# Geometric flows of G2-structures on 3-Sasakian 7-manifolds

Aaron Kennon  Department of Physics UCSB<EMAIL_ADDRESS>and  Jason
D. Lotay  Mathematical Institute, University of Oxford<EMAIL_ADDRESS>
###### Abstract.

A 3-Sasakian structure on a 7-manifold may be used to define two distinct
Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both
metrics are induced by nearly parallel $\mathrm{G}_{2}$-structures which may
also be expressed in terms of the 3-Sasakian structure. Just as Einstein
metrics are critical points for the Ricci flow up to rescaling, nearly
parallel $\mathrm{G}_{2}$-structures provide natural critical points of the
(rescaled) geometric flows of $\mathrm{G}_{2}$-structures known as the
Laplacian flow and Laplacian coflow. We study each of these flows in the
3-Sasakian setting and see that their behaviour is markedly different,
particularly regarding the stability of the nearly parallel
$\mathrm{G}_{2}$-structures. We also compare the behaviour of the flows of
$\mathrm{G}_{2}$-structures with the (rescaled) Ricci flow.

The research of the first author is partially supported by Simons Foundation
Award #488629 (Morrison). The research of the second author is partially
supported by the Simons Collaboration on Special Holonomy in Geometry,
Analysis, and Physics (#724071 Jason Lotay).

###### Contents

  1. 1 Introduction
  2. 2 $\mathrm{G}_{2}$-structures on 3-Sasakian 7-manifolds
  3. 3 Laplacian coflow
  4. 4 Laplacian flow
  5. 5 Ricci flow

##  1\. Introduction

###  1.1. Nearly parallel G2-structures

A $\mathrm{G}_{2}$-structure on a 7-manifold is encoded by a 3-form $\varphi$
satisfying a certain nondegeneracy condition, and such a 3-form determines a
Riemannian metric and orientation. One of the most important types of
$\mathrm{G}_{2}$-structure is a _nearly parallel_ $\mathrm{G}_{2}$-structure
since it defines an Einstein metric with positive scalar curvature, as well as
a real Killing spinor [2, 7]. Moreover, the cone over a 7-manifold with a
nearly parallel $\mathrm{G}_{2}$-structure admits a conical metric with
exceptional holonomy $\operatorname{Spin}(7)$ (and so is Ricci-flat), and thus
nearly parallel $\mathrm{G}_{2}$-structures are also important in the study of
asymptotically conical and conically-singular $\operatorname{Spin}(7)$
manifolds (cf. [14]).

Since the existence of a complete positive Einstein metric will lead to
compactness of the underlying manifold by Myers theorem, it is natural to ask
which compact 7-manifolds admit nearly parallel $\mathrm{G}_{2}$-structures.
Though this general question is currently open, an infinite number of examples
of such compact 7-manifolds are known, including the 7-sphere, the
Aloff–Wallach spaces $N(k,l)$, the Berger space SO(5)/SO(3), and the Stiefel
manifold V5,2 [7]. The largest class of 7-manifolds that are known to admit
nearly parallel $\mathrm{G}_{2}$-structures are the _$3$ -Sasakian_
7-manifolds, which are the focus of this paper.

###  1.2. Geometric flows

Nearly parallel $\mathrm{G}_{2}$-structures are natural to study from the
perspective of several geometric flows. Since a nearly parallel
$\mathrm{G}_{2}$-structure induces a positive Einstein metric, it is natural
to evolve its induced metric $g$ by the _Ricci flow_ :

$\frac{\partial g}{\partial t}=-2\mathrm{Ric}(g).$ (1.1)

The induced metric will define a self-similarly _shrinking_ solution to the
Ricci flow, and thus a critical point after rescaling. However, a
$\mathrm{G}_{2}$-structure contains more information than the metric (since
the same metric is induced by a whole family of $\mathrm{G}_{2}$-structures),
so it is worthwhile to examine flows of $\mathrm{G}_{2}$-structures relevant
to nearly parallel $\mathrm{G}_{2}$-structures, and compare and contrast its
behaviour to the Ricci flow.

Two such geometric flows of $\mathrm{G}_{2}$-structures which have been the
most studied, and we shall examine here, are the _Laplacian flow_ (introduced
by Bryant [5]) and the _Laplacian coflow_ (first considered in [12]111It
should be noted that in [12] the opposite sign for the velocity of the
Laplacian coflow is used.).

