# Flows of G2-structures on contact Calabi–Yau $7$-manifolds

Jason D. Lotay
_University of Oxford_ Henrique N. Sá Earp Julieth Saavedra
_University of Campinas (Unicamp)_

###### Abstract

We study the Laplacian flow and coflow on contact Calabi-Yau $7$-manifolds. We
show that the natural initial condition leads to an ancient solution of the
Laplacian flow with a finite time Type I singularity which is not a soliton,
whereas it produces an immortal (though not eternal and not self-similar)
solution of the Laplacian coflow which has an infinite time singularity, that
is Type IIb unless the transverse Calabi–Yau geometry is flat. The flows in
each case collapse (after normalising the volume) to a lower-dimensional
limit, which is either $\mathbb{R}$ for the Laplacian flow or standard
$\mathbb{C}^{3}$ for the Laplacian coflow. We also study the Hitchin flow in
this setting, which we show coincides with the Laplacian coflow up to
reparametrisation of time, and defines an (incomplete) Calabi–Yau structure on
the spacetime track of the flow.

###### Contents

  1. 1 Introduction
    1. 1.1 Laplacian coflow
    2. 1.2 Laplacian flow
    3. 1.3 Singularities
    4. 1.4 Hitchin flow
    5. 1.5 Outline and main results
  2. 2 Preliminaries on coclosed G2-structures
    1. 2.1 G2-structures and their torsion forms
    2. 2.2 Contact Calabi–Yau manifolds
  3. 3 The Laplacian coflow solution
    1. 3.1 Solving the flow
    2. 3.2 Metric and curvature
    3. 3.3 Full torsion tensor
    4. 3.4 Singularity analysis
  4. 4 The Laplacian flow solution
    1. 4.1 Solving the flow
    2. 4.2 Metric and curvature
    3. 4.3 Full torsion tensor
    4. 4.4 Singularity analysis
  5. 5 The Hitchin flow solution
    1. 5.1 Solving the flow
    2. 5.2 Calabi–Yau structure
  6. A Additional properties of coclosed G2-structures
    1. A.1 Contraction identities
    2. A.2 Hodge Laplacian
    3. A.3 Lie derivative

##  1 Introduction

Geometric flows have proven to be a powerful analytical tool in a variety of
geometric and topological problems. In the case of $\mathrm{G}_{2}$-geometry,
they provide a method to search for metrics with $\mathrm{G}_{2}$ holonomy,
which are then Ricci-flat, on an oriented and spin $7$-manifold $M$. This
corresponds to varying a $\mathrm{G}_{2}$-structure, given by a non-degenerate
$3$-form $\varphi$ on $M$, so that it becomes torsion-free. Flows in
$\mathrm{G}_{2}$-geometry were first introduced by Bryant [Bryant2006] and
have been studied by several authors, see e.g. [Bryant2011, Karigiannis2012,
Lauret2017, Fino2018, Grigorian2019, Karigiannis2020, Loubeau2019, Lotay2015,
Loubeau2021].

A $\mathrm{G}_{2}$-structure $\varphi$ determines a metric $g_{\varphi}$ and
orientation with Riemannian volume form $\mathrm{vol}_{\varphi}$. The _full
torsion tensor_ $T$ of $\varphi$ is a 2-tensor which is equivalent to
$\nabla^{g_{\varphi}}\varphi$, where $\nabla^{g_{\varphi}}$ is the Levi-Civita
connection of $g_{\varphi}$. Pairs $(M^{7},\varphi)$ which satisfy $T\equiv 0$
(i.e. so that $\varphi$ is torsion-free) are called
$\mathrm{G}_{2}$-_manifolds_ and are of particular interest since the holonomy
group of $g_{\varphi}$ in this case is contained in $\mathrm{G}_{2}$. However,
complete examples of $\mathrm{G}_{2}$-manifolds $(M,\varphi)$ are very
difficult to construct, especially when $M$ is required to be compact.
Fernandez and Gray [Fernandez1982] showed that the torsion-free condition is
equivalent to $\varphi$ being both _closed_ and _coclosed_ , i.e, ${\rm
d}\varphi=0$ and ${\rm d}\\!\ast_{\varphi}\\!\varphi=0$, where $*_{\varphi}$
is the Hodge star defined by $g_{\varphi}$ and $\mathrm{vol}_{\varphi}$. This
alternative viewpoint on the torsion-free condition as a system of nonlinear
PDE is fundamental to $\mathrm{G}_{2}$ geometry and to geometric flows.

###  1.1 Laplacian coflow

Our first goal is to study the _Laplacian coflow_ of
$\mathrm{G}_{2}$-structures, introduced by Karigiannis, McKay and Tsui
[Karigiannis2012]111In [Karigiannis2012], the flow (1.1) is written with an
additional minus sign on the right-hand side, but this seems incorrect given
the work on the Laplacian flow [Bryant2011, Lotay2017] and the modified
Laplacian coflow [Grigorian2013]. which is given by

$\frac{\partial\psi_{t}}{\partial t}=\Delta_{\psi_{t}}\psi_{t}:=({\rm d}{\rm
d}^{\ast_{t}}+{\rm d}^{\ast_{t}}{\rm d})\psi_{t},$ (1.1)

where $\psi_{t}:=\ast_{t}\varphi_{t}$ is the Hodge dual of the
$\mathrm{G}_{2}$-structure $\varphi_{t}$ (here, we write
$*_{t}:=*_{\varphi_{t}}$) and $\Delta_{\psi_{t}}$ is the Hodge Laplacian of
$g_{t}:=g_{\varphi_{t}}$ on $4$-forms. If $M$ is compact, critical points of
this flow (1.1) are then Hodge duals of non-degenerate 3-forms which are
closed and coclosed, i.e. torsion-free.

We restrict the Laplacian coflow (1.1) to the case where $\psi_{t}$ is closed,
i.e. the $\mathrm{G}_{2}$-structure $\varphi_{t}$ is coclosed. In this
context, solutions of (1.1) (if they exist) preserve the cohomology class
$[\psi_{t}]=[\psi_{0}]\in H^{4}(M)$, for all $t$, and the flow seeks critical
points in this class. When $M$ is compact, (1.1) can be interpreted as the
gradient flow of _Hitchin’s volume functional[Hitchin2001a]_ and so the volume
of $M$ increases monotonically along the flow (see [Grigorian2013]).

One immediate problem with the Laplacian coflow is that the $4$-form
$\psi=*_{\varphi}\varphi$ is generated by both the 3-forms $\varphi$ and
$-\varphi$: in particular, $\psi$ does not determine the orientation on $M$.
However, it is natural to assume that the orientation on $M$ stays fixed along
the flow, which is determined by a choice of $\mathrm{G}_{2}$-structure dual
to the initial 4-form. Another key problem, which is much more serious, is
that it is not known whether solutions to (1.1) actually exist in general,
even for an arbitrarily short time. There is a modification of the coflow
[Grigorian2013] which does have guaranteed short-time existence, but the
critical points are no longer necessarily closed and coclosed 4-forms, and so
it does not currently appear to be useful as a tool for studying key problems
in $\mathrm{G}_{2}$ geometry.

Finally, even if one has short-time existence and uniqueness of the flow
(1.1), one finds that the torsion tensor is not guaranteed to satisfy a heat-
type equation. Thus the Laplacian coflow is not a _reasonable_ (sometimes also
called _Ricci-like_) flow of ${\rm G}_{2}$-structures in the sense of
[Chen2018], so the analytic theory developed there does not apply. This issue
is linked to the lack of parabolicity of the Laplacian coflow, even in the
direction of closed 4-forms (see [Grigorian2013]).

In Theorem 1.1, we will find immortal families of coclosed
$\mathrm{G}_{2}$-structures solving the Laplacian coflow (1.1).

###  1.2 Laplacian flow

A geometric flow of $\mathrm{G}_{2}$-structures which has received much more
attention is the Laplacian flow introduced by Bryant [Bryant2006]:

$\frac{\partial\varphi_{t}}{\partial t}=\Delta_{\varphi_{t}}\varphi_{t}:=({\rm
d}{\rm d}^{*_{t}}+{\rm d}^{*_{t}}{\rm d})\varphi_{t},$ (1.2)

where we again write $*_{t}$ for the Hodge star induced by the
$\mathrm{G}_{2}$-structure $\varphi_{t}$, and $\Delta_{\varphi_{t}}$ is the
Hodge Laplacian of $g_{t}:=g_{\varphi_{t}}$ on 3-forms. Again, if $M$ is
compact, the Laplacian flow (1.2) has torsion-free $\mathrm{G}_{2}$-structures
as its critical points.

Typically, one restricts the Laplacian flow (1.2) to _closed_
$\mathrm{G}_{2}$-structures, because in that setting one obtains a good
analytic theory, the cohomology class of the flowing
$\mathrm{G}_{2}$-structure $\varphi_{t}$ is preserved (so we are seeking
torsion-free $\mathrm{G}_{2}$-structures in a given class), and the flow (in
the compact setting) can be interpreted as the gradient flow of the Hitchin
volume functional on $[\varphi_{t}]=[\varphi_{0}]$. See [Bryant2011,
Karigiannis2020, Lotay2017, Lotay2015] for more details on the Laplacian flow.
By contrast, it is not known whether starting (1.2) at a _coclosed_
$\mathrm{G}_{2}$-structure may preserve coclosedness, however much one might
naively expect this from a flow which is initially in the direction of coexact
forms. Indeed, to our knowledge, there are no particular examples in the
literature of a Laplacian flow actually preserving the coclosed property. We
also note that we do not currently have any general analytic theory for the
Laplacian flow, except when restricted to closed $\mathrm{G}_{2}$-structures.

In Theorem 1.2, we will obtain ancient families of coclosed
$\mathrm{G}_{2}$-structures solving the Laplacian flow (1.2).

###  1.3 Singularities

For various flows of $\mathrm{G}_{2}$-structures (cf. [Lotay2017, Chen2018])
it has been shown that blow-up of the following quantity characterises the
formation of finite-time singularities:

$\Lambda(x,t)=(|Rm(x,t)|^{2}_{g_{t}}+|T(x,t)|^{4}_{g_{t}}+|\nabla^{g_{t}}T(x,t)|^{2}_{g_{t}})^{\frac{1}{2}}$

for $x\in M$ and time $t$. We then let

$\Lambda(t)=\sup_{x\in M}\Lambda(x,t)$ (1.3)

for convenience and introduce the following terminology, motivated by the
singularity classification in Ricci flow (cf. [Hamilton2006]) and the work in
[Lotay2017, Chen2018].

######  Definition 1.1.

Suppose that $(M^{7},\varphi_{t},\psi_{t},g_{t})$ is a solution to a flow of
$\mathrm{G}_{2}$-structures on a closed manifold on a maximal time interval
$[0,T)$ and let $\Lambda(t)$ be as in (1.3).

If we have a finite-time singularity, i.e. $T<\infty$, we say that the
solution forms

  * •

a _Type I singularity_ (rapidly forming) if
$\sup_{t\in[0,T)}(T-t)\Lambda(t)<\infty$; and otherwise

  * •

a _Type IIa singularity_ (slowly forming) if
$\sup_{t\in[0,T)}(T-t)\Lambda(t)=\infty$.

If we have an _infinite-time_ singularity, where $T=\infty$, then it is

  * •

a _Type IIb singularity_ (slowly forming) if
$\sup_{t\in[0,\infty)}t\Lambda(t)=\infty$; and otherwise

  * •

a _Type III singularity_ (rapidly forming) if
$\sup_{t\in[0,\infty)}t\Lambda(t)<\infty$.

In Theorem 1.3, we will apply this singularity classification to our solutions
to the flows (1.1) and (1.2).

###  1.4 Hitchin flow

The _Hitchin flow_ 222It should be made clear that the Hitchin flow for
$\mathrm{G}_{2}$-structures is not a geometric flow in the usual sense, but it
is nonetheless a valid evolution equation for $\mathrm{G}_{2}$-structures.
[Hitchin2001a] for a family $\varphi_{t}$ of $\mathrm{G}_{2}$-structures on
$M^{7}$ with dual 4-form $\psi_{t}$ is given by solving

$\frac{\partial\psi_{t}}{\partial t}={\rm
d}\varphi_{t}\quad\text{and}\quad{\rm d}\psi_{t}=0$ (1.4)

for $t\in I\subseteq\mathbb{R}$. We note that the Hitchin flow (1.4) is an
evolution equation for _coclosed_ $\mathrm{G}_{2}$-structures. The
significance of the flow is that a solution defines a $4$-form $\Phi$ on the
product $I\times M$ by

$\Phi={\rm d}t\wedge\varphi_{t}+\psi_{t},$ (1.5)

which is closed and so defines a torsion-free $\mathrm{Spin}(7)$-structure on
$I\times M$ (and thus a Ricci-flat metric with holonomy contained in
$\mathrm{Spin}(7)$). Relatively little is known about the Hitchin flow and
since the 7-manifolds we are studying naturally admit coclosed
$\mathrm{G}_{2}$-structures, it is worth examining the Hitchin flow in this
context.

In Theorem 1.4, we will obtain immortal solutions to the Hitchin flow (1.4),
which bear a close relation to our Laplacian coflow (1.1) solutions, and
define an incomplete _Calabi-Yau_ structure on the 8-dimensional spacetime
track of the flow.

###  1.5 Outline and main results

We will consider the flows described above on a _contact Calabi–Yau (cCY)_
7-manifold $(M^{7},g,\eta,\Upsilon)$, where $(M^{7},g)$ is a Sasakian
7-manifold with Riemannian metric $g$, contact form $\eta$ and transverse
Kähler form $\omega={\rm d}\eta\in\Omega^{1,1}(M)$, and
$\Upsilon\in\Omega^{3,0}(M)$ is a transverse holomorphic volume form; here
$(p,q)$ denotes basic bidegree with respect to the horizontal distribution
$\mathcal{D}=\ker\eta$, cf. §2.2. An important class of compact examples of
cCY 7-manifolds are _Calabi–Yau links_ , which are total spaces of
$S^{1}$-(orbi)bundles over Calabi–Yau $3$-orbifolds famously listed by
Candelas-Lynker-Schimmrigk [Candelas1990]; these will be discussed in Example
2.1.

