nearly Kähler, symplectic half-flat, or balanced structures.

Abstract

We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold endowed with an $\mathrm{SU}(3)$-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the $\mathrm{SU}(3)$-structure on $M^6$ and the warping function $f$. Necessary and sufficient conditions for the existence of solution for this flow are given. Finally we describe new long time solutions for this flow where the $\mathrm{SU}(3)$-structure on $M^6$ is nearly K\"ahler, symplectic half-flat or balanced.

Tables

Table 1. Table 1. Some classes of SU ​ ( 3 ) SU 3 \mathrm{SU}(3) -structures

Class	Non-vanishing torsion forms	Structure
$\left\{0\right\}$	–	Calabi-Yau
${𝒲}_1^{-}$	${\sigma }_0$	Nearly Kähler
${𝒲}_2^{-}$	${\sigma }_2$	Symplectic half-flat
${𝒲}_3$	${\nu }_3$	Balanced

Table 2. Table 2. Some classes of G 2 subscript G 2 \mathrm{G}_{2} -structures

Class	Non-vanishing torsion forms	Structure
$𝒫$	$-$	Parallel
${𝒳}_1$	${\tau }_0$	Nearly Parallel
${𝒳}_2$	${\tau }_2$	Closed
${𝒳}_3$	${\tau }_3$	Coclosed of pure type
${𝒳}_4$	${\tau }_1$	Locally conformal parallel
${𝒳}_1\oplus {𝒳}_3$	${\tau }_0,{\tau }_3$	Coclosed

Table 3. Table 3. Relation between torsion forms of the warped G 2 subscript G 2 \mathrm{G}_{2} -structure and the SU ​ ( 3 ) SU 3 \mathrm{SU}(3) -structure

Class	$G_2$-torsion forms	$\mathrm{SU}\left(3\right)$-torsion forms	Class
$𝒫$	${\tau }_0={\tau }_1={\tau }_2={\tau }_3=0$	${\sigma }_i={\pi }_i={\nu }_i=0$	$0$
		$d_{6}f=0$
${𝒳}_2$	${\tau }_0={\tau }_1={\tau }_3=0$	${\pi }_0={\sigma }_0={\pi }_1={\nu }_3=0$	${𝒲}_2^{\pm }\oplus {𝒲}_4$
		$\alpha {\pi }_2-\beta {\sigma }_2=0$
		$\frac{1}{f}{d}_{6}f=-{\nu }_1$
${𝒳}_3$	${\tau }_0={\tau }_1={\tau }_2=0$	${\pi }_0={\sigma }_0={\nu }_1=0$	${𝒲}_2^{\pm }\oplus {𝒲}_3\oplus {𝒲}_5$
		$\alpha {\sigma }_2+\beta {\pi }_2=0$
		$\frac{1}{f}{d}_{6}f=-{\pi }_1$
${𝒳}_4$	${\tau }_0={\tau }_2={\tau }_3=0$	${\sigma }_2={\pi }_2={\nu }_3=0$	${𝒲}_1^{\pm }\oplus {𝒲}_4\oplus {𝒲}_5$
		$\frac{1}{f}{d}_{6}f=\frac{1}{2}{\nu }_1=\frac{1}{3}{\pi }_1$
${𝒳}_1\oplus {𝒳}_3$	${\tau }_1={\tau }_2=0$	$\alpha {\sigma }_0+\beta {\pi }_0=0$	${𝒲}_1^{\pm }\oplus {𝒲}_2^{\pm }\oplus {𝒲}_3\oplus {𝒲}_5$
		$\alpha {\sigma }_2+\beta {\pi }_2=0$
		${\nu }_1=0,\frac{1}{f}{d}_{6}f=-{\pi }_1$

Table 4. Table 4. Solutions of the Laplacian coflow in the SHF-case

Lie algebra	$d{\sigma }_2\left(t\right)$	$({\alpha }_1,\dots ,{\alpha }_6)$	$\beta$	$k$
${𝔤}_{5,1}\oplus ℝ$	$A_{135}=-2{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3-2{\alpha }_5}$	$(\frac{1}{6},-\frac{1}{6},\frac{1}{6},-\frac{1}{6},\frac{1}{6},-\frac{1}{6})$	$\frac{1}{6}$	$-3$
$A_{5,7}^{-1,-1,1}\oplus ℝ$	$A_{146}=A_{236}=-4{\left(1+kt\right)}^{2{\alpha }_5}$	$(0,0,0,0,-\frac{1}{2},\frac{1}{2})$	$\frac{1}{2}$	$-4$
$A_{5,17}^{-a,-a,1}\oplus ℝ$	$A_{135}=A_{245}=-4a^2{\left(1+kt\right)}^{-2{\alpha }_5}$	$(0,0,0,0,\frac{1}{2},-\frac{1}{2})$	$\frac{1}{2}$	$-4a^2$
${𝔤}_{6,N3}$	$A_{135}=-6{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3-2{\alpha }_5}$	$(\frac{1}{6},-\frac{1}{6},\frac{1}{6},-\frac{1}{6},\frac{1}{6},-\frac{1}{6})$	$\frac{1}{6}$	$-9$
${𝔤}_{6,38}^0$	$A_{236}=-6{\left(1+kt\right)}^{2{\alpha }_1-4{\alpha }_3}$	$(-\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6},\frac{1}{6})$	$\frac{1}{6}$	$-9$
${𝔤}_{6,54}^{0,-1}$	$A_{146}=A_{236}=-2{\left(1+kt\right)}^{2{\alpha }_5}$	$(-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2})$	$\frac{3}{2}$	$-1$
	$A_{245}=-2{\left(1+kt\right)}^{2{\alpha }_1+2{\alpha }_3-2{\alpha }_5}$
${𝔤}_{6,118}^{0,-1,-1}$	$A_{135}=A_{245}=-4{\left(1+kt\right)}^{-2{\alpha }_5}$	$(0,0,0,0,\frac{1}{2},-\frac{1}{2})$	$\frac{1}{2}$	$-4$
	$A_{146}=A_{236}=$
	$-2{\left(1+kt\right)}^{2{\alpha }_5}\left(-1+{\left(1+kt\right)}^{2{\alpha }_1-2{\alpha }_3}\right)$

Table 5. Table 5. Solutions of the Laplacian coflow in the balanced case

Lie algebra	$d\ast {\nu }_3\left(t\right)$	$({\alpha }_1,\dots ,{\alpha }_6)$	$\beta$	$k$
${𝔥}_2$	$B_{1234}=-128{\left(1+kt\right)}^{-4{\alpha }_1+2{\alpha }_5}$	$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6})$	$-\frac{1}{6}$	$-192$
${𝔥}_3$	$B_{1234}=-8{\left(1+kt\right)}^{-4{\alpha }_1+2{\alpha }_5}$	$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6})$	$-\frac{1}{6}$	$-12$
${𝔥}_4$	$B_{1234}=-6{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3+2{\alpha }_5}$	$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6})$	$-\frac{1}{6}$	$-9$
${𝔥}_5$	$B_{1234}=-4{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3+2{\alpha }_5}$	$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6})$	$-\frac{1}{6}$	$-6$
${𝔥}_6$	$B_{1234}=-2{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3+2{\alpha }_5}$	$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},-\frac{1}{6},-\frac{1}{6})$	$-\frac{1}{6}$	$-3$
${𝔥}_{19}^{-}$	$B_{1234}=-2{\left(1+kt\right)}^{-2{\alpha }_1-2{\alpha }_3+2{\alpha }_5}$	$(\frac{1}{2},\frac{1}{2},0,0,0,0)$	$-\frac{1}{2}$	$-2$
	$B_{1256}=-2{\left(1+kt\right)}^{-2{\alpha }_1+2{\alpha }_3-2{\alpha }_5}$