####  1.2.1. Laplacian flow

The Laplacian flow evolves the 3-form $\varphi$ defining the
$\mathrm{G}_{2}$-structure by its Hodge Laplacian:

$\frac{\partial\varphi}{\partial
t}=\Delta_{\varphi}\varphi=(\mathrm{d}\mathrm{d}^{*}_{\varphi}\varphi+\mathrm{d}^{*}_{\varphi}\mathrm{d})\varphi.$
(1.2)

(Here, we emphasise the nonlinearity in the formal adjoint
$\mathrm{d}^{*}_{\varphi}$ of the exterior derivative, since the metric and
orientation depend on $\varphi$.) The Laplacian flow has received particular
attention in the context of _closed_ $\mathrm{G}_{2}$-structures (when
$\mathrm{d}\varphi=0$), where it has many attractive features, particularly
with regards to torsion-free $\mathrm{G}_{2}$-structures (when
$\mathrm{d}\varphi=0$ and $\mathrm{d}^{*}_{\varphi}\varphi=0$), which define
Ricci-flat metrics with holonomy contained in $\mathrm{G}_{2}$. For
foundational results and a survey of recent developments in the Laplacian flow
for closed $\mathrm{G}_{2}$-structures see e.g. [11, 17, 18].

A nearly parallel $\mathrm{G}_{2}$-structure defines a self-similarly
_expanding_ solution to the Laplacian flow (1.2), so can be viewed as a
critical point up to rescaling. (We note the immediate difference with the
Ricci flow where the induced metric was a shrinker.) A nearly parallel
$\mathrm{G}_{2}$-structure is, however, not closed but _coclosed_ : the
defining 3-form $\varphi$ satisfies $\mathrm{d}^{*}_{\varphi}\varphi=0$.
Whilst it may seem potentially plausible to study coclosed
$\mathrm{G}_{2}$-structures using the Laplacian flow (1.2), in fact it is not
yet known in general whether this flow even has short time existence starting
at a coclosed $\mathrm{G}_{2}$-structure. An example situation where it has
proved instructive to use the Laplacian flow to study coclosed
$\mathrm{G}_{2}$-structures can be found in [16].

####  1.2.2. Laplacian coflow

Currently the best candidate222The Laplacian coflow for coclosed
$\mathrm{G}_{2}$-structures has many attractive features analogous to the
Laplacian flow for closed $\mathrm{G}_{2}$-structures, but with the
significant difference that the analytic foundations for the Laplacian coflow
are currently lacking: see [9, 11] for a discussion of the analytic issues.
for studying coclosed $\mathrm{G}_{2}$-structures is the Laplacian coflow,
which evolves the closed 4-form $\psi=*_{\varphi}\varphi$ dual to the 3-form
$\varphi$ defining the $\mathrm{G}_{2}$-structure by its Hodge Laplacian:

$\frac{\partial\psi}{\partial
t}=\Delta_{\psi}\psi=(\mathrm{d}^{*}_{\psi}\mathrm{d}+\mathrm{d}\mathrm{d}^{*}_{\psi})\psi=\mathrm{d}\mathrm{d}^{*}_{\psi}\psi,$
(1.3)

using the fact that $\psi$ is closed. (The 4-form $\psi$ induces the metric
just like $\varphi$, but not the orientation, though an orientation can be
fixed by the initial choice of $\mathrm{G}_{2}$-structure.) The Laplacian
coflow preserves the cohomology class $[\psi]$ of $\psi$, where it may be
viewed as the gradient flow of the Hitchin volume functional, and the induced
flow of the metric $g$ defined by $\psi$ is

$\frac{\partial g}{\partial t}=-2\mathrm{Ric}(g)+Q(\mathrm{d}\varphi),$ (1.4)

where $Q$ is a quadratic expression in $\mathrm{d}\varphi$: see [9, 11] for
details. Since $Q$ only depends on first order information on $\psi$, whereas
the Ricci tensor involves second order data, one may view (1.4) as a lower
order perturbation of the Ricci flow (1.1).

However, just as for the Laplacian flow, a nearly parallel
$\mathrm{G}_{2}$-structure defines a self-similarly _expanding_ solution to
the Laplacian coflow (1.3), whereas its induced metric defines a shrinker for
Ricci flow. Hence the “lower order terms” in (1.4) drastically alter the
behaviour of the metric flow in this setting.

We should also note that coclosed $\mathrm{G}_{2}$-structures satisfy a
parametric h-principle (see [6]). Therefore, coclosed G2-structures exist on
any (compact or non-compact) 7-manifold admitting a
$\mathrm{G}_{2}$-structure, which just requires the 7-manifold to be oriented
and spin, and so the Laplacian coflow can potentially be studied on any
oriented spin 7-manifold. By contrast, it is currently not clear how
restrictive the closed condition is for a $\mathrm{G}_{2}$-structure on a
compact manifold.