On a cCY 7-manifold there exists a natural 1-parameter family of coclosed
$\mathrm{G}_{2}$-structures defined, for each $\varepsilon>0$, by

$\varphi=\varepsilon\eta\wedge\omega+\mathop{\mathrm{Re}}\Upsilon,$ (1.6)

with induced metric $g_{\varphi}$ (which equals $g$ when $\varepsilon=1$) and
corresponding dual $4$-form

$\psi=\ast_{\varphi}\varphi=\frac{1}{2}\omega^{2}-\varepsilon\eta\wedge\mathop{\mathrm{Im}}\Upsilon.$
(1.7)

In the present paper, we are interested in flows of
$\mathrm{G}_{2}$-structures on cCY 7-manifolds starting at the natural
coclosed $\mathrm{G}_{2}$-structure in (1.6) or its dual 4-form in (1.7). All
of our results are stated in terms of the following standard data (S):

$\bullet$ $(M^{7},g_{0},\eta_{0},\Upsilon_{0})$ a contact Calabi-Yau
7-manifold; $\bullet$ $\mathcal{D}_{0}=\ker\eta_{0}$ its horizontal
distribution, and $\omega_{0}={\rm d}\eta_{0}$ its transverse Kähler form;
$\bullet$ for $\varepsilon>0$, the $\mathrm{G}_{2}$-structure defined by (1.6)
and (1.7), i.e.
$\varphi_{0}=\varepsilon\eta_{0}\wedge\omega_{0}+\mathop{\mathrm{Re}}\Upsilon_{0}\quad\text{and}\quad\psi_{0}=\frac{1}{2}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0}.$
(S)

In §3, we consider the Laplacian coflow on a cCY 7-manifold and prove the
following.

######  Theorem 1.1 (Laplacian coflow on contact Calabi–Yau 7-manifolds).

In the setup (S), the Laplacian coflow (1.1) on $M^{7}$, with initial data
determined by $\varphi_{0}$, is solved by the following family of coclosed
$\mathrm{G}_{2}$-structures $\varphi_{t}$, with associated metric $g_{t}$,
volume form $\mathrm{vol}_{t}$ and dual 4-form $\psi_{t}$:

$\displaystyle\varphi_{t}$ $\displaystyle=\varepsilon{\rm
p}(t)^{-1}\eta_{0}\wedge\omega_{0}+{\rm
p}(t)^{3}\mathop{\mathrm{Re}}\Upsilon_{0};$ $\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}{\rm
p}(t)^{4}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0};$
$\displaystyle g_{t}$ $\displaystyle=\varepsilon^{2}{\rm
p}(t)^{-6}\eta_{0}^{2}+{\rm p}(t)^{2}g_{\mathcal{D}_{0}};$
$\displaystyle\mathrm{vol}_{t}$ $\displaystyle=\varepsilon{\rm
p}(t)^{3}\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$

where ${\rm p}(t)=(1+10\varepsilon^{2}t)^{1/10}$ and
$t\in(-\frac{1}{10\varepsilon^{2}},\infty)$. Hence, the solution of the
Laplacian coflow is immortal, with a finite time singularity (backwards in
time) at $t=-\frac{1}{10\varepsilon^{2}}$.

######  Remark 1.1.

If $M^{7}$ is compact then the volume of $M$ determined by the
$\mathrm{G}_{2}$-structure $\varphi$ on $M$ is:

$\mathcal{H}(\varphi):=\textrm{Vol}(M,\varphi)=\frac{1}{7}\int_{M}\varphi\wedge\psi.$

Along the Laplacian coflow solution given in Theorem 1.1 for a compact cCY
7-manifold we have that

$\mathcal{H}(\varphi_{t})=(10t+1)^{3/10}\mathcal{H}(\varphi_{0}).$

Hence, the Hitchin functional on the cohomology class $[\psi_{0}]$, which is
just $\mathcal{H}(\varphi_{t})$, tends to infinity as $t\to\infty$ and tends
to $0$ as $t\to-1/10$. In particular, recalling that the Laplacian coflow is
the gradient flow of the Hitchin functional on $[\psi_{0}]$, we observe that
the Hitchin functional is unbounded above and does not have a positive lower
bound on $[\psi_{0}]$.

In $\S$4, we switch attention to the Laplacian flow and prove the following
result.

######  Theorem 1.2 (Laplacian flow on contact Calabi–Yau 7-manifolds).

In the setup (S), the Laplacian flow (1.2) on $M^{7}$, with initial data
determined by $\varphi_{0}$, is solved by the following family of _coclosed_
$\mathrm{G}_{2}$-structures $\varphi_{t}$ on $M^{7}$, with associated metric
$g_{t}$, volume form $\mathrm{vol}_{t}$ and dual 4-form $\psi_{t}$ given by

$\displaystyle\varphi_{t}$ $\displaystyle=\varepsilon{\rm
q}(t)\eta_{0}\wedge\omega_{0}+\mathop{\mathrm{Re}}\Upsilon_{0};$
$\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}\omega_{0}^{2}-\varepsilon{\rm
q}(t)\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0};$ $\displaystyle g_{t}$
$\displaystyle=\varepsilon^{2}{\rm q}(t)^{2}\eta_{0}^{2}+g_{\mathcal{D}_{0}};$
$\displaystyle\mathrm{vol}_{t}$ $\displaystyle=\varepsilon{\rm
q}(t)\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$

where ${\rm q}(t)=(1-8\varepsilon^{2}t)^{-1/2}$ and
$t\in(-\infty,\frac{1}{8\varepsilon^{2}})$. Hence, the solution of the
Laplacian flow is ancient, with a finite time singularity (forwards in time)
at $t=\frac{1}{8\varepsilon^{2}}$.

######  Remark 1.2.

Here, we find that the Laplacian flow solution in Theorem 1.2 has the property
that $[\psi_{t}]$ is not constant unless $[\psi_{0}]=0$. Moreover, when
$[\psi_{0}]\neq 0$, one can detect the finite time singularity of the flow
using the cohomology class $[\psi_{t}]$, which is somewhat reminiscent of the
Kähler–Ricci flow.

######  Remark 1.3.

We can interpret taking $\varepsilon\to 0$ in Theorems 1.1 and 1.2 as
obtaining degenerate eternal solutions to the Laplacian flow or coflow, which
is simply given by the Calabi–Yau structure on the horizontal distribution
$\mathcal{D}_{0}$. In the quasi-regular setting for the contact Calabi–Yau
structure, where $M$ fibres by circles over a Calabi–Yau 3-orbifold $Z$,
sending $\varepsilon$ to $0$ is equivalent to shrinking the circle fibres and
collapsing to $Z$.

If we instead consider $\varepsilon\to\infty$ in Theorems 1.1 and 1.2, we
appear to obtain a degenerate solution to the flow, which is only defined for
non-negative times for the coflow and for non-positive times for the flow.
Again, in the quasi-regular setting, this would be equivalent to expanding the
circle fibres to infinite size (usually interpreted as becoming lines) over
the Calabi–Yau 3-orbifold base.

When $M$ is compact, we can describe the singularities of our flow solutions
in terms of Definition 1.1.

######  Theorem 1.3 (Singularities of the Laplacian flow and coflow).

Suppose we are in the setup (S) and $M$ is compact.

  * (a)

The Laplacian coflow solution in Theorem 1.1 has an infinite-time Type IIb
singularity, unless the transverse metric on $\mathcal{D}_{0}$ is flat, in
which case it has an infinite-time Type III singularity.

  * (b)

The Laplacian flow solution in Theorem 1.2 has a finite-time Type I
singularity.

######  Remark 1.4.

This theorem gives the first examples of compact solutions to the Laplacian
coflow which have an infinite-time Type IIb singularity. It also provides the
first compact solutions to the Laplacian flow which have a finite-time
singularity, and the first Type I singularity which is _not_ a soliton. Such
finite-time singularities are not yet known to occur in the Laplacian flow for
_closed_ $\mathrm{G}_{2}$-structures on compact manifolds, which moreover are
not expected to be Type I because there are no compact shrinking solitons in
this setting.

In Section 5 we study the Hitchin flow for the natural coclosed
$\mathrm{G}_{2}$-structures on cCY 7-manifolds, and obtain the following
result.

######  Theorem 1.4 (Hitchin flow on contact Calabi–Yau 7-manifolds).

In the setup (S), the Hitchin flow (1.4) on $M^{7}$, with initial data
determined by $\varphi_{0}$, is solved by the following family of coclosed
$\mathrm{G}_{2}$-structures $\varphi_{t}$, with associated metric $g_{t}$,
volume $\mathrm{vol}_{t}$ and dual 4-form $\psi_{t}$ given by

$\displaystyle\varphi_{t}$ $\displaystyle=\varepsilon{\rm
r}(t)^{-1}\eta_{0}\wedge\omega_{0}+{\rm
r}(t)^{3}\mathop{\mathrm{Re}}\Upsilon_{0};$ $\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}{\rm
r}(t)^{4}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0};$
$\displaystyle g_{t}$ $\displaystyle=\varepsilon^{2}{\rm
r}(t)^{-6}\eta_{0}^{2}+{\rm r}(t)^{2}g_{\mathcal{D}_{0}};$
$\displaystyle\mathrm{vol}_{t}$ $\displaystyle=\varepsilon{\rm
r}(t)\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$

where ${\rm r}(t)=(1+\frac{5}{2}\varepsilon t)^{1/5}$ and
$t\in(-\frac{2}{5\varepsilon},\infty)$. This Hitchin flow solution coincides
with the Laplacian coflow in Theorem 1.1, up to reparametrisation of time, and
defines an incomplete Calabi–Yau structure on
$(-\frac{2}{5\varepsilon},\infty)\times M$, with Kähler form
$\widehat{\omega}$, metric $\widehat{g}$ with
$\mathrm{Hol}(\widehat{g})\subseteq{\rm SU}(4)$, and holomorphic volume form
$\widehat{\Upsilon}$, as follows:

$\displaystyle\widehat{\omega}$ $\displaystyle=\varepsilon{\rm r}(t)^{-3}{\rm
d}t\wedge\eta_{0}+{\rm r}(t)^{2}\omega_{0};$ $\displaystyle\widehat{g}$
$\displaystyle={\rm d}t^{2}+\varepsilon^{2}{\rm r}(t)^{-6}\eta_{0}^{2}+{\rm
r}(t)^{2}g_{0};$ $\displaystyle\widehat{\Upsilon}$ $\displaystyle=({\rm
r}(t)^{3}{\rm d}t+i\varepsilon\eta_{0})\wedge\Upsilon_{0}.$

######  Remark 1.5.

The relation between contact Calabi–Yau 7-manifolds, the Hitchin flow and
Calabi–Yau structures on 8-manifolds was suggested to the first author by
Simon Salamon. It is also interesting to observe the close connection between
the Laplacian coflow and the Hitchin flow in this setting.

We also include, in Appendix A, some additional computations which emerged
during the course of our research and that we believe will be of further use
in the study of flows of coclosed $\mathrm{G}_{2}$-structures.

######  Remark 1.6.

A natural question, if perhaps a little bold, is whether flows on cCY
$7$-manifolds might actually converge to a torsion-free
$\mathrm{G}_{2}$-structure. While cautioning against excessive optimism, we
notice that there are many compact examples with no apparent topological
obstructions, on which moreover such a structure would induce a metric with
holonomy group $\mathrm{G}_{2}$, cf. Remark 2.2 in §2.2.

Acknowledgements: The authors would like to thank Simon Salamon, Mark Haskins
and Andrés Moreno for some valuable discussions. JL and HSE were supported by
a UK Royal Society Newton Mobility Award
[NMG$\backslash$R1$\backslash$191068]. JL is also partially supported by the
Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics
($\\#724071$ Jason Lotay). HSE has also been supported by the São Paulo
Research Foundation (Fapesp) [2018/21391-1] and the Brazilian National Council
for Scientific and Technological Development (CNPq) [311128/2020-3]. JPS was
supported by the Coordination for the Improvement of Higher Education
Personnel-Brazil (CAPES) [88882.329037/2019-1].

##  2 Preliminaries on coclosed G2-structures

In this section, we collect definitions and important properties regarding
(coclosed) $\mathrm{G}_{2}$-structures which will be useful in the study of
flows on contact Calabi–Yau 7-manifolds.

###  2.1 G2-structures and their torsion forms

A $\mathrm{G}_{2}$-structure on a (orientable and spin) 7-manifold $M$ is a
reduction of the structure group of $TM$ to $\mathrm{G}_{2}$, where
$\mathrm{G}_{2}$ is viewed as the subgroup of $\mathrm{GL}(7,\mathbb{R})$
containing all linear maps preserving the $3$-form

$\varphi_{0}=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}\in\Lambda^{3}(\mathbb{R}^{7})^{*},$

where $e^{i}=dx^{i}$, $e^{ij}=e^{i}\wedge e^{j}$ etc. It is equivalently
determined by a $3$-form $\varphi$ on $M$ such that $(TM,\varphi)$ is
pointwise isomorphic to $(\mathbb{R}^{7},\varphi_{0})$. A
$\mathrm{G}_{2}$-structure $\varphi$ determines a Riemannian metric
$g_{\varphi}$ and an orientation given by the Riemannian volume form
$\mathrm{vol}_{\varphi}$ so that

$6g_{\varphi}(X,Y)\mathrm{vol}_{\varphi}=(X\lrcorner\varphi)\wedge(Y\lrcorner\varphi)\wedge\varphi$

for any tangent vectors $X,Y$ on $M$. For simplicity, we write
$g=g_{\varphi}$. The metric and the orientation determine a Hodge star
operator $\ast_{\varphi}$, so we have the dual $4$-form
$\psi=\ast_{\varphi}\varphi$ of the $\mathrm{G}_{2}$-structure $\varphi$.

A $\mathrm{G}_{2}$-structure gives rise to a decomposition of differential
forms on $M$ corresponding to irreducible $\mathrm{G}_{2}$ representations. In
particular, the spaces $\Omega^{2}$ and $\Omega^{3}$ of $2$-forms and
$3$-forms decompose as

$\displaystyle\Omega^{2}$
$\displaystyle=\Omega^{2}_{7}\oplus\Omega^{2}_{14}\quad\text{and}\quad\Omega^{3}=\Omega^{3}_{1}\oplus\Omega_{7}^{3}\oplus\Omega^{3}_{27},$
(2.1)

where $\Omega^{k}_{l}$ has (pointwise) dimension $l$ and this decomposition is
orthogonal with respect to the metric $g$. Via the Hodge star, this defines
corresponding decompositions of the 4-forms $\Omega^{4}$ and 5-forms
$\Omega^{5}$. If we let $S^{2}$ denote the symmetric 2-tensors on $M$ then, as
in [Bryant2006], we define a linear operator ${\rm
j}_{\varphi}:\Omega^{3}\rightarrow S^{2}$ by

$\displaystyle{\rm j}_{\varphi}(\gamma)(X,Y)$
$\displaystyle=\ast_{\varphi}((X\lrcorner\varphi)\wedge(Y\lrcorner\varphi)\wedge\gamma),$
(2.2)

where $X,Y$ are tangent vectors on $M$. Then ${\rm j}_{\varphi}$ is surjective
with kernel equal to $\Omega^{3}_{7}$ (see e.g. [Karigiannis2007]*Proposition
2.17). We can also define an injective linear operator ${\rm
i}_{\varphi}:S^{2}\to\Omega^{3}_{1}\oplus\Omega^{3}_{27}$ as in [Bryant2006]
which is (up to scaling) a right inverse for ${\rm j}_{\varphi}$, and is given
locally using summation notation by

$\displaystyle{\rm i}_{\varphi}(h)$
$\displaystyle=\frac{1}{2}h_{i}^{l}\varphi_{ljk}e^{ijk},$ (2.3)

where $h\in S^{2}$ is given locally by $h_{ij}e^{i}e^{j}$. We note that ${\rm
i}_{\varphi}(g)=3\varphi$, ${\rm j}_{\varphi}(\varphi)=6g$ and

${\rm j}_{\varphi}\circ{\rm i}_{\varphi}(h)=2(\mathop{\mathrm{tr}}\nolimits
h)g+4h,$ (2.4)

where $\mathop{\mathrm{tr}}\nolimits h=g^{ij}h_{ij}$ is the trace of $h$ with
respect to $g$. For later use, we let $S^{2}_{0}$ denote the trace-free
symmetric 2-tensors on $M$ and note that ${\rm
i}_{\varphi}:S^{2}_{0}\to\Omega^{3}_{27}$ is an isomorphism.