Equations

$$\frac{\partial }{\partial t}\phi (t)={\Delta }_7\phi (t),$$

$$\frac{\partial }{\partial t}{\ast }_7\phi (t)=-{\Delta }_7{\ast }_7\phi (t),$$

$$(-1)^{n(n-1)/2}\Big{(}\frac{\imath}{2}\Big{)}^{n}\Psi\wedge\overline{\Psi}=\frac{1}{n!}\,\omega^{n},$$

$$g_{\omega ,{\psi }_{\pm }}(X,Y)vo{l}_6=-3({\iota }_X)\omega \wedge ({\iota }_Y{\psi }_{+})\wedge {\psi }_{+},$$

$$\begin{array}[]{c}\omega=e^{12}+e^{34}+e^{56},\\[10.0pt] \psi_{+}=e^{135}-e^{146}-e^{236}-e^{245},\quad\psi_{-}=-e^{246}+e^{235}+e^{145}+e^{136},\end{array}$$

$$\tau \in {\mathcal{W}}_1^{\pm }\oplus {\mathcal{W}}_2^{\pm }\oplus {\mathcal{W}}_3\oplus {\mathcal{W}}_4\oplus {\mathcal{W}}_5,$$

$$\begin{array}[]{ll}\Omega^{3}_{+}(M^{6})=\{f\psi_{+}|\,f\in\mathcal{C}^{\infty}(M^{6})\},&\quad\Omega^{3}_{6}(M^{6})=\{\eta\wedge\omega|\,\eta\in\Omega^{1}(M^{6})\}=\{\gamma\in\Omega^{3}(M^{6})|\ast_{6}J\gamma=\gamma\},\\[7.0pt] \Omega^{3}_{-}(M^{6})=\{f\psi_{-}|\,f\in\mathcal{C}^{\infty}(M^{6})\},&\quad\Omega^{3}_{12}(M^{6})=\{\gamma\in\Omega^{3}(M^{6})|\,\gamma\wedge\omega=0,\gamma\wedge\psi_{\pm}=0\},\end{array}$$

$$\begin{array}[]{lll}d\omega&=&\frac{-3}{2}\sigma_{0}\psi_{+}+\frac{3}{2}\pi_{0}\psi_{-}+\nu_{1}\wedge\omega+\nu_{3},\\[5.0pt] d\psi_{+}&=&\pi_{0}\,\omega^{2}+\pi_{1}\wedge\psi_{+}-\pi_{2}\wedge\omega,\\[5.0pt] d\psi_{-}&=&\sigma_{0}\,\omega^{2}+\pi_{1}\wedge\psi_{-}-\sigma_{2}\wedge\omega,\end{array}$$

$$g_{\phi }(X,Y)vo{l}_7=\frac{1}{6}({\iota }_X\phi )\wedge ({\iota }_Y\phi )\wedge \phi ,$$

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

$$\mathcal{T}\in {\mathcal{X}}_1\oplus {\mathcal{X}}_2\oplus {\mathcal{X}}_3\oplus {\mathcal{X}}_4,$$

$$\begin{array}[]{ll}\Omega^{3}_{1}(M^{7})=\{f\varphi|\,f\in\mathcal{C}^{\infty}(M^{7})\},&\quad\Omega^{3}_{27}(M^{7})=\{\gamma\in\Omega^{3}(M^{7})|\,\gamma\wedge\varphi=\gamma\wedge\ast_{7}\varphi=0\}.\\[10.0pt] \Omega^{3}_{7}(M^{7})=\{\ast_{7}(\eta\wedge\varphi)|\,\eta\in\Omega^{1}(M^{7})\},&\end{array}$$

$$d\phi ={\tau }_0{\ast }_7\phi +3{\tau }_1\wedge \phi +{\ast }_7{\tau }_3,d({\ast }_7\phi )=4{\tau }_1{\ast }_7\phi +{\tau }_2\wedge \phi ,$$

$$\begin{array}[]{ll}\tau_{0}=\dfrac{1}{7}\ast_{7}(d\varphi\wedge\varphi),&\qquad\tau_{2}=-\ast_{7}d\ast_{7}\varphi+4\ast_{7}(\tau_{1}\wedge\ast_{7}\varphi),\\[5.0pt] \tau_{1}=\dfrac{-1}{12}\ast_{7}(\ast_{7}d\varphi\wedge\varphi),&\qquad\tau_{3}=\ast_{7}d\varphi-\tau_{0}\varphi-3\ast_{7}(\tau_{1}\wedge\varphi).\\ \end{array}$$

$$W=B\times F$$

$$\phi =f\omega \wedge ds+(\alpha {\psi }_{+}-\beta {\psi }_{-}),$$

$$g_{\phi }=g_{\omega ,{\psi }_{\pm }}+f^{2}d{s}^2,vo{l}_7=fvo{l}_6\wedge ds.$$

$${\ast }_7\eta$$

$${\ast }_7(\eta \wedge ds)$$

$${\ast }_7\phi =\frac{1}{2}{\omega }^2+f(\alpha {\psi }_{-}+\beta {\psi }_{+})\wedge ds.$$

$$\begin{array}[]{lll}\tau_{0}&=&\frac{12}{7}(\alpha\pi_{0}-\beta\sigma_{0}),\\ &&\\ \tau_{1}&=&\frac{1}{2}(\alpha\sigma_{0}+\beta\pi_{0})fds+\frac{1}{6}\eta_{1},\\ &&\\ \tau_{2}&=&-\alpha\sigma_{2}-\beta\pi_{2}+\frac{f}{3}\ast_{6}\left(\eta_{2}\wedge\omega^{2}\right)\wedge ds-\frac{1}{3}\ast_{6}\left(\eta_{2}\wedge(\alpha\psi_{-}+\beta\psi+)\right),\\ &&\\ \tau_{3}&=&\left[\frac{2}{7}(\alpha\pi_{0}-\beta\sigma_{0})f\omega-\frac{f}{2}\ast_{6}\left(\eta_{3}\wedge(\alpha\psi_{+}-\beta\psi_{-})\right)+f(\alpha\pi_{2}-\beta\sigma_{2})\right]\wedge ds-\frac{1}{2}\ast_{6}(\eta_{3}\wedge\omega)-\\[10.00002pt] &&\frac{3}{14}(\alpha\pi_{0}-\beta\sigma_{0})(\alpha\psi_{+}-\beta\psi_{-})-\ast_{6}\nu_{3},\end{array}$$

$${\eta }_1=\frac{1}{f}{d}_{6}f+{\pi }_1+{\nu }_1,{\eta }_2=\frac{1}{f}{d}_{6}f+{\pi }_1-2{\nu }_1,{\eta }_3=\frac{1}{f}{d}_{6}f-{\pi }_1+{\nu }_1.$$