Given any $\mathrm{G}_{2}$-structure $\varphi$, there exist unique
differential forms $\tau_{0}\in\Omega^{0}$, $\tau_{1}\in\Omega^{1}$,
$\tau_{2}\in\Omega_{14}^{2}$ and $\tau_{3}\in\Omega^{3}_{27}$, using the
decomposition of forms in (2.1), such that (see e.g. [Bryant2006]*Proposition
1)

$\displaystyle{\rm d}\varphi$
$\displaystyle=\tau_{0}\psi+3\tau_{1}\wedge\varphi+\ast\tau_{3},$ (2.5)
$\displaystyle{\rm d}\psi$
$\displaystyle=4\tau_{1}\wedge\psi+\tau_{2}\wedge\varphi.$ (2.6)

Together, the forms $\\{\tau_{0},\tau_{1},\tau_{2},\tau_{3}\\}$ are called the
_intrinsic torsion forms_ of the $\mathrm{G}_{2}$-structure $\varphi$. The
_full torsion tensor_ $T$ is defined locally by the formula

$\nabla_{i}\varphi_{jkl}=T_{i}^{m}\psi_{mjkl}$ (2.7)

and may be expressed using the torsion forms as (see e.g.
[Karigiannis2007]*Theorem 2.27)

$T=\frac{\tau_{0}}{4}g-\tau_{1}^{\sharp}\lrcorner\varphi-\frac{1}{2}\tau_{2}-\frac{1}{4}{\rm
j}_{\varphi}(\tau_{3}).$ (2.8)

By (2.6), a $\mathrm{G}_{2}$-structure is coclosed if and only if $\tau_{1}=0$
and $\tau_{2}=0$. Hence, the full torsion tensor of a coclosed
$\mathrm{G}_{2}$-structure simplifies to the symmetric $2$-tensor

$T=\frac{\tau_{0}}{4}g-\frac{1}{4}{\rm j}_{\varphi}(\tau_{3})\in S^{2}.$ (2.9)

###  2.2 Contact Calabi–Yau manifolds

In this section we review the properties of contact Calabi–Yau (cCY)
7-manifolds and the natural coclosed $\mathrm{G}_{2}$-structures on them. The
original sources for cCY manifolds are [Tomassini2008, Habib2015]; for a more
detailed exposition and study of $\mathrm{G}_{2}$-geometry on cCY
$7$-manifolds, see [Calvo-Andrade2020].

######  Definition 2.1.

A _contact Calabi–Yau_ (cCY) manifold is a quadruple
$(M^{2n+1},g,\eta,\Upsilon)$ such that

  * •

$(M,g)$ is a $(2n+1)$-dimensional Sasakian manifold with contact form $\eta$;

  * •

$\Upsilon$ is a nowhere vanishing transversal form on $\mathcal{D}=\ker\eta$
of type $(n,0)$ with

$\Upsilon\wedge{\overline{\Upsilon}}=c_{n}\omega^{n},\quad{\rm d}\Upsilon=0,$

where $c_{n}=(-1)^{\frac{n(n+1)}{2}}{\rm i}^{n}$ and $\omega={\rm d}\eta$. We
shall write:

$\mathop{\mathrm{Re}}\Upsilon:=\frac{\Upsilon+{\overline{\Upsilon}}}{2},\quad\mathop{\mathrm{Im}}\Upsilon:=\frac{\Upsilon-{\overline{\Upsilon}}}{2{\rm
i}}.$

######  Remark 2.1.

A contact Calabi–Yau manifold $(M,g,\eta,\Upsilon)$ has a transverse
Calabi–Yau geometry on $\mathcal{D}=\ker\eta$, given by $g|_{\mathcal{D}}$,
$\omega$ and $\Upsilon$. When the Sasakian geometry is regular or quasi-
regular, $M$ is an $S^{1}$-(orbi)bundle over a Calabi–Yau orbifold, but the
Sasakian geometry can also be irregular, and then there is no
$S^{1}$-fibration structure on $M$ compatible with the contact Calabi–Yau
geometry.

We now see how to relate the contact Calabi–Yau geometry in 7 dimensions to
$\mathrm{G}_{2}$ geometry (cf. [Habib2015]*Corollary 6.8 and [lotay2021]).

######  Proposition 2.1.

Let $(M^{7},g,\eta,\Upsilon)$ be a contact Calabi-Yau 7-manifold. Then $M$
carries a 1-parameter of coclosed $\mathrm{G}_{2}$-structures defined by
(1.6), for $\varepsilon>0$, where $\omega={\rm d}\eta$. Furthermore, ${\rm
d}\varphi=\varepsilon\omega\wedge\omega$ and its corresponding dual $4$-form
is given by (1.7).

Notice that $\psi$ is manifestly closed, as $\omega$ and $\Upsilon$ are closed
and $\omega\wedge\Upsilon=0$ by bidegree considerations, i.e. $\varphi$ is
coclosed, and its torsion is encoded in ${\rm
d}\varphi=\varepsilon\omega\wedge\omega.$ Various other properties of the
$\varepsilon$-family (1.6) are studied in [lotay2021].

We now provide a concrete means for finding examples of contact Calabi–Yau
7-manifolds, which are circle (orbi)bundles over Calabi–Yau 3-orbifolds.

######  Example 2.1 (Calabi–Yau links).

Given $w=(w_{0},\dots,w_{4})\in\mathbb{Q}^{5}$, the zero set of a $w$-weighted
homogeneous polynomial $f\in\mathbb{C}[z_{0},\dots,z_{4}]$ of degree
$d=\sum_{i=0}^{4}w_{i}$ is an affine hypersurface in $\mathbb{C}^{5}$ with an
isolated singularity at $0$.

Its link $M_{f}$ on a local $9$-sphere is a compact and $2$-connected smooth
cCY $7$-manifold, fibering by circles over a Calabi–Yau $3$-orbifold
$Z\subseteq\mathbb{P}^{4}(w)$ by the weighted Hopf fibration [Calvo-
Andrade2020]*Theorem 1.1:

${M_{f}^{7}}$${S^{9}}$${Z}$${\mathbb{P}^{4}(w)}$

In particular, $Z$ can be taken to be any of the weighted Calabi–Yau $3$-folds
listed in [Candelas1990]. For a detailed survey on Calabi–Yau links, see
[Calvo-Andrade2020]*§2. The $\mathbb{C}$-family of Fermat quintics yields the
simplest examples, and indeed the only ones for which the base $Z$ is smooth.

######  Remark 2.2.

Regarding the possibility of a torsion-free $\mathrm{G}_{2}$-structure on a
Calabi–Yau link, we observe a number of favourable topological circumstances,
in the terms of [Joyce2000]*§10.2.

Since it admits natural $\mathrm{G}_{2}$-structures such as in (S), every cCY
$7$-manifold is obviously orientable and spin. In particular, CY links are
$2$-connected, so trivially $\pi_{1}(M_{f})$ is finite and there is no
obstruction coming from an intersection form on $H^{2}(M_{f})$. Moreover, CY
links are compact, so if a torsion-free $\mathrm{G}_{2}$-structure exists at
all, then its induced Riemannian metric will have holonomy group precisely
$\mathrm{G}_{2}$. Finally, since every such CY link $M_{f}\to
Z\subseteq\mathbb{P}^{4}(w)$ admits solutions to the heterotic Bianchi
identity [lotay2021]*Theorem 1, its first Pontryagin class $p_{1}(M_{f})$
coincides with the pullback of the second Chern class $c_{2}(Z)$ of the
weighted projective $3$-orbifold, which is certainly not trivial in general,
e.g. for the Fermat quintic itself.

##  3 The Laplacian coflow solution

In this section we solve the Laplacian coflow (1.1) of
$\mathrm{G}_{2}$-structures on a contact Calabi–Yau 7-manifold by an explicit
ansatz, and we study the behaviour of the metric and torsion along the flow.

###  3.1 Solving the flow

We want to consider the Laplacian coflow starting at the natural coclosed
$\mathrm{G}_{2}$-structure defined by (1.6) and (1.7) on a cCY setup (S):

$\varphi_{0}=\varepsilon\eta_{0}\wedge\omega_{0}+\mathop{\mathrm{Re}}\Upsilon_{0}\quad\text{and}\quad\psi_{0}=\frac{1}{2}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0}.$
(3.1)

To this end, we consider the family of $\mathrm{G}_{2}$-structures given by

$\varphi_{t}=f_{t}h_{t}^{2}\eta_{0}\wedge\omega_{0}+h_{t}^{3}\mathop{\mathrm{Re}}\Upsilon_{0},$
(3.2)

for functions $f_{t},h_{t}$ depending only on time, with

$f_{0}=\varepsilon\quad\text{and}\quad h_{0}=1.$ (3.3)

The induced metric and associated volume form are then given by

$g_{t}=f_{t}^{2}\eta_{0}^{2}+h_{t}^{2}g_{\mathcal{D}_{0}}\quad\text{and}\quad\mathrm{vol}_{t}=f_{t}h_{t}^{6}\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$
(3.4)

with

$\mathrm{vol}_{\mathcal{D}}=\frac{1}{3!}\omega_{0}^{3}=\frac{i}{8}\Upsilon_{0}\wedge{\overline{\Upsilon}}_{0}=\frac{1}{4}\mathop{\mathrm{Re}}\Upsilon_{0}\wedge\mathop{\mathrm{Im}}{\overline{\Upsilon}}_{0}.$
(3.5)

It follows from (3.2), (3.4), and (3.5) that

$\psi_{t}=\frac{1}{2}h_{t}^{4}\omega_{0}^{2}-f_{t}h_{t}^{3}\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0}.$
(3.6)

We observe that (3.1) is indeed the instance at $t=0$ of (3.2) and (3.6).

######  Lemma 3.1.

Let $\varphi_{t}$ be a $\mathrm{G}_{2}$-structure as in (3.2). Then we have

${\rm d}\varphi_{t}=f_{t}h_{t}^{2}\omega_{0}^{2},\qquad{\rm
d}\psi_{t}=0\quad\text{and}\quad\ast_{t}{\rm
d}\varphi_{t}=2f_{t}^{2}\eta_{0}\wedge\omega_{0}.$ (3.7)

Hence, the torsion forms of $\varphi_{t}$ as in (2.5) are

$(\tau_{0})_{t}=\frac{6f_{t}}{7h_{t}^{2}},\quad(\tau_{1})_{t}=0,\quad(\tau_{2})_{t}=0,\quad(\tau_{3})_{t}=\frac{8}{7}f_{t}^{2}\eta_{0}\wedge\omega_{0}-\frac{6}{7}f_{t}h_{t}\mathop{\mathrm{Re}}\Upsilon_{0}.$
(3.8)

###### Proof.

From (3.2), (3.6), and Definition 2.1, we easily see that

${\rm d}\varphi_{t}=f_{t}h_{t}^{2}\omega_{0}^{2}\quad\text{and}\quad{\rm
d}\psi_{t}=0,$ (3.9)

since ${\rm d}\eta_{0}=\omega_{0}$ and $\omega_{0}\wedge\Upsilon_{0}=0$. We
also note from (3.4) and (3.9) that

$\begin{split}\ast_{t}{\rm
d}\varphi_{t}&=\ast_{t}(f_{t}h_{t}^{2}\omega_{0}\wedge\omega_{0})=\ast_{t}(2f_{t}h_{t}^{-2}\cdot\textstyle\frac{1}{2}h_{t}^{4}\omega_{0}^{2})=2f_{t}h_{t}^{-2}\cdot
f_{t}h_{t}^{2}\eta_{0}\wedge\omega_{0}\\\
&=2f_{t}^{2}\eta_{0}\wedge\omega_{0},\end{split}$

which then yields (3.7).

We can now compute the torsion forms of $\varphi_{t}$ as follows. First,

$\begin{split}(\tau_{0})_{t}&=\frac{1}{7}\ast_{t}(\varphi_{t}\wedge{\rm
d}\varphi_{t})=\frac{1}{7}\ast_{t}\big{(}(f_{t}h_{t}^{2}\eta_{0}\wedge\omega_{0}+h_{t}^{3}\mathop{\mathrm{Re}}\Upsilon_{0})\wedge
f_{t}h_{t}^{2}\omega_{0}^{2}\big{)}\\\
&=\frac{1}{7}\ast_{t}f_{t}^{2}h_{t}^{4}\eta_{0}\wedge\omega_{0}^{3}=\frac{6f_{t}}{7h_{t}^{2}}.\end{split}$
(3.10)

We know that $(\tau_{1})_{t}=0$ and $(\tau_{2})_{t}=0$ since $\varphi_{t}$ is
coclosed. Furthermore, from (3.9) and (3.10), we have

$\begin{split}(\tau_{3})_{t}&=\ast_{t}{\rm
d}\varphi_{t}-(\tau_{0})_{t}\varphi_{t}\\\
&=2f_{t}^{2}\eta_{0}\wedge\omega_{0}-\frac{6f_{t}}{7h_{t}^{2}}(f_{t}h_{t}^{2}\eta_{0}\wedge\omega_{0}+h_{t}^{3}\mathop{\mathrm{Re}}\Upsilon_{0})\\\
&=\frac{8}{7}f_{t}^{2}\eta_{0}\wedge\omega_{0}-\frac{6}{7}f_{t}h_{t}\mathop{\mathrm{Re}}\Upsilon_{0}.\end{split}$

Equation (3.8) then follows. ∎

We can now prove Theorem 1.1, by finding $f_{t},h_{t}$ so that (3.6) is a
solution of the Laplacian coflow (1.1).