$$d\phi$$

$${\ast }_7(d\phi )$$

$$d({\ast }_7\phi )$$

$${\tau }_0=0$$

$${\tau }_1=0$$

$${\tau }_2=0$$

$${\tau }_3=0$$

$$(LF)\{\begin{array}{l}\frac{\partial }{\partial t}\phi (t)={\Delta }_t\phi (t),\\ d_7\phi (t)=0,\end{array}(LcF)\{\begin{array}{l}\frac{\partial }{\partial t}({\ast }_t\phi (t))=-{\Delta }_t({\ast }_t\phi (t)),\\ d_7({\ast }_t\phi (t))=0,\end{array}$$

$$\phi (t)=f(t)\omega (t)\wedge ds+(\alpha {\psi }_{+}(t)+\beta {\psi }_{-}(t)).$$

$$\begin{array}[]{l}\dfrac{\partial}{\partial t}\varphi(t)=\left(\dfrac{\partial}{\partial t}f(t)\,\omega(t)+f(t)\dfrac{\partial}{\partial t}\omega(t)\right)\wedge ds+\alpha\dfrac{\partial}{\partial t}\psi_{+}(t)-\beta\dfrac{\partial}{\partial t}\psi_{-}(t),\\[7.0pt] \dfrac{\partial}{\partial t}(\ast_{7}\varphi(t))=\left[\dfrac{\partial}{\partial t}f(t)\,(\beta\psi_{+}(t)+\alpha\psi_{-}(t))+f(t)\left(\beta\dfrac{\partial}{\partial t}\psi_{+}(t)+\alpha\dfrac{\partial}{\partial t}\psi_{-}(t)\right)\right]\wedge ds+\dfrac{1}{2}\dfrac{\partial}{\partial t}\omega^{2}(t).\end{array}$$

Full text

1. 1 $\mathrm{SU}(3)$ and $\mathrm{G}_{2}$-structures

2. 1.1 $\mathrm{SU}(3)$-structures

3. 1.2 $\mathrm{G}_{2}$-structures

4. 2 Warped $\mathrm{G}_{2}$-structures

5. 3 The Laplacian flow and coflow of warped $\mathrm{G}_{2}$-structure of the form $M^{6}\times_{f}S^{1}$

6. 4 New solutions to the Laplacian coflow

7. 4.1 The nearly Kähler case

8. 4.2 The symplectic half-flat case

9. 4.3 The balanced case

Introduction

The first author to consider flows of $\mathrm{G}_{2}$-structures was Bryant in 2006, [3]. Concretely he considered the Laplacian flow of a $\mathrm{G}_{2}$-structure:

[TABLE]

starting from $\varphi_{0}$ a closed 3-form defining the G2-structure. $\Delta_{7}$ is the corresponding Hodge Laplacian, given by $\Delta_{7}=\,\ast_{7}\,d_{7}\ast_{7}d_{7}-d_{7}\ast_{7}d_{7}\,\ast_{7}$.

In the last years there has been a lot of fundamental works on this issue. In [5] it was proved the short time existence and uniqueness of solution on compact manifolds. The first examples of long time solutions to this flow were described in [8]. These examples consist on non compact nilpotent Lie groups endowed with a one-parameter family of closed $\mathrm{G}_{2}$-structures such that satisfy the Laplacian flow equation for all $t\in(a,+\infty)$ with $a<0$.

Recent papers by Lotay and Wei [17, 18, 19] derived important properties of the Laplacian flow as long time existence or convergence results. Even more recently Fino and Raffero on [11] obtained sufficient conditions for the existence of solution of this flow on warped products of the form $M^{6}\times_{f}S^{1}$ with $M^{6}$ a 6-dimensional manifold endowed with an $\mathrm{SU}(3)$-structure. Recall that, if $(B,g_{B})$ and $(F,g_{F})$ are Riemannian manifolds and $f$ is a non-vanishing differentiable function on $B$, then the warped product $W=B\times_{f}F$ consists on the product manifold $B\times F$ endowed with the metric $g=\pi_{1}^{*}(g_{B})+f^{2}\pi_{2}^{*}(g_{F})$ where $\pi_{1}$ and $\pi_{2}$ are the projections of $W$ onto $B$ and $F$ respectively. They also reinterpret the flow as a set of evolution equations on $M^{6}$ involving the differential forms defining the $\mathrm{SU}(3)$-structure and the warping function $f$. More details about the Laplacian flow of a closed $\mathrm{G}_{2}$-structure can be found in the reviews [9, 16] and the references therein. Another interesting result concerning this flow was due to Xu and Ye in [23], where they proved long time existence and uniqueness of solution for this flow starting near a torsion free $\mathrm{G}_{2}$-structure.

In this work we consider the so-called Laplacian “coflow” of $\mathrm{G}_{2}$-structures. This coflow was introduced by Karigiannis, McKay and Tsui in [15] and can be considered as the analogous of the Laplacian flow of a closed $\mathrm{G}_{2}$-structure where the 3-form $\varphi_{0}$ is now considered to be coclosed instead of closed. Equivalently this flow can be stated as:

[TABLE]

where the 4-form $\ast_{7}\varphi_{0}$ is closed and $\ast_{7}$ denotes the Hodge star operator. These authors considered more natural to define this flow with a minus sign in order to make it more likely to the heat equation. In order to obtain solutions they consider 7-dimensional manifolds $M^{6}\times L^{1}$ with $L^{1}=\mathbb{R}$ or $S^{1}$ where $M^{6}$ is endowed with a Calabi-Yau or a nearly Kähler structure. Grigorian in [13] introduced the modified Laplacian coflow, which consists on a modified version of the Laplacian coflow, proving short time existence and uniqueness of solution for this modified flow. He also derives the modified Laplacian coflow for warped $\mathrm{G}_{2}$-structures of the form $M^{6}\times_{f}L^{1}$ obtaining solution for $M^{6}$ being Calabi-Yau or nearly Kähler. Long time solutions for the Laplacian coflow on non compact nilpotent Lie groups were described in [1]. In this work we present long time solutions for the coflow on warped products where the base manifolds are Lie groups endowed with metrics belonging to the Gray-Hervella classes $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$.

The paper is structured as follows. In Section 1 we give an introduction to $\mathrm{SU}(3)$ and $\mathrm{G}_{2}$-structures. Section 2 is devoted to $\mathrm{G}_{2}$-structures of the form $M^{6}\times_{f}S^{1}$ ($M^{6}$ being compact and endowed with an SU(3)-structure) whose induced metric describes a warped product. In particular in Theorem 2.3 we give an explicit description of the torsion forms of such a $\mathrm{G}_{2}$-structure in terms of the torsion forms of the $\mathrm{SU}(3)$-structure on the base manifold and the warping function. In Section 3 we reinterpret the Laplacian flow and coflow of a $\mathrm{G}_{2}$-structure as a set of evolution equations of the $\mathrm{SU}(3)$-structure and we describe the Laplacian coflow operator of the warped $\mathrm{G}_{2}$-structure by means of the torsion forms of the $\mathrm{SU}(3)$-structure and the warping function. In particular for the Laplacian flow we reobtain the equations due to Fino and Raffero in [11]. Finally the goal of Section 4 is to obtain new examples of long time solutions of the Laplacian coflow constructed as warped products where the base manifolds are 6-dimensional and they are endowed with nearly Kähler, symplectic half-flat or balanced SU(3)-structures.