###### Proof of Theorem 1.1.

On the cCY setup (S), consider the family of $\mathrm{G}_{2}$-structures given
by (3.2) and (3.6). We may then compute the Laplacian of $\psi_{t}$ using
(3.9) and Definition 2.1:

$\begin{split}\Delta_{t}\psi_{t}&={\rm d}\ast_{t}{\rm d}\varphi_{t}={\rm
d}(2f_{t}^{2}\eta_{0}\wedge\omega_{0})=2f_{t}^{2}\omega_{0}\wedge\omega_{0}.\end{split}$
(3.11)

Differentiating (3.6) with respect to $t$ and using (3.11), we can compute
$\frac{\partial\psi}{\partial t}=\Delta_{t}\psi_{t}$ and then equate the
coefficients of $\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0}$ and
$\omega_{0}^{2}$ (since the flow preserves the class of (3.6)) to obtain

$\frac{d}{dt}h_{t}^{4}=4f_{t}^{2},\quad\frac{d}{dt}f_{t}h_{t}^{3}=0.$ (3.12)

Therefore, from (3.3) and the second equation in (3.12) we conclude that

$f_{t}=\varepsilon h_{t}^{-3}.$ (3.13)

Substituting (3.13) into (3.12), we have

$\frac{d}{dt}h_{t}^{4}=4\varepsilon^{2}h_{t}^{-6}.$ (3.14)

The ODE (3.14) can be easily solved and so, together with (3.3) and (3.13), we
find that

$\displaystyle
f_{t}=\varepsilon(1+10\varepsilon^{2}t)^{-3/10}\quad\text{and}\quad
h_{t}=(1+10\varepsilon^{2}t)^{1/10}.$ (3.15)

In conclusion, we have found a solution to the Laplacian coflow (1.1) in the
cCY setup (S), with initial condition (3.1), which induces the following
family of $\mathrm{G}_{2}$-structures and their duals, metrics and volume
forms:

$\displaystyle\varphi_{t}$
$\displaystyle=\varepsilon(1+10\varepsilon^{2}t)^{-1/10}\eta_{0}\wedge\omega_{0}+(1+10\varepsilon^{2}t)^{3/10}\mathop{\mathrm{Re}}\Upsilon_{0};$
(3.16) $\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}(1+10\varepsilon^{2}t)^{2/5}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0};$
(3.17) $\displaystyle g_{t}$
$\displaystyle=\varepsilon^{2}(1+10\varepsilon^{2}t)^{-3/5}\eta_{0}^{2}+(1+10\varepsilon^{2}t)^{1/5}g_{\mathcal{D}_{0}};$
(3.18) $\displaystyle\mathrm{vol}_{t}$
$\displaystyle=\varepsilon(1+10\varepsilon^{2}t)^{3/10}\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$
(3.19)

where $\mathcal{D}_{0}=\ker\eta_{0}$, defined for all
$t\in(-\frac{1}{10\varepsilon^{2}},\infty)$. ∎

######  Remark 3.1.

We notice that the solution to the Laplacian coflow we have obtained is
_immortal_ (i.e. exists for all positive time), but it is not eternal, since
it fails to exist for $t\leq-\frac{1}{10\varepsilon^{2}}$. If $M$ is compact,
this will be the unique solution of the form (3.2) satisfying (3.3), and we
expect that it will be the unique solution starting at (3.1) in general, but
such a general theory is lacking as mentioned in $\S$1.1.

######  Remark 3.2.

We see from (3.19) that the volume form is strictly increasing pointwise in
time. We therefore confirm that, if $M$ is compact, then its volume is indeed
strictly increasing in time, tending to infinity:

$\text{Vol}(M,g_{t})\to\infty\quad\text{as }t\to\infty.$

We also notice that $\psi_{t}$ does indeed stay in the cohomology class
$[\psi_{0}]$.

However, suppose the contact Calabi–Yau structure is quasi-regular, so that
$M$ is an $S^{1}$-(orbi)bundle over a Calabi–Yau 3-orbifold $Z$. We then
observe that, by (3.11),

$\Delta_{t}\psi_{t}=2f_{t}^{2}\omega_{0}\wedge\omega_{0},$

which is a transverse 4-form that will descend to $Z$ and cannot be exact on
$Z$. Hence, the Laplacian coflow does not induce a flow of ${\rm
SU}(3)$-structures on $Z$ with forms staying in some fixed cohomology class.

###  3.2 Metric and curvature

It is worth studying aspects of the asymptotic behaviour of the solution to
the Laplacian coflow we have found in terms of the metric geometry.

It is straightforward to see from (3.18) that, as $t\to\infty$, the direction
dual to $\eta_{0}$ on $M$ collapses whilst the transverse geometry given by
$\mathcal{D}_{0}$ expands. On the other hand, the reverse situation occurs as
$t\to-\frac{1}{10\varepsilon^{2}}$.

######  Example 3.1.

For illustration, let us suppose that $M$ is compact and that the cCY
structure is quasi-regular, so that $M$ is an $S^{1}$-(orbi)bundle over a
compact Calabi–Yau 3-orbifold $Z$ with metric $g_{Z}$. Then, as $t\to\infty$
the circle fibres collapse, whilst the base $Z$ expands to become
$\mathbb{C}^{3}$ with the flat metric, since the coefficient of
$g_{\mathcal{D}_{0}}=g_{Z}$ in (3.18) tends to infinity. As
$t\to-\frac{1}{10\varepsilon^{2}}$, on the other hand, the base $Z$ collapses
to a point and the circles expand to become the real line $\mathbb{R}$.

In the analysis so far we have neglected the fact that the Laplacian coflow is
the gradient flow of the volume functional, and so the volume must always be
increasing along the flow. To remedy this, we may rescale the family
$(M,g_{t})$ so that the volume is fixed and obtain the following.

######  Lemma 3.2.

Let $M$ be compact. After normalising $(M,g_{t})$ to a fixed volume, the
Laplacian coflow solution collapses to $\mathbb{R}$, as
$t\to-\frac{1}{10\varepsilon^{2}}$, and to $\mathbb{C}^{3}$ with the flat
metric, as $t\to\infty$.

###### Proof.

Fixing the volume to be constant is equivalent to multiplying the metric by
$(1+10\varepsilon^{2}t)^{-3/35}$ by (3.19), which gives

$(1+10\varepsilon^{2}t)^{-3/35}g_{t}=(1+10\varepsilon^{2}t)^{-24/35}\eta_{0}^{2}+(1+10\varepsilon^{2}t)^{4/35}g_{\mathcal{D}_{0}}.$

The result then follows. ∎

We now take the standard approach to understanding the behaviour of volume
along a family of compact manifolds, by normalising the diameter. From the
formula (3.18) for the metric $g_{t}$, we immediately obtain:

######  Lemma 3.3.

Let $M$ be compact. After normalising $(M,g_{t})$ to unit diameter, the
Laplacian coflow solution is volume-collapsing to a 1-dimensional space
$M_{-}$ as $t\to-\frac{1}{10\varepsilon^{2}}$ and to a 6-dimensional space
$M_{+}$ as $t\to\infty$. If the Sasakian structure is quasi-regular, so that
$M$ is an $S^{1}$-orbibundle over a Calabi–Yau 3-orbifold $Z$, then
$M_{-}=S^{1}$ and $M_{+}=Z$.

In the light of what is known for the Laplacian flow of closed
$\mathrm{G}_{2}$-structures, cf. [Lotay2015]*Theorem 8.1, we might expect that
a condition on the uniform continuity of the metrics along the Laplacian
coflow of coclosed $\mathrm{G}_{2}$-structures, together with a (pointwise)
bound on the torsion tensor, would lead to long-time existence results. Since
no such general theory currently exists, it is therefore useful to examine the
uniform continuity properties of the solution to the Laplacian coflow we have
found.

######  Lemma 3.4.

Let $\varphi_{t}$ be the solution to the Laplacian coflow given by (3.16).
Then the associated metric $g_{t}$ is uniformly continuous (in $t$) on any
compact interval contained in $(-\frac{1}{10\varepsilon^{2}},\infty)$, but it
is not uniformly continuous on $(-\frac{1}{10\varepsilon^{2}},T)$ or
$(T,\infty)$ for any $T$.

###### Proof.

We immediately see from the formula (3.18) for the metrics $g_{t}$ that, since
the 2-tensors $\eta_{0}^{2}$ and $g_{\mathcal{D}_{0}}$ are fixed, the uniform
continuity in $t$ of $g_{t}$ on an interval $I$ is equivalent to the uniform
continuity of both $(1+10\varepsilon^{2}t)^{-3/5}$ and
$(1+10\varepsilon^{2}t)^{1/5}$ on $I$. The result follows. ∎

We now study the behaviour of the Riemann curvature tensor along our Laplacian
coflow solution, based on some computations of the Riemannian geometry of cCY
$7$-manifolds carried out in detail in [lotay2021]*§3.1.

######  Proposition 3.5.

Let $\varphi_{t}$ be the solution to the Laplacian coflow given by (3.16) with
associated metric $g_{t}$ as in (3.18). Let $Rm_{t}$ denote the Riemann
curvature tensor of $g_{t}$ and let $Rm_{0}^{\mathcal{D}_{0}}$ denote the
curvature of the transverse connection on $\mathcal{D}_{0}$ induced by the
Levi-Civita connection of $g_{0}$. Then

$|Rm_{t}|_{g_{t}}^{2}=(1+10\varepsilon^{2}t)^{-2/5}|Rm_{0}^{\mathcal{D}_{0}}|_{g_{0}}^{2}+c_{0}\varepsilon^{4}(1+10\varepsilon^{2}t)^{-2}$

for some constant $c_{0}>0$.

###### Proof.

If we write $g_{t}$ as in (3.4) for functions $f_{t},h_{t}$ given in (3.15),
we may observe that if we let

$\overline{g}_{t}=\frac{f_{t}^{2}}{h_{t}^{2}}\eta_{0}^{2}+g_{\mathcal{D}_{0}}$
(3.20)

then $g_{t}=h_{t}^{2}\overline{g}_{t}$. Hence, if $\overline{Rm}_{t}$ is the
Riemann curvature tensor of $\overline{g}_{t}$ in (3.20), we deduce that

$|Rm_{t}|_{g_{t}}=\frac{1}{h_{t}^{2}}|\overline{Rm}_{t}|_{\overline{g}_{t}}.$
(3.21)

It is a direct consequence of [lotay2021]*Proposition 3.2 that the transverse
connection on $\mathcal{D}_{0}$ induced by $\overline{g}_{t}$ is the same as
for $g_{0}$, and hence its curvatures are independent of $t$. It follows from
[lotay2021]*Proposition 3.8 that the remaining components of the Riemann
curvature of $\overline{g}_{t}$ scale by $f_{t}^{2}/h_{t}^{2}$ in comparison
to the curvature of the standard Sasakian metric $g$. We deduce that

$|\overline{Rm}_{t}|^{2}_{\overline{g}_{t}}=|Rm_{0}^{\mathcal{D}_{0}}|_{g_{0}}^{2}+\frac{c_{0}f_{t}^{4}}{h_{t}^{4}}$
(3.22)

for some constant $c_{0}>0$. The assertion now follows from (3.15), (3.21),
and (3.22). ∎

######  Remark 3.3.

We see from Proposition 3.5 that the curvature of the metric $g_{t}$ given by
the solution to the Laplacian coflow decays to $0$ as $t\to\infty$ but blows
up as $t$ approaches the finite negative time singularity at
$-\frac{1}{10\varepsilon^{2}}$. However, if we consider $t|Rm_{t}|_{g_{t}}$,
this will tend to infinity as $t\to\infty$ unless the curvature
$Rm^{\mathcal{D}_{0}}_{0}$ of the transverse connection is zero. However, we
see that $(1+10\varepsilon^{2}t)|Rm_{t}|_{g_{t}}$ will remain bounded as
$t\to-\frac{1}{10\varepsilon^{2}}$. We will return to these observations
later.

###  3.3 Full torsion tensor

We now wish to analyse the behaviour of the torsion along our solution to the
Laplacian coflow. We begin by writing down the full torsion tensor for the
solution.

######  Proposition 3.6.

The full torsion tensor $T_{t}$ of the solution to the Laplacian coflow in
(3.16)–(3.19) is

$\begin{split}T_{t}&=\frac{1}{4}(\tau_{0})_{t}g_{t}-\frac{1}{4}{\rm
j}_{\varphi_{t}}\big{(}(\tau_{3})_{t}\big{)}\\\
&=\frac{3\varepsilon}{14}(1+10\varepsilon^{2}t)^{-1/2}g_{t}\\\
&\quad-\frac{1}{14}{\rm
j}_{\varphi_{t}}\left(4\varepsilon^{2}(1+10\varepsilon^{2}t)^{-3/5}\eta_{0}\wedge\omega_{0}-3\varepsilon(1+10\varepsilon^{2}t)^{-1/5}\mathop{\mathrm{Re}}\Upsilon_{0}\right)\\\
&=-\frac{3}{2}\varepsilon^{3}(1+10\varepsilon^{2}t)^{-11/10}\eta_{0}^{2}+\frac{1}{2}\varepsilon(1+10\varepsilon^{2}t)^{-3/10}g_{\mathcal{D}_{0}}.\end{split}$
(3.23)

###### Proof.

This first two expressions for $T_{t}$ follow from (2.9), Lemma 3.1 and
(3.15).