1. $\mathrm{SU}(3)$ and $\mathrm{G}_{2}$-structures

In this section we review some preliminaries concerning $\mathrm{SU}(3)$ and $\mathrm{G}_{2}$-structures. More concretely we present these structures, their corresponding $\mathrm{SU}(3)$ and $\mathrm{G}_{2}$ type decomposition of the space of differential forms and finally their torsion forms.

1.1. $\mathrm{SU}(3)$-structures

An $\mathrm{SU}(n)$-structure on a differentiable manifold $M^{2n}$ consists on a triple $(g,J,\Psi)$ where $(g,J)$ is an almost Hermitian structure on $M^{2n}$ and $\Psi$ is a complex $(n,0)$ form, satisfying

[TABLE]

with $\overline{\Psi}$ the conjugated form of $\Psi$ and $\omega$ the Kähler form of the almost Hermitian structure. An $\mathrm{SU}(n)$-structure can equivalently be described by the triple $(\omega,\psi_{+},\psi_{-})$ where $\psi_{+}$ and $\psi_{-}$ are, respectively the real and the imaginary part of the complex form $\Psi$. In what follows we will focus on $\mathrm{SU}(3)$-structures on 6-dimensional manifolds. Note that in this case, the metric $g_{\omega,\psi_{\pm}}$ can be recovered from $(\omega,\psi_{+},\psi_{-})$ as

[TABLE]

where $\iota$ denotes the contraction operator, $vol_{6}=\frac{1}{3!}\omega^{3}$ and $X,Y\in\mathfrak{X}(M^{2n})$.

The presence of such a structure on a manifold $M^{6}$ can also be characterized by the existence of a local basis of 1-forms $\{e^{1},\dots,e^{6}\}$ such that $(\omega,\psi_{+},\psi_{-})$ can be described as:

[TABLE]

where we denote, as usual in the related literature, $e^{ij}$ the wedge product $e^{i}\wedge e^{j}$ and $e^{ijk}$ the wedge product $e^{i}\wedge e^{j}\wedge e^{k}$. In the following, a basis in which the $\mathrm{SU}(3)$-structure has the expression \tagform@1 will be called an adapted basis.

In [6] it is described how the intrinsic torsion of an $\mathrm{SU}(3)$-structure, namely $\tau$, lies in a space of the form

[TABLE]

where $\mathcal{W}_{i}$ denote the irreducible components under the action of the group $\mathrm{SU}(3)$. This torsion can be described by the exterior derivatives of $\omega,\psi_{+}$ and $\psi_{-}$ and also in terms of the so called torsion forms. This latter description is given in [4] where the authors consider the natural action of the group $\mathrm{SU}(3)$ on $\Omega^{k}(M^{6})$, the space of $k$-forms on $M^{6}$. Thus, the different spaces of forms $\Omega^{k}(M^{6})$ can be splitted into $\mathrm{SU}(3)$ irreducible subspaces as follows:

$\Omega^{1}(M^{6})$ is irreducible,

$\Omega^{2}(M^{6})=\Omega^{2}_{1}(M^{6})\oplus\Omega^{2}_{6}(M^{6})\oplus\Omega^{2}_{8}(M^{6})$,

with

$\Omega^{2}_{1}(M^{6})=\{f\omega|f\in\mathcal{C}^{\infty}(M^{6})\},$

$\Omega^{2}_{6}(M^{6})=\{\ast_{6}J(\eta\wedge\psi_{+})|\eta\in\Omega^{1}(M^{6})\}=\{\sigma\in\Omega^{2}(M^{6})|J\sigma=\sigma\}$,

$\Omega^{2}_{8}(M^{6})=\{\sigma\in\Omega^{2}(M^{6})|\sigma\wedge\psi_{+}=0,\ast_{6}J\sigma=-\sigma\wedge\omega\}=\{\sigma\in\Omega^{2}(M^{6})|J\sigma=-\sigma,\sigma\wedge\omega^{2}=0\}$;

and

$\Omega^{3}(M^{6})=\Omega^{3}_{+}(M^{6})\oplus\Omega^{3}_{-}(M^{6})\oplus\Omega^{3}_{6}(M^{6})\oplus\Omega^{3}_{12}(M^{6})$,

with

[TABLE]

where $\ast_{6}$ denotes the Hodge star operator associated to the induced metric $g_{\omega,\psi_{\pm}}$ and the volume form $vol_{6}$. Notice that $\Omega^{k}_{d}(M^{6})$ denotes the $\mathrm{SU}(3)$-irreducible space of $k$-forms having dimension $d$. Decompositions of the spaces of $k$-forms for $k=4,5$ and $6$ need not to be detailled since they can be achieved via the Hodge star operator, $\ast_{6}\Omega^{k}_{d}(M^{6})=\Omega^{6-k}_{d}(M^{6})$.

With all these previous descriptions the derivatives of $\omega,\psi_{+}$ and $\psi_{-}$ can be decomposed into summands belonging to the $\mathrm{SU}(3)$-invariant spaces as follows (see [4] for details):

[TABLE]

where $\sigma_{0},\pi_{0}\in\mathcal{C}^{\infty}(M^{6}),\pi_{1},\nu_{1}\in\Omega^{1}(M^{6}),\pi_{2},\sigma_{2}\in\Omega^{2}_{8}(M^{6})$ and $\nu_{3}\in\Omega^{3}_{12}(M^{6})$ are the torsion forms of the $\mathrm{SU}(3)$-structure.

Some classes of $\mathrm{SU}(3)$-structures that are useful for our purposes are given in Table 1.

1.2. $\mathrm{G}_{2}$-structures

A $\mathrm{G}_{2}$-structure on a 7-dimensional differentiable manifold consists on a three form $\varphi$ such that it defines a metric, namely $g_{\varphi}$, a volume form $vol_{7}$ and a 2-fold vector cross product, see [7, 14]. The metric $g_{\varphi}$ can be recovered from $\varphi$ as

[TABLE]

with $X,Y\in\mathfrak{X}(M^{7})$. The presence of such structure on a manifold $M^{7}$ can be characterized by the existence of an adapted basis, i.e. a local basis of 1-forms $\{e^{1},\dots,e^{7}\}$ such that $\varphi$ can be described as:

[TABLE]

Concerning the intrinsic torsion of a $\mathrm{G}_{2}$-structure, namely $\mathcal{T}$, in [7] it is described how this torsion lies in a space of the form

[TABLE]

where $\mathcal{X}_{i}$ denotes the irreducible components under the action of the group $\mathrm{G}_{2}$. Thus, we can distinguish between 16 different classes of $\mathrm{G}_{2}$-structures, the so-called Fernández-Gray classes, which can be characterized by the behavior of the exterior derivative of $\varphi$ and $\ast_{7}\varphi$ where $\ast_{7}$ is the Hodge star operator induced by the $\mathrm{G}_{2}$-structure. In [3] it is given a description of the derivatives of $\varphi$ and $\ast_{7}\varphi$ as summands belonging to the different $\mathrm{G}_{2}$-invariant spaces $\mathcal{X}_{i}$.