For the final expression, we first let $\xi_{0}$ be the dual vector field to
$\eta_{0}$, we let $X,Y\in\mathcal{D}_{0}$ and consider $\varphi_{t}$ as in
(3.2) for functions $f_{t}$ and $h_{t}$. We recall that, since
$(\omega_{0},\Upsilon_{0})$ defines an ${\rm SU}(3)$-structure on
$\mathcal{D}_{0}$ we have that
$\omega_{0}\wedge\mathop{\mathrm{Re}}\Upsilon_{0}=0$,

$\displaystyle(X\lrcorner\mathop{\mathrm{Re}}\Upsilon_{0})\wedge(Y\lrcorner\mathop{\mathrm{Re}}\Upsilon_{0})\wedge\omega_{0}$
$\displaystyle=2g_{\mathcal{D}_{0}}(X,Y)\mathrm{vol}_{\mathcal{D}_{0}}$ (3.24)
$\displaystyle(X\lrcorner\omega_{0})\wedge(Y\lrcorner\mathop{\mathrm{Re}}\Upsilon_{0})\wedge\mathop{\mathrm{Re}}\Upsilon_{0}$
$\displaystyle=-2g_{\mathcal{D}_{0}}(X,Y)\mathrm{vol}_{\mathcal{D}_{0}}.$
(3.25)

We then see from (2.2), (3.2), (3.4), and (3.24) that

$\displaystyle{\rm
j}_{\varphi_{t}}(\eta_{0}\wedge\omega_{0})(\xi_{0},\xi_{0})$
$\displaystyle=\ast_{t}(f_{t}^{2}h_{t}^{4}\eta_{0}\wedge\omega_{0}^{3})=6\frac{f_{t}}{h_{t}^{2}},$
(3.26) $\displaystyle{\rm
j}_{\varphi_{t}}(\eta_{0}\wedge\omega_{0})(\xi_{0},X)$ $\displaystyle=0,$
(3.27) $\displaystyle{\rm j}_{\varphi_{t}}(\eta_{0}\wedge\omega_{0})(X,Y)$
$\displaystyle=\ast_{t}(2h_{t}^{6}g_{\mathcal{D}_{0}}(X,Y)\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}})=\frac{2}{f_{t}}g_{\mathcal{D}_{0}}(X,Y).$
(3.28)

Similarly, using (3.25),

$\displaystyle{\rm
j}_{\varphi_{t}}(\mathop{\mathrm{Re}}\Upsilon_{0})(\xi_{0},\xi_{0})$
$\displaystyle=0,$ (3.29) $\displaystyle{\rm
j}_{\varphi_{t}}(\mathop{\mathrm{Re}}\Upsilon_{0})(\xi_{0},X)$
$\displaystyle=0,$ (3.30) $\displaystyle{\rm
j}_{\varphi_{t}}(\mathop{\mathrm{Re}}\Upsilon_{0})(X,Y)$
$\displaystyle=*_{t}(4f_{t}h_{t}^{5}g_{\mathcal{D}_{0}}(X,Y)\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}})=\frac{4}{h_{t}}g_{\mathcal{D}_{0}}(X,Y),$
(3.31)

Applying (3.26)–(3.31) along with the particular expressions for $f_{t}$ and
$h_{t}$ in (3.15) then gives that

${\rm
j}_{\varphi_{t}}\left(4\varepsilon^{2}(1+10\varepsilon^{2}t)^{-3/5}\eta_{0}\wedge\omega_{0}-3\varepsilon(1+10\varepsilon^{2}t)^{-1/5}\mathop{\mathrm{Re}}\Upsilon_{0}\right)\\\
=24\varepsilon^{3}(1+10\varepsilon^{2}t)^{-11/10}\eta_{0}^{2}-4\varepsilon(1+10\varepsilon^{2}t)^{-3/10}g_{\mathcal{D}_{0}}.$
(3.32)

Substituting (3.32) into (3.23) gives the claimed final expression for
$T_{t}$. ∎

Given the description of the full torsion tensor in Proposition 3.6, we can
compute its norm, the norm of its gradient and its divergence as follows.

######  Proposition 3.7.

Let $T_{t}$ be the full torsion tensor of the solution to the Laplacian coflow
in (3.16)–(3.19). Then

$\displaystyle|T_{t}|_{g_{t}}^{2}$
$\displaystyle=\frac{15}{4}\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1},$
$\displaystyle|\nabla_{t}T_{t}|_{g_{t}}^{2}$
$\displaystyle=c_{0}\varepsilon^{4}(1+10\varepsilon^{2}t)^{-2},$
$\displaystyle\mathrm{div}_{t}T_{t}$ $\displaystyle=0,$

where $c_{0}>0$ is a constant, $\nabla_{t}$ is the Levi-Civita connection of
$g_{t}$ and $\mathrm{div}_{t}$ is the divergence with respect to the metric
$g_{t}$.

###### Proof.

We know from Proposition 3.6 that

$T_{t}=-2\varepsilon^{3}(1+10\varepsilon^{2}t)^{-11/10}\eta_{0}^{2}+\frac{1}{2}\varepsilon(1+10\varepsilon^{2}t)^{-1/2}g_{t}.$
(3.33)

On the other hand, from (3.18),

$|\varepsilon^{2}\eta_{0}^{2}|_{g_{t}}^{2}=(1+10\varepsilon^{2}t)^{6/5}\quad\text{and}\quad|g_{\mathcal{D}_{0}}|^{2}_{g_{t}}=6(1+10\varepsilon^{2}t)^{-2/5}.$

We then deduce the first claimed equation:

$|T_{t}|_{g_{t}}^{2}=\frac{9}{4}\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1}+\frac{3}{2}\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1}=\frac{15}{4}\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1}.$

For the remaining claimed equations, we know that $\nabla_{t}g_{t}=0$ and that
$(\nabla_{t})_{\xi_{0}}\xi_{0}=0$ where $\xi_{0}$ is the vector field dual to
$\eta_{0}$, since $\xi_{0}$ generates a local symmetry of the
$\mathrm{G}_{2}$-structure $\varphi_{t}$, and hence the metric $g_{t}$, for
all $t$. We deduce from (3.33) that $\mathrm{div}_{t}T_{t}=0$ as claimed and
that the only non-zero terms in $\nabla_{t}T_{t}$ arise from
$(\nabla_{t})_{X}\eta_{0}^{2}$ for $X\in\mathcal{D}_{0}$.

We now see from [lotay2021]*Proposition 3.2 that, for a metric of the form
$g_{t}=f_{t}^{2}\eta_{0}^{2}+h_{t}^{2}g_{\mathcal{D}_{0}}$, and
$X\in\mathcal{D}_{0}$ so that $g_{t}(X,X)=1$, we have
$\big{(}(\nabla_{t})_{X}\eta_{0}\big{)}^{\sharp}\in\mathcal{D}_{0}$, and

$|(\nabla_{t})_{X}(f_{t}^{2}\eta_{0}^{2})|_{g_{t}}^{2}=\frac{c_{0}f_{t}^{2}}{h_{t}^{4}},$

for some constant $c_{0}>0$. This is because the component of the connection
form relevant to $(\nabla_{t})_{X}\eta_{0}$ scales by $f_{t}/h_{t}$ and the
metric on $\mathcal{D}_{0}$ scales by $h_{t}^{2}$. Choosing $f_{t}$ and
$h_{t}$ as in (3.15), we finally obtain

$\displaystyle|(\nabla_{t})_{X}T_{t}|^{2}_{g_{t}}$
$\displaystyle=|2\varepsilon(1+10\varepsilon^{2}t)^{-1/2}(\nabla_{t})_{X}(\varepsilon^{2}(1+10\varepsilon^{2}t)^{-3/5}\eta_{0}^{2})|^{2}_{g_{t}}$
$\displaystyle=4\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1}\cdot
c_{0}\varepsilon^{2}(1+10\varepsilon^{2}t)^{-1}$
$\displaystyle=4c_{0}\varepsilon^{4}(1+10\varepsilon^{2}t)^{-2}.\qed$

######  Remark 3.4.

We see from Proposition 3.7 that $|T_{t}|_{g_{t}}^{2}$ and
$|\nabla_{t}T_{t}|_{g_{t}}$ have the same dependence on $t$, as we would
expect from the general theory in the study of flows of
$\mathrm{G}_{2}$-structures, and that there is a component of
$|Rm_{t}|_{g_{t}}$ with this same dependence on $t$, by Proposition 3.5.
However, there is a component of $|Rm_{t}|_{g_{t}}$ which has a different
dependence on $t$, as long as the tranverse geometry on the contact Calabi–Yau
is not flat: this is unsurprising because, in particular, having small torsion
does not imply that the metric is close to flat.

###  3.4 Singularity analysis

We can now combine all of the estimates we have obtained to describe the
infinite time singularity for the Laplacian coflow in terms of the types in
Definition 1.1, which proves Theorem 1.3(a).

######  Proposition 3.8.

In a cCY setup (S), suppose moreover $M^{7}$ is compact, and let
$K:=\sup_{M}|Rm_{0}^{\mathcal{D}_{0}}|_{g_{0}}$. Then there is a constant
$c_{0}>0$, independent of $\varepsilon$, such that the quantity $\Lambda(t)$
in (1.3), along the Laplacian coflow solution given by Theorem 1.1, is given
by

$\Lambda(t)=\big{(}K^{2}(1+10\varepsilon^{2}t)^{-2/5}+c_{0}\varepsilon^{4}(1+10\varepsilon^{2}t)^{-2}\big{)}^{1/2}.$

Hence, the Laplacian coflow has a Type IIb infinite time singularity, unless
$g_{\mathcal{D}_{0}}$ is flat, in which case it has a Type III infinite time
singularity.

###### Proof.

The formula for $\Lambda(t)$ follows immediately from Proposition 3.5 and
Proposition 3.7. We then see that $t\Lambda(t)\to\infty$ as $t\to\infty$
unless $K=0$, from which the result then follows by Definition 1.1. ∎

######  Remark 3.5.

As remarked earlier, these are the first examples of Laplacian coflow
solutions which have Type IIb singularities. The other examples are all
homogeneous, arising from nilmanifold and solvmanifold constructions, and in
those cases one always has a Type III singularity if it does not converge to a
flat torsion-free ${\rm G}_{2}$-structure.

##  4 The Laplacian flow solution

We now want to consider the Laplacian flow (1.2) starting at one of the
natural coclosed $\mathrm{G}_{2}$-structures on a contact Calabi–Yau
7-manifold. Whilst the class of coclosed $\mathrm{G}_{2}$-structures is not
expected to be preserved, in general, along the Laplacian flow, we will see
that it is in our setting and that we can solve the flow explicitly. By again
studying the evolution of the metric and torsion along an ansatz for the flow,
we will see that the behaviour is rather different from the Laplacian coflow,
having a finite time singularity but existing for all negative times.

###  4.1 Solving the flow

Again, we choose the natural coclosed $\mathrm{G}_{2}$-structure $\varphi_{0}$
on a cCY setup (S), cf. (3.1), as the initial condition for our flow. We are
therefore again led to consider the $\mathrm{G}_{2}$-structures $\varphi_{t}$
in (3.2) depending on functions $f_{t}$, $h_{t}$ (which only depend on time)
satisfying the initial conditions (3.3) to prove Theorem 1.2 as follows.

###### Proof of Theorem 1.2.

We compute the Laplacian of $\varphi_{t}$ using (3.4) and (3.11):

$\begin{split}\Delta_{\varphi_{t}}\varphi_{t}&=\ast_{\varphi_{t}}(\Delta_{\psi_{t}}\psi_{t})=\ast_{\varphi_{t}}(2f_{t}^{2}\omega_{0}^{2})=\ast_{\varphi_{t}}\left(4\frac{f_{t}^{2}}{h_{t}^{4}}\cdot\frac{h_{t}^{4}\omega_{0}^{2}}{2}\right)=4\frac{f_{t}^{2}}{h_{t}^{4}}\cdot
f_{t}h_{t}^{2}\eta_{0}\wedge\omega_{0}\\\
&=\frac{4f_{t}^{3}}{h_{t}^{2}}\eta_{0}\wedge\omega_{0}.\end{split}$ (4.1)

We deduce from (3.2) and (4.1) that the Laplacian flow (1.2) preserves the
class of coclosed $\mathrm{G}_{2}$-structures in (3.2). Moreover, the
Laplacian flow is equivalent to the following pair of ODEs:

$\frac{d}{dt}(f_{t}h_{t}^{2})=\frac{4f_{t}^{3}}{h_{t}^{2}},\quad\frac{d}{dt}(h_{t}^{3})=0.$
(4.2)

We deduce from the second equation in (4.2) and (3.3) that $h_{t}=1$. We
therefore see that the first equation in (4.2) becomes

$\frac{d}{dt}f_{t}=4f_{t}^{3}.$ (4.3)

From (4.3) and (3.3) we conclude that

$f_{t}=\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\quad\text{and}\quad h_{t}=1.$
(4.4)

We conclude from (3.2), (3.4), and (3.6) that we have a solution $\varphi_{t}$
to the Laplacian flow starting at $\varphi_{0}$ in (3.1) which induces the
following data:

$\displaystyle\varphi_{t}$
$\displaystyle=\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\eta_{0}\wedge\omega_{0}+\mathop{\mathrm{Re}}\Upsilon_{0},$
(4.5) $\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}\omega_{0}^{2}-\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0},$
(4.6) $\displaystyle g_{t}$
$\displaystyle=\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}\eta_{0}^{2}+g_{\mathcal{D}_{0}},$
(4.7) $\displaystyle\mathrm{vol}_{t}$
$\displaystyle=\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$
(4.8)

for $t\in(-\infty,\frac{1}{8\varepsilon^{2}})$. ∎

######  Remark 4.1.

Our solution to the Laplacian flow is _ancient_ (i.e. exists for all negative
times), but not eternal, forming a finite-time singularity at
$t=\frac{1}{8\varepsilon^{2}}>0$. Again, we cannot guarantee that this is the
unique solution to the Laplacian flow starting at (3.1), even if we assume $M$
is compact, because we do not have general analytic theory for the Laplacian
flow when not restricted to closed $\mathrm{G}_{2}$-structures.

######  Remark 4.2.

Even though the Laplacian flow no longer has the interpretation as the
gradient flow of the volume functional in this setting, we observe from (4.8)
that the volume form of $M$ in the metric $g_{t}$ is still pointwise strictly
increasing in time, just as for the Laplacian coflow. Moreover, if $M$ is
compact, we conclude that the volume is strictly increasing in time, but now
tending to infinity in _finite_ time:

$\text{Vol}(M,g_{t})\to\infty\quad\text{as $t\to\frac{1}{8\varepsilon^{2}}$.}$

We also notice from (4.7) that the cohomology class of $\psi_{t}$ satisfies

$[\psi_{t}]=(1-8\varepsilon^{2}t)^{-1/2}[\psi_{0}]$

and so it is not constant, unless $\psi_{0}$ is exact. Moreover, the
cohomology class $[\psi_{t}]$ degenerates precisely at the singular time
$t=\frac{1}{8\varepsilon^{2}}$, whenever $[\psi_{0}]\neq 0$.

###  4.2 Metric and curvature

Here, we see from (4.7) that the transverse geometry stays constant along the
Laplacian flow, and the direction dual to $\eta_{0}$ in $M$ expands as $t$
increases to $\frac{1}{8\varepsilon^{2}}$ and collapses as $t\to-\infty$.

######  Example 4.1.

Let $M^{7}$ be a compact quasi-regular cCY 7-manifold, so that it is a circle
bundle over a compact Calabi–Yau 3-orbifold $Z$ with metric $g_{Z}$. Then, for
our Laplacian flow solution, we see from (4.7) that the induced metric on $Z$
is fixed along the flow. However, the circle fibres expand to infinite size
(and so become copies of $\mathbb{R}$) as $t\to\frac{1}{8\varepsilon^{2}}$,
and that $M$ collapses to $Z$ as $t\to-\infty$.

As in the Laplacian coflow case, we can account for the volume increasing
along the flow by normalising the volume of the evolving metrics to be the
same as $g_{0}$, yielding

$(1-8\varepsilon^{2}t)^{1/7}g_{t}=\varepsilon^{2}(1-8\varepsilon^{2}t)^{-6/7}\eta_{0}^{2}+(1-8\varepsilon^{2}t)^{1/7}g_{Z}.$

This immediately gives the following result.