To obtain this description it is considered the natural action of the group $\mathrm{G}_{2}$ on $\Omega^{k}(M^{7})$. Thus, the different spaces of forms $\Omega^{k}(M^{7})$ can be splitted into $\mathrm{G}_{2}$-irreducible subspaces as follows:

$\Omega^{1}(M^{7})$ is irreducible,

$\Omega^{2}(M^{7})=\Omega^{2}_{7}(M^{7})\oplus\Omega^{2}_{14}(M^{7})$,

with

$\Omega^{2}_{7}(M^{7})=\{\ast_{7}(\eta\wedge\ast_{7}\varphi)|\,\eta\in\Omega^{1}(M^{7})\}=\{\sigma\in\Omega^{2}(M^{7})|\sigma\wedge\varphi=2\ast_{7}\sigma\}$,

$\Omega^{2}_{14}(M^{7})=\{\sigma\in\Omega^{2}(M^{7})|\,\sigma\wedge\varphi=-\ast_{7}\sigma\}$;

and

$\Omega^{3}(M^{7})=\Omega^{3}_{1}(M^{7})\oplus\Omega^{3}_{7}(M^{7})\oplus\Omega^{3}_{27}(M^{7})$,

with

[TABLE]

Similarly to the previous case, $\Omega^{k}_{d}(M^{7})$ denotes the $\mathrm{G}_{2}$-irreducible space of $k$-forms which has dimension $d$. For the rest of dimensions ($k=4,5,6$ and $7$) use the relation: $\ast_{7}\Omega^{k}_{d}(M^{7})=\Omega^{7-k}_{d}(M^{7})$.

Thus, the derivatives of $\varphi$ and $\ast_{7}\varphi$ can be decomposed into summands belonging to the $\mathrm{G}_{2}$-invariant spaces as follows (see [3]):

[TABLE]

where $\tau_{0}\in\mathcal{C}^{\infty}(M^{7}),\tau_{1}\in\Omega^{1}(M^{7}),\tau_{2}\in\Omega^{2}_{14}(M^{7})$ and $\tau_{3}\in\Omega^{3}_{27}(M^{7})$ are the torsion forms.

In particular:

[TABLE]

The principal Fernández-Gray classes are given in Table 2:

2. Warped $\mathrm{G}_{2}$-structures

Consider two Riemannian manifolds, namely $(F,g_{F})$ and $(B,g_{B})$, and $f$ a non-vanishing real differentiable function on $B$. The warped product, denoted as $B\times_{f}F$, consists on the product manifold

[TABLE]

endowed with the metric $g_{f}=\pi_{1}^{*}(g_{B})+f^{2}\pi_{2}^{*}(g_{F})$ with $\pi_{1}$ and $\pi_{2}$ being the projections of $W$ onto $B$ and $F$ respectively.

Starting from an $\mathrm{SU}(3)$-structure $(\omega,\psi_{\pm})$ over $M^{6}$, and considering a function $f\in\mathcal{C}^{\infty}(M^{6})$ it is possible to construct a $\mathrm{G}_{2}$-structure $\varphi$ over $M^{7}=M^{6}\times S^{1}$ such that:

[TABLE]

with $s$ the coordinate on $S^{1}$ and $\alpha,\,\beta\in\mathbb{R}$ satisfying $\alpha^{2}+\beta^{2}=1$. Thus, the metric and the volume form of this $\mathrm{G}_{2}$-structure are given in terms of the $\mathrm{SU}(3)$-structure by:

[TABLE]

Observe that $g_{\varphi}=g_{f}$, so $M^{7}$ is in fact a warped product. In what follows we will call warped $\mathrm{G}_{2}$-structure to this $\mathrm{G}_{2}$-structure \tagform@5.

Remark 2.1**.**

If we consider the pair $(\alpha,\beta)=(1,0)$, this definition of warped $\mathrm{G}_{2}$-structure is exactly the one already given in [11].

The metrics $g_{\omega,\psi_{\pm}}$ and $g_{\varphi}$ on the base manifold $M^{6}$ and the warped product $M^{6}\times_{f}S^{1}$ respectively define two star operators $\ast_{6}$ and $\ast_{7}$ related by the following:

Lemma 2.2** (Lemma 3.2, [11]).**

Let $\eta\in\Omega^{k}(M^{6})$ be a differential $k$-form on $M^{6}$, and let $\ast_{6}$ and $\ast_{7}$ be the Hodge star operator determined by the $\mathrm{SU}(3)$-structure and the warped $\mathrm{G}_{2}$-structure, respectively. Then

[TABLE]

Hence from \tagform@5 and the previous lemma it can be checked that

[TABLE]

Remark 2.3**.**

The key idea of this section is to study how the $\mathrm{G}_{2}$-geometry of the warped product $M^{6}\times_{f}S^{1}$ forces conditions on the $\mathrm{SU}(3)$-geometry of the base $M^{6}$. Having this idea in mind, we are going to describe the torsion forms \tagform@4 of the warped $\mathrm{G}_{2}$-structure in terms of the torsion forms of the $\mathrm{SU}(3)$-structure and the warping function.

In the spirit of [20, Theorem 3.4] we can prove:

Theorem 2.4**.**

Let $(M^{6},\omega,\psi_{\pm})$ be an $\mathrm{SU}(3)$-manifold with torsion forms $\pi_{0},\sigma_{0},\pi_{1},\nu_{1},\pi_{2},\sigma_{2}$ and $\nu_{3}$. Then, the torsion forms \tagform@4 of a warped $\mathrm{G}_{2}$-manifold $(M^{7}=M^{6}\times_{f}S^{1},\,\varphi)$ are given by

[TABLE]

where $\eta_{i}$ are the following 1-forms:

[TABLE]

Proof.

The result holds after long computations where the definition of the spaces $\Omega^{k}_{d}(M^{6})$ are used. As hint, let us write down the expressions for $d\varphi,\,\ast_{7}(d\varphi)$ and $d(\ast_{7}\varphi)$. From \tagform@5 and \tagform@6 one gets:

[TABLE]

[TABLE]

[TABLE]

Finally, from \tagform@4 and using Lemma 2.2 the result is achieved after long and standard computations.

∎

Most of the Fernández-Gray classes of $\mathrm{G}_{2}$-structures are characterized in terms of the cancellation of some of their torsion forms (see Table 2). Using expressions \tagform@7, the cancellations of $\tau_{0},\,\tau_{1},\,\tau_{2}$ and $\tau_{3}$ are expressed by using the $\mathrm{SU}(3)$-torsion forms of the base $M^{6}$ and the warping function $f$.

Corollary 2.5**.**

Let $(M^{6},\omega,\psi_{\pm})$ be an $\mathrm{SU}(3)$-manifold. Thus, the torsion forms of the warped $\mathrm{G}_{2}$-structure satisfy:

[TABLE]

In Table 3 we show how the $\mathrm{G}_{2}$-geometry of the warped product $M^{6}\times_{f}S^{1}$ forces conditions on the $\mathrm{SU}(3)$-geometry of the base $M^{6}$.

Remark 2.6**.**

From Corollary 2.5, $\tau_{3}=0$ implies $\tau_{0}=0$, therefore nearly Parallel structures can not be achieved as warped $\mathrm{G}_{2}$-structures of the form \tagform@5.

3. The Laplacian flow and coflow of warped $\mathrm{G}_{2}$-structure of the form $M^{6}\times_{f}S^{1}$

Recall the definition of the Laplacian flow and coflow, that are respectively:

[TABLE]

where $\varphi(t)$ is a one-parameter family of $\mathrm{G}_{2}$-structures and $\Delta_{t}$, $\ast_{t}$ denote the Laplacian and the Hodge star operator induced by $\varphi(t)$ for every $t$.

Our objective in this section is to particularize the Laplacian flow and coflow considering one-parameter families of $\mathrm{G}_{2}$-structures obtained as warped products, i.e.