######  Lemma 4.1.

Let $M$ be compact. After normalising the volume of $(M,g_{t})$ to be fixed,
the Laplacian flow solution collapses to $\mathbb{R}$ as
$t\to\frac{1}{8\varepsilon^{2}}$ and collapses to $\mathbb{C}^{3}$ with the
flat metric as $t\to-\infty$.

We can also normalise the diameter and immediately obtain the following from
(4.7).

######  Lemma 4.2.

Let $M$ be compact. Then, after normalising $(M,g_{t})$ to unit diameter, the
Laplacian flow is volume-collapsing to a $6$-dimensional space $M_{-}$ as
$t\to-\infty$ and to a $1$-dimensional space $M_{+}$ as
$t\to\frac{1}{8\varepsilon^{2}}$. If the Sasakian structure is quasi-regular,
so that $M$ is an $S^{1}$-orbibundle over a Calabi–Yau 3-orbifold $Z$, then
$M_{-}=Z$ and $M_{+}=S^{1}$.

We see that there is a finite-time singularity for our flow at
$t=\frac{1}{8\varepsilon^{2}}$. As we know from the study of the Laplacian
flow for closed $\mathrm{G}_{2}$-structures, the formation of singularities
should be related to the lack of uniform continuity of the evolving metrics
and the blow-up of curvature, so we examine these properties for our solution.

######  Proposition 4.3.

Let $\varphi_{t}$ be the solution to the Laplacian flow given by (4.5) with
associated metric $g_{t}$ as in (4.7). Then $g_{t}$ is uniformly continuous
(in $t$) on $(-\infty,T]$, for any $T<\frac{1}{8\varepsilon^{2}}$, but it is
_not_ uniformly continuous on $[T,\frac{1}{8\varepsilon^{2}})$, for any $T$.

###### Proof.

This is immediate from the formula (4.7) for the evolving metric $g_{t}$,
which shows that its uniform continuity on an interval $I$ is equivalent to
the uniform continuity of $(1-8\varepsilon^{2}t)^{-1}$ on $I$. ∎

Moreover, one can immediately express the norm of curvature along the flow,
from the proof of Proposition 3.5 and (4.4), as follows.

######  Proposition 4.4.

Let $\varphi_{t}$ be the solution to the Laplacian flow given by (4.5) with
associated metric $g_{t}$ as in (4.7). Let $Rm_{t}$ denote the Riemann
curvature tensor of $g_{t}$ and let $Rm_{0}^{\mathcal{D}_{0}}$ denote the
curvature of the transverse connection on $\mathcal{D}_{0}$ induced by the
Levi-Civita connection of $g_{0}$. Then

$|Rm_{t}|_{g_{t}}^{2}=|Rm_{0}^{\mathcal{D}_{0}}|_{g_{0}}^{2}+c_{0}\varepsilon^{4}(1-8\varepsilon^{2}t)^{-2}$

for some constant $c_{0}>0$.

######  Remark 4.3.

We observe, by Proposition 4.4, that $|Rm_{t}|_{g_{t}}$ blows up as
$t\to\frac{1}{8\varepsilon^{2}}$ (the finite-time singularity), but that
$(1-8\varepsilon^{2}t)|Rm_{t}|_{g_{t}}$ remains bounded as
$t\to\frac{1}{8\varepsilon^{2}}$. We also notice that $|Rm_{t}|_{g_{t}}$
remains bounded as $t\to-\infty$, but that $-t|Rm_{t}|_{g_{t}}$ always blows
up at $t\to-\infty$, unless the tranverse Calabi–Yau geometry is flat.

###  4.3 Full torsion tensor

We now study the behaviour of the full torsion tensor along our solution of
the Laplacian flow.

######  Proposition 4.5.

The full torsion tensor $T_{t}$ of the solution to the Laplacian flow in
(4.5)–(4.8) is

$\begin{split}T_{t}&=\frac{1}{4}(\tau_{0})_{t}g_{t}-\frac{1}{4}{\rm
j}_{\varphi_{t}}((\tau_{3})_{t})\\\
&=\frac{3}{14}\varepsilon(1-8\varepsilon^{2}t)^{-1/2}g_{t}\\\
&\qquad-\frac{1}{14}{\rm
j}_{\varphi_{t}}(4\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}\eta_{0}\wedge\omega_{0}-3\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\mathop{\mathrm{Re}}\Upsilon_{0})\\\
&=-\frac{3}{2}\varepsilon^{3}(1-8\varepsilon^{2}t)^{-3/2}\eta_{0}^{2}+\frac{1}{2}\varepsilon(1-8\varepsilon^{2}t)^{-1/2}g_{\mathcal{D}_{0}}.\end{split}$
(4.9)

###### Proof.

The first two descriptions of $T_{t}$ in (4.9) follow from (2.9), Lemma 3.1
and (4.4).

We then see from (3.26)–(3.28) that

${\rm
j}_{\varphi_{t}}(\eta_{0}\wedge\omega_{0})=6\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\eta_{0}^{2}+\frac{2}{\varepsilon}(1-8\varepsilon^{2}t)^{1/2}g_{\mathcal{D}_{0}},$
(4.10)

and applying (3.29)–(3.31) yields

${\rm
j}_{\varphi_{t}}(\mathop{\mathrm{Re}}\Upsilon_{0})=4g_{\mathcal{D}_{0}}.$
(4.11)

Combining (4.10) and (4.11) shows that

${\rm
j}_{\varphi_{t}}(4\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}\eta_{0}\wedge\omega_{0}-3\varepsilon(1-8\varepsilon^{2}t)^{-1/2}\mathop{\mathrm{Re}}\Upsilon_{0})\\\
=24\varepsilon^{3}(1-8\varepsilon^{2}t)^{-3/2}\eta_{0}^{2}-4\varepsilon(1-8\varepsilon^{2}t)^{-1/2}g_{\mathcal{D}_{0}}.$
(4.12)

Substituting (4.7) and (4.12) into (4.9) gives the result. ∎

Our next result describes the norm of the full torsion tensor and is gradient
along the flow.

######  Proposition 4.6.

Let $T_{t}$ be the full torsion tensor of the solution to the Laplacian flow
in (4.5)–(4.8). Then

$\displaystyle|T_{t}|^{2}_{g_{t}}$
$\displaystyle=\frac{15}{4}\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1},$
$\displaystyle|\nabla_{t}T_{t}|_{g_{t}}^{2}$
$\displaystyle=c_{0}\varepsilon^{4}(1-8\varepsilon^{2}t)^{-2},$
$\displaystyle\mathrm{div}_{t}T_{t}$ $\displaystyle=0,$

where $c_{0}>0$ is a constant, $\nabla_{t}$ is the Levi-Civita connection of
$g_{t}$ and $\mathrm{div}_{t}$ is the divergence with respect to the metric
$g_{t}$.

###### Proof.

The proof is almost identical to that of Proposition 3.7, so we only prove the
formula for $|T_{t}|_{g_{t}}^{2}$. We observe from the formula (4.7) for the
metric $g_{t}$ that the transverse metric is not changing along the flow, but
that

$|\varepsilon^{2}\eta_{0}^{2}|_{g_{t}}^{2}=(1-8\varepsilon^{2}t)^{2}.$

Hence, we see from Proposition 4.5 that

$|T_{t}|_{g_{t}}^{2}=\frac{9}{4}\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}+\frac{3}{2}\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}=\frac{15}{4}\varepsilon^{2}(1-8\varepsilon^{2}t)^{-1}$

as claimed. The equation for $|\nabla_{t}T_{t}|_{g_{t}}^{2}$ and the vanishing
of $\mathrm{div}_{t}T_{t}$ follow exactly as in the proof of Proposition 3.7.
∎

######  Remark 4.4.

We again see that $|T_{t}|_{g_{t}}^{2}$ and $|\nabla_{t}T_{t}|_{g_{t}}$ blow
up at the same rate as we approach the finite time singularity at
$t=\frac{1}{8\varepsilon^{2}}$, as we would expect, and again that there is a
component of $|Rm_{t}|_{g_{t}}$ that instead stays bounded as
$t\to\frac{1}{8\varepsilon^{2}}$. Moreover, multiplying each of those
quantities by $(1-8\varepsilon^{2}t)$ leads to functions that are bounded as
$t\to\frac{1}{8\varepsilon^{2}}$, which will be significant later.

###  4.4 Singularity analysis

As for the Laplacian coflow, we can now put our estimates together to obtain a
description of the finite time singularity in the language of Definition 1.1,
which proves Theorem 1.3(b).

######  Proposition 4.7.

In a cCY setup (S), suppose moreover $M^{7}$ is compact, and let
$K:=\sum_{M}|Rm_{0}^{\mathcal{D}_{0}}|_{g_{0}}$. Then there is a constant
$c_{0}>0$, independent of $\varepsilon$, such that the quantity $\Lambda(t)$
in (1.3), along the Laplacian flow solution given by Theorem 1.2, is given by

$\Lambda(t)=(K^{2}+c_{0}\varepsilon^{4}(1-8\varepsilon^{2}t)^{-2})^{1/2}.$

Hence, the Laplacian flow has a Type I finite-time singularity at
$t=\frac{1}{8\varepsilon^{2}}$.

###### Proof.

The formula for $\Lambda(t)$ follows from Propositions 4.4 and 4.5. We then
see that $(\frac{1}{8\varepsilon^{2}}-t)\Lambda(t)$ remains bounded as
$t\to\frac{1}{8\varepsilon^{2}}$, and thus the result follows from Definition
1.1. ∎

##  5 The Hitchin flow solution

In this section we solve the Hitchin flow (1.4) for the natural coclosed
$\mathrm{G}_{2}$-structures on a contact Calabi–Yau 7-manifold and show a
relationship to Calabi–Yau structures in (real) dimension 8.

###  5.1 Solving the flow

As the initial condition for the Hitchin flow, we again choose the natural
coclosed $\mathrm{G}_{2}$-structure $\varphi_{0}$ as in (3.1) on a contact
Calabi-Yau $7$-manifold.

######  Proposition 5.1.

On a cCY setup (S), the following family of coclosed
$\mathrm{G}_{2}$-structures $\varphi_{t}$ with associated 4-form $\psi_{t}$,
metric $g_{t}$, and volume $\mathrm{vol}_{t}$, satisfies the Hitchin flow
(1.4), with initial condition (3.1):

$\displaystyle\varphi_{t}$ $\displaystyle=\varepsilon(1+\frac{5}{2}\varepsilon
t)^{-1/5}\eta_{0}\wedge\omega_{0}+(1+\frac{5}{2}\varepsilon
t)^{3/5}\mathop{\mathrm{Re}}\Upsilon_{0};$ (5.1) $\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{2}(1+\frac{5}{2}\varepsilon
t)^{4/5}\omega_{0}^{2}-\varepsilon\eta_{0}\wedge\mathop{\mathrm{Im}}\Upsilon_{0};$
(5.2) $\displaystyle g_{t}$
$\displaystyle=\varepsilon^{2}(1+\frac{5}{2}\varepsilon
t)^{-6/5}\eta_{0}^{2}+(1+\frac{5}{2}\varepsilon t)^{2/5}g_{\mathcal{D}_{0}};$
(5.3) $\displaystyle\mathrm{vol}_{t}$
$\displaystyle=\varepsilon(1+\frac{5}{2}\varepsilon
t)^{3/5}\eta_{0}\wedge\mathrm{vol}_{\mathcal{D}_{0}},$ (5.4)

defined for all $t\in(-\frac{2}{5\varepsilon},\infty)$.

###### Proof.

We consider the ansatz for our $\mathrm{G}_{2}$-structures $\varphi_{t}$
solving the Hitchin flow as in (3.2), depending on functions $f_{t},h_{t}$ of
time only, with dual $4$-forms $\psi_{t}$ as in (3.6), with the constraints on
$f_{0},h_{0}$ as in (3.3).

Under this assumption, we can compute

${\rm d}\varphi_{t}=f_{t}h_{t}^{2}\omega_{0}^{2}\quad\text{and}\quad{\rm
d}\psi_{t}=0,$

so that the Hitchin flow (1.4) is then equivalent to solving

$\frac{d}{dt}(h_{t}^{4})=2f_{t}h_{t}^{2}\quad\text{and}\quad\frac{d}{dt}(f_{t}h_{t}^{3})=0$
(5.5)

using (3.6). The second equation in (5.5), together with (3.3), yields

$f_{t}=\varepsilon h_{t}^{-3}.$ (5.6)

Substituting (5.6) into the first equation in (5.5) then gives

$\frac{d}{dt}(h_{t}^{4})=2\varepsilon h_{t}^{-1}.$ (5.7)

The ODE (5.7) can be solved using (3.3) and thus, using (5.6), we can conclude
that the solution to (5.5) is

$\displaystyle f_{t}$ $\displaystyle=\varepsilon(1+\frac{5}{2}\varepsilon
t)^{-3/5}\quad\text{and}\quad h_{t}=(1+\frac{5}{2}\varepsilon t)^{1/5}.\qed$

###  5.2 Calabi–Yau structure

We can now show that the solution to the Hitchin flow in Proposition 5.1 leads
to a Calabi–Yau structure on the spacetime track of the flow.

######  Lemma 5.2.

The following 2-form $\widehat{\omega}$, metric $\widehat{g}$ and complex
$4$-form $\widehat{\Upsilon}$ define a Calabi–Yau structure on
$(-\frac{3}{5\varepsilon},\infty)\times M^{7}$:

$\displaystyle\widehat{\omega}$
$\displaystyle=\varepsilon(1+\frac{5}{2}\varepsilon t)^{-3/5}{\rm
d}t\wedge\eta_{0}+(1+\frac{5}{2}\varepsilon t)^{2/5}\omega_{0};$ (5.8)
$\displaystyle\widehat{g}$ $\displaystyle={\rm
d}t^{2}+\varepsilon^{2}(1+\frac{5}{2}\varepsilon
t)^{-6/5}\eta_{0}^{2}+(1+\frac{5}{2}\varepsilon t)^{2/5}g_{0};$ (5.9)
$\displaystyle\widehat{\Upsilon}$ $\displaystyle=((1+\frac{5}{2}\varepsilon
t)^{3/5}{\rm d}t+i\varepsilon\eta_{0})\wedge\Upsilon_{0}.$ (5.10)

The metric $\widehat{g}$ is incomplete and has holonomy contained in
$\mathrm{SU}(4)$.

###### Proof.

It is straightforward to check that $\widehat{\omega}$ in (5.8) and
$\widehat{\Upsilon}$ in (5.10) induce the metric $\widehat{g}$ in (5.9), they
satisfy the algebraic conditions to define an $\mathrm{SU}(4)$ structure and
they are both closed, and thus define a Calabi–Yau structure as claimed. The
incompleteness of the metric is clear. ∎

######  Remark 5.1.

We know that the metric induced by the Hitchin flow is given by
$\widehat{g}={\rm d}t^{2}+g_{t}$, where $g_{t}$ is given in (5.3), which
agrees with the formula in (5.9).