[TABLE]

From the previous expression, we derive the following:

[TABLE]

Now we focus on the 3-form $\Delta_{7}\varphi$, resp. the 4-form $\Delta_{7}\ast_{7}\varphi$. For a generic G2-structure, considering the formulas given in \tagform@3 of the exterior derivatives of $\varphi$ and $\ast_{7}\varphi$, a description of the Laplacian in terms of the torsion forms can be given as

[TABLE]

Since the Laplacian commutes with the Hodge star operator, $\Delta_{7}\ast_{7}=\ast_{7}\Delta_{7},$ combining \tagform@7 and \tagform@10 it is also posible to describe $\Delta_{7}\ast_{7}\varphi$ of a warped $\mathrm{G}_{2}$-structure in terms of the torsion forms of the $\mathrm{SU}(3)$-structure and the warping function $f$ for particular classes of $\mathrm{G}_{2}$-structures.

Provided that we are interested in the Laplacian flow, resp. coflow, we consider the 3-form $\Delta_{7}\varphi$, resp. the 4-form $\Delta_{7}\ast_{7}\varphi$, when $\varphi$ is closed, resp. coclosed. Let us start with the closed ones:

Proposition 3.1**.**

Let $\varphi$ be a warped closed $\mathrm{G}_{2}$-structure \tagform@5 on $M^{6}\times_{f}S^{1}$ where $(\omega,\psi_{\pm})$ is an $\mathrm{SU}(3)$-structure on $M^{6}$. Then $\Delta_{7}\varphi$ has the following expression:

[TABLE]

where $\alpha\pi_{2}-\beta\sigma_{2}=0$.

In the particular case that the warping function $f$ is constant $(d_{6}f=0)$, then

[TABLE]

Proof.

Since $\varphi$ is closed, $\tau_{0}=\tau_{1}=\tau_{3}=0$ and by \tagform@10

[TABLE]

where in view of \tagform@7

[TABLE]

For the case $f$ constant, since $\frac{1}{f}d_{6}f=-\nu_{1}$ (see Table  3) then $\nu_{1}=0$ and the result holds. ∎

Consider now coclosed $\mathrm{G}_{2}$-structures:

Proposition 3.2**.**

Let $\varphi$ be a warped coclosed $\mathrm{G}_{2}$-structure \tagform@5 on $M^{6}\times_{f}S^{1}$ where $(\omega,\psi_{\pm})$ is an $\mathrm{SU}(3)$-structure on $M^{6}$. Then $\Delta_{7}\ast_{7}\varphi$ has the following expression:

[TABLE]

where $\alpha\sigma_{i}+\beta\pi_{i}=0$ for $i=0,2$.

Moreover, if $f$ is constant, then

[TABLE]

Proof.

The condition $\varphi$ being coclosed is equivalent to $\tau_{1}=\tau_{2}=0$ and as a consequence of \tagform@10:

[TABLE]

Now, using \tagform@7:

[TABLE]

and the result follows. In order to prove \tagform@11, observe that $\pi_{1}=0$ according to Table 3.

∎

Remark 3.3**.**

In what follows, and similarly as in [11], we restrict our attention to the case when the warping function $f$ is constant over the base manifold $M^{6}$.

In order to obtain solutions of the Laplacian flow of a warped closed $\mathrm{G}_{2}$-structure, combining the expressions \tagform@9 and Proposition 3.1, we can set the system of equations that must be satisfied:

Proposition 3.4**.**

For a closed warped $\mathrm{G}_{2}$-structure \tagform@5, the equation of the Laplacian flow (LF) is equivalent to:

[TABLE]

where $\alpha\pi_{2}(t)-\beta\sigma_{2}(t)=0$.

Remark 3.5**.**

*For the particular case of $(\alpha,\beta)=(1,0)$, we recover the system already studied by Fino and Raffero in [11, Prop. 5.2]. *

Similarly, for the coflow, we get the following system of equations:

Proposition 3.6**.**

For a coclosed warped $\mathrm{G}_{2}$-structure \tagform@5, the equation of the Laplacian coflow (LcF) is equivalent to:

[TABLE]

where $\alpha\sigma_{i}(t)+\beta\pi_{i}(t)=0$ for $i=0,2$.

Corollary 3.7**.**

For the particular case of $(\alpha,\beta)=(0,1)$, the Laplacian coflow becomes:

[TABLE]

Remark 3.8**.**

For the Laplacian coflow we chose the parameters $(\alpha,\beta)$ to be $(0,1)$ in order to obtain equations depending on the torsion forms $\sigma_{0},\sigma_{2}$ and $\nu_{3}$ (see \tagform@2) which are the ones that appear in the canonical definitions of the $\mathrm{SU}(3)$-structures, nearly Kähler, symplectic half-flat and balanced, respectively (see equations \tagform@18, \tagform@21 and \tagform@27 in the next sections).

4. New solutions to the Laplacian coflow

Our main objective is to provide new long-time solutions $\varphi(t)$ for the Laplacian coflow \tagform@12. In what follows we will consider one parameter families of warped G2-structures \tagform@8 on $G\times S^{1}$, being $G$ a Lie group. The underlying $\mathrm{SU}(3)$-structures $(\omega(t),\psi_{+}(t),\psi_{-}(t))$ are left-invariant and can be locally described as

[TABLE]

where $\{x^{i}(t)\}$ denotes for every $t$ a local adapted basis and $x^{ij}$ stands for $x^{i}(t)\wedge x^{j}(t)$ and $x^{ijk}$ stands for $x^{i}(t)\wedge x^{j}(t)\wedge x^{k}(t)$. Our ansatz consists on stating that

[TABLE]

where $f_{i}(t)$ are differentiable non-vanishing real functions satisfying $f_{i}(0)=1$ and $\{h^{1},\ldots,h^{6}\}$ is an adapted basis for the SU(3)-structure for $t=0$. Notice that \tagform@14 defines in fact a global basis since we are considering parallelizable manifolds.

Direct computations show:

[TABLE]

[TABLE]

with $\mathcal{J}=\{(1,2,3,4),(1,2,5,6),(3,4,5,6)\}$.

[TABLE]

with $\mathcal{I}=\{(1,4,6),(2,3,6),(2,4,5)\}$.

As we mentioned before, the $\mathrm{G}_{2}$-geometry of the warped product imposes conditions on the $\mathrm{SU}(3)$-geometry of the base $M^{6}$. Concretely, the $\mathrm{G}_{2}$-structure is coclosed if and only if the corresponding $\mathrm{SU}(3)$-structure lies on the space $\mathcal{W}_{1}^{\pm}\oplus\mathcal{W}_{2}^{\pm}\oplus\mathcal{W}_{3}\oplus\mathcal{W}_{5}$ (see Table 3). Notice that if we consider a one-parameter family of SU(3)-structures $(\omega(t),\,\psi_{\pm}(t))$ belonging to the previous space for any $t$, then the corresponding warped G2-structure will remain coclosed for any $t$. Now we particularize \tagform@12 for some interesting cases of $\mathrm{SU}(3)$-structures lying on this space.