We also know by the definition of the Hitchin flow that the induced torsion-
free $\mathrm{Spin}(7)$-structure on $(-\frac{5}{2\varepsilon},\infty)$ is
given by the 4-form

$\Phi={\rm d}t\wedge\varphi_{t}+\psi_{t},$

where $\varphi_{t}$, $\psi_{t}$ are given in (5.1)–(5.2). It is
straightforward to check from (5.8) and (5.10) that

$\Phi=\frac{1}{2}\widehat{\omega}^{2}+\mathop{\mathrm{Re}}\widehat{\Upsilon}.$

Thus the Hitchin flow solution in Proposition 5.1 does indeed induce the
Calabi–Yau structure in Lemma 5.2, which then completes the proof of Theorem
1.4.

##  Appendix A Additional properties of coclosed G2-structures

We gather in this Appendix some basic formulae, as well as some original
material on the Hodge Laplacian and the Lie derivative of coclosed ${\rm
G}_{2}$-structures which emerged over the course of research leading to this
paper. The latter computations, albeit not essential to our main matter, might
nonetheless be useful to readers interested e.g. in soliton formation along
flows of coclosed ${\rm G}_{2}$-structures.

###  A.1 Contraction identities

Contractions of $\varphi$ with $\varphi$:

$\displaystyle\varphi_{abc}\varphi^{abc}$ $\displaystyle=42,$ (A.1)
$\displaystyle\varphi_{abj}\varphi^{ab}_{\,\,\,k}$ $\displaystyle=6g_{jk},$
(A.2) $\displaystyle\varphi_{apq}\varphi^{a}_{\,\,jk}$
$\displaystyle=g_{pj}g_{qk}-g_{pk}g_{qj}+\psi_{pqjk}.$ (A.3)

Contractions of $\varphi$ with $\psi$:

$\displaystyle\varphi_{ijk}\psi_{a}^{\,\,ijk}$ $\displaystyle=0,$
$\displaystyle\varphi_{ijq}\psi^{ij}_{\,\,\,kl}$
$\displaystyle=4\varphi_{qkl},$ (A.4)
$\displaystyle\varphi_{ipq}\psi^{i}_{\,\,jkl}$
$\displaystyle=g_{pj}\varphi_{qkl}-g_{jq}\psi_{pkl}+g_{pk}\psi_{jql}$
$\displaystyle+g_{kq}\psi_{jpl}+g_{pl}\psi_{jkq}-g_{lq}\psi_{jkp}.$ (A.5)

Contractions of $\psi$ with $\psi$:

$\displaystyle\psi_{abcd}\psi^{ab}_{\,\,\,\,mn}$
$\displaystyle=4g_{cm}g_{dn}-4g_{cn}g_{dm}+2\psi_{abmn},$ (A.6)
$\displaystyle\psi_{abcd}\psi_{m}^{\,\,\,bcd}$ $\displaystyle=24g_{am},$ (A.7)
$\displaystyle\psi_{abcd}\psi^{abcd}$ $\displaystyle=168$

The full torsion tensor is a $2$-tensor $T$ satisfying

$\displaystyle\nabla_{i}\varphi_{jkl}$
$\displaystyle=T_{i}^{\,\,\,m}\psi_{mjkl},$ $\displaystyle T_{i}^{\,\,j}$
$\displaystyle=\frac{1}{24}\nabla_{i}\varphi_{lmn}\psi^{jlmn},$ (A.8)

and

$\nabla_{m}\psi_{ijkl}=-(T_{mi}\varphi_{jkl}-T_{mj}\varphi_{ikl}-T_{mk}\varphi_{jil}-T_{ml}\varphi_{jki}).$
(A.9)

###  A.2 Hodge Laplacian

To study the Laplacian coflow, we naturally need to analyse the Hodge
Laplacian $\Delta_{\psi}$ of the dual $\psi=\ast\varphi$ of a coclosed
$\mathrm{G}_{2}$-structure $\varphi$. Then ${\rm
d}\varphi=\tau_{0}\psi+\ast\tau_{3}$ by (2.5)–(2.6), so

$\displaystyle\Delta_{\psi}\psi=$ $\displaystyle{\rm d}{\rm d}^{\ast}\psi+{\rm
d}^{\ast}{\rm d}\psi={\rm d}\ast{\rm d}\varphi$ $\displaystyle=$
$\displaystyle{\rm d}(\tau_{0}\varphi+\tau_{3})={\rm
d}\tau_{0}\wedge\varphi+\tau_{0}^{2}\psi+\tau_{0}\ast\tau_{3}+{\rm
d}\tau_{3}.$ (A.10)

The following proposition, adapted from [Grigorian2013]*Proposition 2.3, Lemma
4.5, (4.30), (5.8), provides useful properties of a coclosed
$\mathrm{G}_{2}$-structure and its full torsion tensor $T$. In the statement,
for $h,k\in S^{2}$ given locally by $h=h_{ij}dx^{i}dx^{j}$ and
$k=k_{ij}dx^{i}dx^{j}$, we define the inner product $\langle h,k\rangle$ and
the circ product $h\circ k\in S^{2}$ by

$\displaystyle\langle
h,k\rangle=h_{ij}k_{ab}g^{ia}g^{jb}\quad\text{and}\quad(h\circ
k)_{ab}=\varphi_{amn}\varphi_{bpq}h^{mp}k^{nq}.$ (A.11)

We define the divergence $\mathrm{div}h\in\Omega^{1}$ and curl
$\mathrm{Curl}h$ (which is a 2-tensor) of $h\in S^{2}$ by

$\displaystyle(\mathrm{div}h)_{a}=\nabla^{b}h_{ba}\quad\text{and}\quad(\mathrm{Curl}h)_{ab}=(\nabla_{m}h_{an})\varphi_{b}^{\,\,mn}.$
(A.12)

We also let $S^{t}$ denote the transpose of a 2-tensor $S$.

######  Proposition A.1.

Suppose we have a coclosed $\mathrm{G}_{2}$-structure $\varphi$ on a manifold
$M$ and recall the notation in (2.3), (A.11) and (A.12).

  * (a)

If $\mu={\rm i}_{\varphi}(h)\in\Omega^{3}_{27}$ with $h\in S^{2}_{0}$ then

$\displaystyle{\rm d}\mu$
$\displaystyle=-\frac{1}{2}(\mathrm{div}h)^{\flat}\wedge\varphi+*{\rm
i}_{\varphi}(k)$

where

$k=\frac{1}{2}\big{(}\mathrm{Curl}h+(\mathrm{Curl}h)^{t}\big{)}+\frac{1}{2}T\circ
h+\frac{1}{2}\big{(}Th+(Th)^{t}\big{)}-\frac{1}{2}(\mathop{\mathrm{tr}}\nolimits
T)h-\frac{1}{6}\langle T,h\rangle g.$

  * (b)

The full torsion tensor $T$ satisfies the following identities:

$\begin{array}[]{rcl}\mathrm{div}T&=&\nabla(\mathop{\mathrm{tr}}\nolimits
T),\\\ {\rm Ric}&=&\mathrm{Curl}T-T^{2}+(\mathop{\mathrm{tr}}\nolimits
T)T,\end{array}\begin{array}[]{rcl}\mathrm{Curl}T&=&(\mathrm{Curl}T)^{t},\\\
R&=&(\mathop{\mathrm{tr}}\nolimits T)^{2}-|T|^{2}.\end{array}$ (A.13)

The following lemma, which decomposes the Hodge Laplacian of $\psi$, is
originally proved in [Grigorian2013]*Proposition 4.6, although with a
different orientation for the $\mathrm{G}_{2}$-structure. Since signs are
important for applications, we work out the proof again, in our convention.

######  Lemma A.2.

Let $\varphi$ be a coclosed $\mathrm{G}_{2}$-structure on a manifold $M$ with
associated metric $g$. Then,

$\displaystyle\Delta_{\psi}\psi$
$\displaystyle=(\frac{2}{3}R+\frac{4}{3}|T|^{2})\psi\oplus({\rm
d}\mathop{\mathrm{tr}}\nolimits T)\wedge\varphi$
$\displaystyle\quad\oplus\ast_{\varphi}{\rm i}_{\varphi}\Big{(}-{\rm
Ric}+\frac{1}{14}(R-2|T|^{2})g+\mathop{\mathrm{tr}}\nolimits(T)T-2T^{2}-\frac{1}{2}T\circ
T\Big{)}.$

To prove this lemma, we will require the following elementary facts relating
$\tau_{3}={\rm i}_{\varphi}(h)$ and $T$. By (2.4) and (2.9),

$h=\frac{1}{7}(\mathop{\mathrm{tr}}\nolimits T)g-T$ (A.14)

and, applying (A.11) to $T$ and $g$,

$T\circ g=(\mathop{\mathrm{tr}}\nolimits T)g-T.$ (A.15)

We also use the following elementary result.

######  Lemma A.3 ([moreno2019tesis]*Lemma 25).

Let $\varphi$ be a coclosed $\mathrm{G}_{2}$-structure, let $\tau_{3}={\rm
i}_{\varphi}(h)$, using the map in (2.3), and recall (A.11)–(A.12).

  * (a)

$\mathrm{div}h=\frac{1}{7}\nabla(\mathop{\mathrm{tr}}\nolimits
T)-\mathrm{div}T=-\frac{6}{7}\nabla(\mathop{\mathrm{tr}}\nolimits T)$.

  * (b)

$\frac{1}{2}(\mathrm{Curl}h+(\mathrm{Curl}h)^{\rm t})=-\mathrm{Curl}T$ and
$\mathop{\mathrm{tr}}\nolimits(\mathrm{Curl}T)=0$.

  * (c)

$T\circ h=\frac{1}{7}\big{(}(\mathop{\mathrm{tr}}\nolimits
T)^{2}g-(\mathop{\mathrm{tr}}\nolimits T)T\big{)}-T\circ T$ and
$\mathop{\mathrm{tr}}\nolimits(T\circ T)=(\mathop{\mathrm{tr}}\nolimits
T)^{2}-|T|^{2}.$

###### Proof of Lemma A.2.

We can decompose $\Delta_{\psi}\psi$ as:

$\Delta_{\psi}\psi=Y^{\flat}\wedge\varphi\oplus*{\rm
i}_{\varphi}(s)=\frac{3}{7}(\mathop{\mathrm{tr}}\nolimits s)\psi\oplus
Y^{\flat}\wedge\varphi\oplus\ast{\rm i}_{\varphi}(\overline{s}),$

where $Y$ is a vector field and $\overline{s}$ is the trace-free part of $s\in
S^{2}$. Now, we apply Proposition A.1(a) to $\tau_{3}={\rm i}_{\varphi}(h)$
and use Lemma A.3 to obtain

$\begin{split}{\rm d}\tau_{3}&=\frac{3}{7}{\rm
d}(\mathop{\mathrm{tr}}\nolimits T)\wedge\varphi\\\ &+\ast{\rm
i}_{\varphi}\Big{(}-\mathrm{Curl}T+\frac{1}{2}T\circ
h+\frac{1}{2}\big{(}Th+(Th)^{t}\big{)}-\frac{1}{2}(\mathop{\mathrm{tr}}\nolimits
T)h-\frac{1}{6}\langle T,h\rangle g\Big{)}.\end{split}$ (A.16)

Using the identities $\tau_{0}=\frac{4}{7}\mathop{\mathrm{tr}}\nolimits(T)$,
$3\psi=\ast{\rm i}_{\varphi}(g)$, and (A.16) in (A.2), we obtain

$\displaystyle\Delta_{\psi}\psi$ $\displaystyle={\rm
d}(\mathop{\mathrm{tr}}\nolimits T)\wedge\varphi$ $\displaystyle+\ast{\rm
i}_{\varphi}\Big{(}-\mathrm{Curl}T+\frac{1}{2}(T\circ
h)+\frac{1}{2}\big{(}Th+(Th)^{t}\big{)}+\frac{1}{14}(\mathop{\mathrm{tr}}\nolimits
T)h+\frac{16}{147}(\mathop{\mathrm{tr}}\nolimits T)^{2}g-\frac{1}{6}\langle
T,h\rangle g\Big{)}.$

Now, replacing $h$ using (A.14), as well as using (A.15), Proposition A.1 and
Lemma A.3, we have

$\displaystyle\Delta_{\psi}\psi$ $\displaystyle={\rm
d}(\nabla\mathop{\mathrm{tr}}\nolimits T)^{\flat}\wedge\varphi+\ast{\rm
i}_{\varphi}\Big{(}-\mathrm{Curl}T-\frac{1}{2}T\circ
T-T^{2}+\frac{1}{6}(\mathop{\mathrm{tr}}\nolimits
T)^{2}g+\frac{1}{6}|T|^{2}g\Big{)}.$

Therefore

$\displaystyle\mathop{\mathrm{tr}}\nolimits(s)=$
$\displaystyle\frac{2}{3}\big{(}(\mathop{\mathrm{tr}}\nolimits
T)^{2}+|T|^{2}\big{)}=\frac{2}{3}R+\frac{4}{3}|T|^{2},$
$\displaystyle\overline{s}=$ $\displaystyle-{\rm
Ric}+\frac{1}{14}(R-2|T|^{2})g+\mathop{\mathrm{tr}}\nolimits(T)T-2T^{2}-\frac{1}{2}T\circ
T,$ $\displaystyle Y=$ $\displaystyle\nabla\mathop{\mathrm{tr}}\nolimits T.$

The result then follows. ∎

###  A.3 Lie derivative

The Lie derivative of the Hodge dual of a coclosed $\mathrm{G}_{2}$-structure
naturally appears when studying geometric flows of coclosed
$\mathrm{G}_{2}$-structures in the context of “gauge-fixing” the action by
diffeomorphisms (as in DeTurck’s trick) and in the study of solitons. An
analogous expression for the Lie derivative of a closed
$\mathrm{G}_{2}$-structure can be found in [Lotay2017]*Lemma 9.7 and proved
useful for the study of solitons in that setting. In the statement of the
formula we derive below, for a vector field $X$ on $M$, we let $\mathrm{div}X$
denote its divergence and let $\mathrm{Curl}X$ be the vector field given
locally by

$(\mathrm{Curl}X)^{l}=\varphi_{ijk}(\nabla_{i}X^{j})g^{kl}.$

######  Theorem A.1.

Let $\varphi$ be a coclosed $\mathrm{G}_{2}$-structure on $M^{7}$, with
associated metric $g$, and let $X$ be a vector field on $M$. Then, if
$\psi=*_{\varphi}\varphi$,

$\mathcal{L}_{X}\psi=\frac{4}{7}(\mathrm{div}X)\psi\oplus(\mathrm{Curl}X)^{\flat}\wedge\varphi\oplus\ast{\rm
i}_{\varphi}\Big{(}\frac{3}{49}(\mathrm{div}X)g-\frac{3}{14}(\mathcal{L}_{X}g)\Big{)}\in\Omega_{1}^{4}\oplus\Omega_{7}^{4}\oplus\Omega_{27}^{4},$
(A.17)

where ${\rm i}_{\varphi}$ is given by (2.3). In particular, any infinitesimal
symmetry $X$ of $\psi$ must be a Killing vector field of $g$ with no curl,
i.e.,

$\mathcal{L}_{X}\psi=0\quad\Longleftrightarrow\quad\mathcal{L}_{X}g=0\quad\text{and}\quad\mathrm{Curl}(X)=0.$

###### Proof.

Since $\varphi$ is coclosed, i.e. ${\rm d}\psi=0$, we have

$\mathcal{L}_{X}\psi={\rm d}(X\lrcorner\psi)+X\lrcorner{\rm d}\psi={\rm
d}(X\lrcorner\psi).$

Let $\alpha=X\lrcorner\psi$, so that locally $\alpha_{ijk}=X^{l}\psi_{lijk}$
and

$(\mathcal{L}_{X}\psi)_{ijkl}=({\rm
d}\alpha)_{ijkl}=\nabla_{i}\alpha_{jkl}-\nabla_{j}\alpha_{ikl}+\nabla_{k}\alpha_{jil}-\nabla_{l}\alpha_{jki}.$

Denoting by $\pi_{l}^{k}:\Omega^{k}\rightarrow\Omega_{l}^{k}$ the orthogonal
projections, we decompose $\mathcal{L}_{X}\psi$ as

$\mathcal{L}_{X}\psi=\pi_{1}^{4}(\mathcal{L}_{x}\psi)+\pi_{7}^{4}(\mathcal{L}_{x}\psi)+\pi_{27}^{4}(\mathcal{L}_{X}\psi)=a\psi+W^{\flat}\wedge\varphi+\ast
i_{\varphi}(h),$ (A.18)

where $a\in\Omega^{0}$, and $h$ is a trace-free symmetric $2$-tensor on $M$.
We compute $a$ as follows:

$\begin{split}a&=\frac{1}{7}\langle\mathcal{L}_{X}\psi,\psi\rangle=\frac{1}{168}(\nabla_{i}\alpha_{jkl}-\nabla_{j}\alpha_{ikl}+\nabla_{k}\alpha_{jil}-\nabla_{l}\alpha_{jki})\psi^{ijkl}\\\
&=\frac{1}{42}\nabla_{i}\alpha_{jkl}\psi^{ijkl}=\frac{1}{42}\nabla_{i}(\alpha_{jkl}\psi^{ijkl})-\frac{1}{42}\alpha_{jkl}\nabla_{i}\psi^{ijkl}\\\
&=\frac{24}{42}\nabla_{i}(X^{m}g_{mi})-\frac{1}{42}X^{m}\psi_{mjkl}(\nabla_{i}\psi^{ijkl})=\frac{4}{7}\nabla_{i}X_{i}\\\
&=\frac{4}{7}\mathrm{div}X,\end{split}$ (A.19)

where we used (A.4) and that $T$ is symmetric. To compute $W^{\flat}$, note
that

$\langle\ast((\ast\mathcal{L}_{X}\psi)\wedge\varphi),e^{m}\rangle=4\langle
W^{\flat},e^{m}\rangle,$

thus

$\displaystyle 4W^{m}$
$\displaystyle=\ast\big{(}(\ast\mathcal{L}_{X}\psi)\wedge\varphi\wedge
e^{m}\big{)}=\ast\big{(}\langle\varphi\wedge
e^{m},\mathcal{L}_{X}\psi\rangle\mathrm{vol}\big{)}=\frac{1}{3!}(\mathcal{L}_{X}\psi)^{ijkm}\varphi_{ijk}.$

Therefore, we obtain

$\begin{split}W^{m}&=\frac{1}{4!}(\mathcal{L}_{X}\psi)^{ijkm}\varphi_{ijk}=\frac{1}{3!}(g^{si}\nabla_{s}\alpha^{jkm}\varphi_{ijk})\\\
&=\frac{1}{3!}(g^{si}\nabla_{s}(X^{l}\psi_{l}^{\,jkm}\varphi_{ijk})-X^{l}\psi_{l}^{\,jkm}g^{si}\nabla_{s}\varphi_{ijk})\\\
&=\frac{1}{3!}(4g^{si}\nabla_{s}(X^{l}\varphi_{il}^{\,\,\,m})-X^{l}\psi_{l}^{jkm}g^{si}T_{sa}g^{an}\psi_{nijk})\\\
&=\frac{4}{3!}g^{si}\nabla_{s}(X^{l}\varphi_{il}^{\,\,\,m})=\frac{4}{3!}g^{si}\nabla_{s}X^{l}\varphi_{il}^{\,\,\,m}+\frac{4}{3!}X^{l}g^{si}\nabla_{s}\varphi_{ir}^{\,\,\,m}\\\
&=\frac{4}{3!}\varphi_{ir}^{\,\,\,m}\nabla^{i}X^{l}+\frac{4}{3!}X^{l}g^{si}\nabla_{s}\varphi_{ir}^{\,\,\,m}\\\
&=\frac{4}{3!}\varphi_{ila}(\nabla^{i}X^{l})g^{am}+\frac{4}{3!}X^{l}g^{si}T_{sn}g^{nl}\psi_{lirb}g^{bm}\\\
&=\frac{4}{3!}(\mathrm{Curl}X)^{m}.\end{split}$ (A.20)

Finally, to compute $h$, observe that

$\begin{split}&(\mathcal{L}_{X}\psi)_{mnpi}\psi_{j}^{mnp}+(\mathcal{L}_{X}\psi)_{mnpj}\psi_{i}^{mnp}\\\
&=a(\psi_{mnpi}\psi_{j}^{mnp}+\psi_{mnpj}\psi_{i}^{mnp})+(\ast{\rm
i}_{\varphi}(h))_{mnpi}\psi_{j}^{mnp}+(\ast{\rm
i}_{\varphi}(h))_{mnpj}\psi_{i}^{mnp},\end{split}$ (A.21)

where

$(\ast{\rm
i}_{\varphi}(h))_{mnpi}=h_{m}^{q}\psi_{qnpi}-h_{n}^{q}\psi_{mqpi}+h_{p}^{q}\psi_{mnqi}-h_{i}^{q}\psi_{mnpq}.$

Equation (A.21) becomes

$\begin{split}&a\psi_{mnpi}\psi_{j}^{mnp}+(h_{m}^{q}\psi_{qnpi}-h_{n}^{q}\psi_{mqpi}+h_{p}^{q}\psi_{mnqi}-h_{i}^{q}\psi_{mnpq})\psi_{j}^{mnp}\\\
&+a\psi_{mnpj}\psi_{i}^{mnp}+(h_{m}^{q}\psi_{qnpj}-h_{n}^{q}\psi_{mqpj}+h_{p}^{q}\psi_{mnqj}-h_{j}^{q}\psi_{mnpq})\psi_{i}^{mnp}\\\
&=-48ag_{ij}+h_{m}^{q}(4g_{qj}g_{im}-4g_{qm}g_{ij}+4g_{qi}g_{jm}-4g_{qm}g_{ij})\\\
&\quad-
h_{n}^{q}(4g_{qj}g_{in}-4g_{qn}g_{ij}+4g_{qi}g_{jn}-4g_{qn}g_{ij})+24h_{i}^{q}g_{qj}\\\
&\quad+h_{p}^{q}(4g_{qj}g_{ip}-4g_{qp}g_{ji}+4g_{qi}g_{jp}-4g_{qp}g_{ij})+24h_{j}^{q}g_{qi}\\\
&=-48ag_{ij}+56h_{ij}.\end{split}$ (A.22)

We can calculate the left-hand side of (A.22) as follows:

$\displaystyle(\mathcal{L}_{X}\psi)_{mnpi}\psi_{j}^{mnp}+(\mathcal{L}_{X}\psi)_{mnpj}\psi_{i}^{mnp}$
(A.23) $\displaystyle=$
$\displaystyle(\nabla_{m}\alpha_{npi}-\nabla_{n}\alpha_{mpi}+\nabla_{p}\alpha_{nmi}-\nabla_{i}\alpha_{npm})\psi_{j}^{mnp}$
$\displaystyle+(\nabla_{m}\alpha_{npj}-\nabla_{n}\alpha_{mpj}+\nabla_{p}\alpha_{mmj}-\nabla_{j}\alpha_{npm})\psi_{i}^{mnp}$
$\displaystyle=$ $\displaystyle
3(\nabla_{m}\alpha_{npj}\psi^{mnp}_{i}+\nabla_{m}\alpha_{npi}\psi_{j}^{m}np)-\nabla_{i}\alpha_{npm}\psi_{j}^{mnp}-\nabla_{j}\alpha_{npm}\psi_{i}^{mnp}$
$\displaystyle=$ $\displaystyle
3\underset{I(i)}{\underline{\nabla_{m}(\alpha_{npi}\psi_{j}^{mnp})}}-3\underset{II(i)}{\underline{\alpha_{npi}\nabla_{m}\psi_{j}^{mnp}}}-\underset{III(i)}{\underline{\nabla_{i}(\alpha_{mnp}\psi_{j}^{mnp})}}+\underset{IV(i)}{\underline{\alpha_{mnp}\nabla_{i}\psi_{j}^{mnp}}}$
$\displaystyle+3\underset{I(j)}{\underline{\nabla_{m}(\alpha_{npj}\psi_{i}^{mnp})}}-3\underset{II(j)}{\underline{\alpha_{npj}\nabla_{m}\psi_{i}^{mnp}}}-\underset{III(j)}{\underline{\nabla_{j}(\alpha_{mnp}\psi_{i}^{mnp})}}+\underset{IV(j)}{\underline{\alpha_{mnp}\nabla_{j}\psi_{i}^{mnp}}}.$

For a coclosed $\mathrm{G}_{2}$-structure, $T$ is symmetric, so we have the
following expression in terms of the ‘matrix product’ of $2$-tensors:

$\displaystyle
X^{l}T_{i}^{m}\varphi_{ljm}=(X\lrcorner\varphi)_{jm}T_{i}^{m}=(T\cdot(X\lrcorner\varphi))_{ji}=-((X\lrcorner\varphi)\cdot
T)_{ij}.$ (A.24)

Using moreover (A.6), (A.7), (A.9), we obtain

$\displaystyle I(i)$
$\displaystyle=\nabla_{m}(\alpha_{npi}\psi_{j}^{mnp})=\nabla_{m}(X^{l}\psi_{lnpi}\psi_{j}^{mnp})$
$\displaystyle=\nabla_{m}(X^{l}(4g_{lj}g_{i}^{\,\,m}-4g_{l}^{\,\,m}g_{ij}+2\psi_{lij}^{\,\,\,\,m}))$
$\displaystyle=4\nabla_{i}X_{j}-4(\mathrm{div}X)g_{ij}+2\nabla_{m}(X^{l}\psi_{lij}^{\,\,\,\,m}),$
$\displaystyle I(j)+I(i)$
$\displaystyle=4\nabla_{j}X_{i}-4(\mathrm{div}X)g_{ij}+\nabla_{m}(X^{l}\psi_{lji}^{\,\,\,\,m})+4\nabla_{i}X_{j}-4(\mathrm{div}X)g_{ji}+\nabla_{m}(X^{l}\psi_{lij}^{\,\,\,\,m})$
$\displaystyle=4(\nabla_{i}X_{j}+\nabla_{j}X_{i})-8(\mathrm{div}X)g_{ij}.$

Furthermore,

$\displaystyle II(i)+II(j)$
$\displaystyle=\alpha_{npi}\nabla_{m}\psi_{j}^{mnp}+\alpha_{npj}\nabla_{m}\psi_{i}^{mnp}=X^{l}\psi_{lnpi}\nabla_{m}\psi_{j}^{mnp}+X^{l}\psi_{lnpj}\nabla_{m}\psi_{i}^{mnp}$
$\displaystyle=X^{l}\psi_{lnpi}(-T_{mj}\varphi^{mnp}+T_{m}^{m}\varphi_{j}^{np}-T_{m}^{n}\varphi_{j}^{mp}+T_{m}^{p}\varphi_{j}^{mn})$
$\displaystyle\quad+X^{l}\psi_{lnpj}(-T_{mi}\varphi^{mnp}+T_{m}^{m}\varphi_{i}^{np}-T_{m}^{n}\varphi_{i}^{mp}+T_{m}^{p}\varphi_{i}^{mn})$
$\displaystyle=4(X\lrcorner\varphi\cdot T)_{ij}-4(T\cdot
X\lrcorner\varphi)_{ij}=4[X\lrcorner\varphi,T]_{ij},$ $\displaystyle
III(i)+III(j)$
$\displaystyle=\nabla_{i}(X^{l}\psi_{lmnp}\psi_{j}^{mnp})+\nabla_{j}(X^{l}\psi_{lmnp}\psi_{i}^{mnp})$
$\displaystyle=24(\nabla_{i}X_{j}+\nabla_{j}X_{i}),$ $\displaystyle IV(i)$
$\displaystyle=\alpha_{mnp}\nabla_{i}\psi_{j}^{mnp}=X^{l}\psi_{lmnp}\nabla_{i}\psi_{j}^{mnp}$
$\displaystyle=X^{l}\psi_{lmnp}(-T_{ij}\varphi^{mnp}+T_{i}^{m}\varphi_{j}^{np}-T_{i}^{n}\varphi_{j}^{mp}+T_{i}^{p}\varphi_{j}^{mn})=12(X\lrcorner\varphi\cdot
T)_{ij},$ $\displaystyle IV(i)+IV(j)$ $\displaystyle=12(X\lrcorner\varphi\cdot
T)_{ij}+12(X\lrcorner\varphi\cdot T)_{ji}=12[X\lrcorner\varphi,T]_{ij}.$

So, using (A.19), (A.22) and the above expressions, we obtain

$-48ag_{ij}+56h_{ij}=-12(\nabla_{i}X_{j}+\nabla_{j}X_{i})-24(\mathrm{div}X)g_{ij}$

which, upon re-arranging, gives

$\begin{split}h_{ij}&=\frac{3}{49}\mathrm{div}(X)g_{ij}-\frac{3}{14}(\nabla_{i}X_{j}+\nabla_{j}X_{i})\\\
&=\frac{3}{49}(\mathrm{div}X)g_{ij}-\frac{3}{14}(\mathcal{L}_{X}g)_{ij}.\end{split}$
(A.25)

Hence, substituting (A.19), (A.20) and (A.25) into (A.18) we obtain (A.17). ∎

## References