4.1. The nearly Kähler case

Recall that a nearly Kähler SU(3)-structure satisfies

[TABLE]

In particular, $\sigma_{2}=\nu_{3}=0$. Particularizing \tagform@12 for $\sigma_{2}(t)=\nu_{3}(t)=0,$ we get

[TABLE]

Observe that with this particular ansatz, the left-hand side of the first equation above is a combination of the 4-forms $x^{1234},x^{1256}$ and $x^{3456}$ (see \tagform@16); however, it can be easily proven that if $\eta$ is a one-form, then $\eta\wedge\psi_{-}(t)$ never belongs to the space generated by $x^{1234},x^{1256}$ and $x^{3456}$, unless $\eta=0$. Therefore, we need $d_{6}\sigma_{0}(t)=0$, which means that $\sigma_{0}(t)$ is constant as a differentiable function on $M^{6}$.

Now, the previous system simplifies as:

[TABLE]

Let us solve this system (as before, we denote $f_{i}(t)f_{j}(t)$ simply as $f_{ij}$).

Lemma 4.1**.**

If $\dfrac{\partial\omega^{2}(t)}{\partial t}=-3\sigma_{0}(t)^{2}\,\omega^{2}(t)$, then, $f_{12}=f_{34}=f_{56},$ where $f_{i}(t)$ are the functions in \tagform@14.

Proof.

Using the symplectic operator $L:\Omega^{q}(M)\to\Omega^{q+2}(M)$ defined by $L(\eta)=\eta\wedge\omega$, the previous equation can be expressed as:

[TABLE]

It happens that $L$ is injective for $q\leq n-1$, being $\dim M=2n$ [2]. Since in our case $n=3$, we have that

[TABLE]

Using \tagform@15, $\dfrac{\partial\omega(t)}{\partial t}=-\frac{3}{2}\sigma_{0}(t)^{2}\,\omega(t)$ if and only if

[TABLE]

which is equivalent to say

[TABLE]

In particular,

[TABLE]

where $c_{i}$ are constants. Since $f_{i}(0)=1$, we obtain that $f_{12}=f_{34}=f_{56}.$ ∎

For the second equation we get:

Lemma 4.2**.**

If $\frac{f^{\prime}(t)}{f(t)}\psi_{+}(t)+\dfrac{\partial\psi_{+}(t)}{\partial t}=-3\sigma_{0}(t)^{2}\,\psi_{+}(t)$, then, $f_{1}(t)=f_{2}(t),\,\,f_{3}(t)=f_{4}(t),\,\,f_{5}(t)=f_{6}(t),$ where $f_{i}(t)$ are the functions in \tagform@14.

Proof.

Arguing as before, if $\frac{f^{\prime}(t)}{f(t)}\psi_{+}(t)+\dfrac{\partial\psi_{+}(t)}{\partial t}=-3\sigma_{0}(t)^{2}\,\psi_{+}(t)$, then:

[TABLE]

In particular, observe that:

[TABLE]

where $c$ is a constant and we have used the fact that $f_{i}(0)=1$. So:

[TABLE]

[TABLE]

[TABLE]

where for the last equivalence we have used that $f_{i}(t)$ are continuous functions satisfying $f_{i}(0)=1$. ∎

We can combine the two previous results to conclude that $f_{i}(t)=f_{j}(t)$ for $i,j=1,\ldots,6$. If we denote $f_{i}(t)=F(t)$ for all $i=1,\dots,6$, then $(\omega(t),\psi_{\pm}(t))$ has the particular form:

[TABLE]

Lemma 4.3**.**

Let $(\omega(t),\psi_{\pm}(t))$ be the one-parameter family of $\mathrm{SU}(3)$-structures given in \tagform@20 where $(\omega,\psi_{\pm})$ is a nearly Kähler structure. Then $(\omega(t),\psi_{\pm}(t))$ is nearly Kähler for all $t$ if and only if $\sigma_{0}(t)=\frac{\sigma_{0}}{F(t)}$.

Proof.

Equation \tagform@20 implies that $d\omega(t)=F^{2}(t)d\omega,$ and $d\psi_{-}(t)=F^{3}(t)d\psi_{-}.$ Since $(\omega,\psi_{\pm})$ is nearly Kähler, one has

[TABLE]

or equivalently

[TABLE]

Therefore, \big{(}\omega(t),\psi_{\pm}(t)\big{)} is nearly Kähler for all $t$ if and only if $\sigma_{0}(t)=\frac{\sigma_{0}}{F(t)},$ and the result follows. ∎

In the next result we show how to solve the Laplacian coflow in this particular case.

Proposition 4.4**.**

Let $M^{6}$ be a manifold endowed with a nearly Kähler structure $(\omega,\psi_{\pm})$. Then the one-parameter family of warped $\mathrm{G}_{2}$-structures on $M^{6}\times_{f}S^{1}$ given by

[TABLE]

is a solution of the Laplacian coflow for $t\in\left(-\infty,\frac{2}{3\sigma^{2}_{0}}\right)$, being f(t)=c\Big{(}1-\frac{3\sigma_{0}^{2}}{2}\,t\Big{)}^{1/2},\,\,c\in\mathbb{R}^{*}.

Proof.

From Lemmas 4.1, 4.2 and 4.3, the system \tagform@19 with $(\omega(t),\psi_{\pm}(t))$ nearly Kähler for all $t$ is equivalent to

[TABLE]

whose solution is

[TABLE]

and the result follows. ∎

Remark 4.5**.**

Not many examples of nearly Kähler manifolds are known. Recently, new complete examples on $S^{6}$ and $S^{3}\times S^{3}$ have been described in [12] and [21]. Next we solve the Laplacian coflow using an explicit example of nearly Kähler structure appeared in [21].

Example 4.6**.**

Consider the sphere $S^{3}$, viewed as the Lie group $\mathrm{SU}(2)$ with the basis of left-invariant one-forms $\{\lambda^{1},\lambda^{2},\lambda^{3}\}$ satisfying

[TABLE]

Thus, $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ is the Lie algebra of $S^{3}\times S^{3}$ and its structure equations are:

[TABLE]

with $\{\nu^{i}\}$ the basis of left-invariant $1$-forms on the second sphere. The pair $(\omega,\psi_{+})$ with

[TABLE]

[TABLE]

where $\omega$ is the Kähler form and $\psi_{+}$ is the real part of the complex (3,0)-form, defines a nearly Kähler SU(3)-structure on $S^{3}\times S^{3}$. Observe that the basis $\{\lambda^{i},\,\nu^{i}\}$ is not adapted to the SU(3)-structure.

Consider $\{h^{1},\dots,h^{6}\}$ the basis of left-invariant $1$-forms on $S^{3}\times S^{3}$ given by

[TABLE]

This basis is adapted to the SU(3)-structure and $(\omega,\psi_{+})$ turns out to be nearly Kähler with $\sigma_{0}=-2$. Therefore, in view of Proposition 4.4, the one-parameter family of warped $\mathrm{G}_{2}$-structures on $(S^{3}\times S^{3})\times_{f}S^{1}$ given by

[TABLE]

and

[TABLE]

where $f(t)=c\,\left(1-6t\right)^{\frac{1}{2}}$, is a solution of the Laplacian coflow for all t\in\big{(}-\infty,\frac{1}{6}\big{)}.

4.2. The symplectic half-flat case

Recall that a symplectic half-flat SU(3)-structure satisfies

[TABLE]

In particular, $\sigma_{0}=\nu_{3}=0$. Particularizing \tagform@12 for $\sigma_{0}(t)=\nu_{3}(t)=0,$ we get

[TABLE]

Now, we get necessary conditions in order to solve the Laplacian coflow. Arguing similarly as Lemma 4.1 and providing that $\sigma_{0}(t)=0$, it is straightforward to see that the first equation of \tagform@22 holds if and only if

[TABLE]

The following technical result, that makes use of equation \tagform@17, states how to solve the coflow in the symplectic half-flat case:

Lemma 4.7**.**

Consider a warped coclosed $\mathrm{G}_{2}$-structure $\varphi$ on $M^{6}\times_{f}S^{1}$ where $(\omega,\psi_{\pm})$ is a symplectic half-flat $\mathrm{SU}(3)$-structure. Then $\varphi(t)$, given by \tagform@8, is a solution of the coflow \tagform@22 using the ansatz \tagform@14 if and only if $f(t),\,f_{1}(t),\,f_{3}(t)$ and $f_{5}(t)$ satisfy:

[TABLE]

where functions $A_{135}(t),A_{146}(t),A_{236}(t),A_{245}(t)$ are such that

[TABLE]

and $(\omega(t),\,\psi_{\pm}(t))$ is symplectic half-flat for all $t$.

In order to obtain examples and inspired in the solutions given in Proposition 4.4, we will consider the functions $f_{i}(t)$ of potential type, i.e.

[TABLE]

with $\alpha_{i}$ and $k$ real numbers. Thus the solutions of the coflow are of the form:

[TABLE]

where the basis $\{h^{1},\,\ldots,\,h^{6}\}$ is defined in \tagform@14.

Next we solve the Laplacian coflow on unimodular solvable Lie algebras.

Example 4.8**.**

Consider the Lie algebra $\mathfrak{e}(1,1)\oplus\mathfrak{e}(1,1)$ whose structure equations are

[TABLE]

The corresponding connected and simply connected Lie group $G$ admits a left-invariant symplectic half-flat structure which is given canonically by \tagform@1 in basis $\{h^{i}\}$. Let us consider a one-parameter family of $\mathrm{SU}(3)$-structures given by \tagform@13 with $x^{i}(t)=f_{i}(t)h^{i}$ being $f_{i}(t)$ of potential type as in \tagform@25. The structure equations of $\mathfrak{e}(1,1)\oplus\mathfrak{e}(1,1)$ with respect to the time-dependent basis $\{x^{i}(t)\}$ are

[TABLE]

In order to obtain solutions for the Laplacian coflow, and in view of \tagform@23, we can set

[TABLE]

With these values, we impose the preservation of the symplectic half-flat condition. It is easy to verify that $d\omega(t)=0$ for all $t$; $\psi_{+}(t)$ remains closed if and only if $\alpha_{1}=\alpha_{3}=0$, since

[TABLE]

So, $(\omega(t),\psi_{\pm}(t))$ is symplectic half-flat for all $t$ if and only if $\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=0.$ Observe that the structure equations are simply:

[TABLE]

Finally, to solve the second equation of \tagform@22 we make use of \tagform@24. Since $(\omega(t),\psi_{\pm}(t))$ is symplectic half-flat for all $t$, $\sigma_{2}(t)=-\ast_{t}d\psi_{-}(t)$, see \tagform@2, and therefore

[TABLE]

*which means that $A_{ijk}(t)=-2$. We obtain the system *

[TABLE]

which can be solved taking

[TABLE]

Therefore, the one-parameter family of $\mathrm{G}_{2}$-structures on $G\times_{f}S^{1}$ given by \tagform@26

[TABLE]

is a solution of the Laplacian coflow for all $t\in\mathbb{R}$.

In [10], the authors classify the 6-dimensional unimodular solvable Lie algebras admitting symplectic half-flat $\mathrm{SU}(3)$-structure and show that all the corresponding solvable Lie groups admit a co-compact discrete subgroup. In addition to the Lie algebra $\mathfrak{e}(1,1)\oplus\mathfrak{e}(1,1)$, in terms of an adapted basis $\{h^{i}\}_{i=1}^{6}$ to the $\mathrm{SU}(3)$-structure, the structure equations of these algebras are the following:

[TABLE]

In Table 4 we present long time solutions to the Laplacian coflow for $\mathrm{G}_{2}$-structures obtained as warped products of solvmanifolds endowed with symplectic half-flat $\mathrm{SU}(3)$-structures. These solutions can be obtained as follows: consider Lemma 4.7 with the potential functions given in \tagform@25 and a warping function also of potential type

[TABLE]

Thus, using \tagform@24, we obtain a linear system of equations in $\alpha_{i},\beta$ and $k$ that can be easily solved. Known the values of $\alpha_{i},\beta$ and $k$ and considering \tagform@26 we can give an explicit description of the solutions of the Laplacian coflow for each example. We also include the value of $d\sigma_{2}(t)$ in each case, necessary to compute the parameters of the solutions.

4.3. The balanced case

Recall that a balanced SU(3)-structure satisfies

[TABLE]

In particular, $\sigma_{0}=\sigma_{2}=0$. Particularizing \tagform@12 for $\sigma_{0}(t)=\sigma_{2}(t)=0,$ we get

[TABLE]

In this case, we can apply Lemma 4.2 with $\sigma_{0}(t)=0$ (compare the second equations in \tagform@19 and \tagform@28) obtaining the same conclusion, i.e., $f_{2k}(t)=f_{2k-1}(t)$ for $k=1,2,3$. Now, similarly to Lemma 4.7, we can set:

Lemma 4.9**.**

Consider a warped coclosed $\mathrm{G}_{2}$-structure $\varphi$ on $M^{6}\times_{f}S^{1}$ where $(\omega,\psi_{\pm})$ is a balanced $\mathrm{SU}(3)$-structure. Then $\varphi(t)$, given by \tagform@8, is a solution of the coflow \tagform@28 using the ansatz \tagform@14 if and only if $f(t),\,f_{1}(t),\,f_{3}(t)$ and $f_{5}(t)$ satisfy:

[TABLE]

where functions $B_{1234}(t),\,B_{1256}(t),\,B_{3456}(t)$ are such that

[TABLE]

and $(\omega(t),\,\psi_{\pm}(t))$ is balanced for all $t$.

The examples that we present in this case are the 6-dimensional nilpotent Lie algebras admitting balanced $\mathrm{SU}(3)$-structures, that are classified in [22]. In terms of an adapted basis to the balanced $\mathrm{SU}(3)$-structure, the structure equations are:

[TABLE]

We present long time solutions for the Laplacian coflow of $\mathrm{G}_{2}$-structures obtained as warped products of balanced nilmanifolds endowed with $\mathrm{SU}(3)$-structures. These solutions remain balanced for any $t$. As before, with the notation in Lemma 4.9 and functions of potential type \tagform@25 giving an explicit description of these solutions is equivalent to obtain the values of the parameters $\alpha_{i},\beta$ and $k$. Solving the corresponding linear equations these values are given in Table 5. The solutions $\varphi(t)$ of the coflow are of the form \tagform@26. We also include the value of $d\ast\nu_{3}(t)$ in each case, necessary to compute the parameters of the solutions.

Acknowledgments

The authors would like to thank Anna Fino, Jorge Lauret and Luis Ugarte for useful comments on the subject. This work has been partially supported by the projects MTM2017-85649-P (AEI/Feder, UE), and E22-17R “Álgebra y Geometría” (Gobierno de Aragón/FEDER)