###  1.3. 3-Sasakian 7-manifolds

A 3-Sasakian 7-manifold is a Riemannian 7-manifold $M$ so that the metric cone
over it is hyperkähler. One can use the 3-Sasakian structure to define
two333In fact, there are three natural nearly parallel
$\mathrm{G}_{2}$-structures inducing the 3-Sasakian metric, but these are
permuted by the symmetries in the 3-Sasakian structure. The same does _not_
occur for the squashed Einstein metric. distinct nearly parallel
$\mathrm{G}_{2}$-structures (up to scale), one of which induces the original
3-Sasakian Einstein metric on $M$, and the other induces the so-called
squashed Einstein metric on $M$. This is most easily seen in the example of
the 7-sphere, where the 3-Sasakian metric is the round metric, and the
squashed Einstein metric is obtained by rescaling the 3-sphere fibres relative
to the 4-sphere base in the Hopf fibration of the 7-sphere.

###  1.4. Stability

Our primary goal is to study the stability of nearly parallel G2-structures on
3-Sasakian 7-manifolds under the Laplacian flow and Laplacian coflow, and to
compare the behavior of these flows to the Ricci flow near their induced
Einstein metrics.

For geometric flows, one is primarily interested in the question of dynamical
stability of a critical point, i.e. when the flow starting near a critical
point will flow back to it. An easier and weaker thing to check is linear
stability: whether the critical point is stable for the linearized flow at
that point. In some situations, one can infer dynamical stability from linear
stability: e.g. for complete positive Einstein metrics in Ricci flow, linear
stability plus an integrability assumption implies a weak form of dynamical
stability [13].

In the context of nearly parallel $\mathrm{G}_{2}$-structures on 7-manifolds
$M$, it was shown in [19] that if the third Betti number $b^{3}(M)\neq 0$,
then under the Ricci flow any Einstein metric induced by a nearly parallel
$\mathrm{G}_{2}$ structure is linearly unstable and therefore dynamically
unstable. As 3-Sasakian 7-manifolds $M$ necessarily have $b^{3}(M)=0$, this
class of examples admitting nearly parallel $\mathrm{G}_{2}$-structures is
particularly interesting for Ricci flow in light of this result.

In this article, when discussing stability we will always be referring to
dynamical stability.

###  1.5. Main results

On any 3-Sasakian 7-manifold we introduce two disjoint 3-parameter families of
coclosed $\mathrm{G}_{2}$-structures defined in terms of the 3-Sasakian
structure. These families of $\mathrm{G}_{2}$-structures each include exactly
one of the natural nearly parallel $\mathrm{G}_{2}$-structures we discussed
above (and their rescalings). We refer the reader to §2 for details.

Our main results concern the behaviour of the Laplacian coflow, the Laplacian
flow and the Ricci flow for these families of coclosed
$\mathrm{G}_{2}$-structures and their induced metrics, which we show are
preserved by the flows. (Note, in particular, that the Laplacian flow is shown
to preserve the coclosed condition in this setting.)

Our most significant result is for the Laplacian coflow (1.3).

######  Theorem 1.1.

The Laplacian coflow starting at any initial coclosed
$\mathrm{G}_{2}$-structure in either of our families converges, after
rescaling, to the nearly parallel $\mathrm{G}_{2}$-structure in that family.
In particular, the nearly parallel $\mathrm{G}_{2}$-structures are both stable
within their families.

Comparing the Laplacian flow (1.2) and Laplacian coflow (1.3), one might
naively expect them to have similar behaviour as their velocities are Hodge
dual. However, in our setting, we have the following, which contrasts sharply
with our Laplacian coflow result.

######  Theorem 1.2.

Both nearly parallel $\mathrm{G}_{2}$-structures are unstable sources within
their families under the rescaled Laplacian flow, so coclosed
$\mathrm{G}_{2}$-structure in our families which are not nearly parallel
cannot flow to either of them.

Finally, for the Ricci flow (1.1), we have the following, which differs again
from our previous two results.

######  Theorem 1.3.

Along the rescaled Ricci flow for our families of metrics, the 3-Sasakian
metric is stable, whereas the squashed Einstein metric is a saddle point and
so unstable.

This result again shows that, whilst the Ricci flow and the induced flow of
metrics (1.4) from the Laplacian coflow are closely related, their behaviour
can be markedly different.

###  1.6. Summary

We begin in §2 by discussing background on 3-Sasakian geometry, the nearly
parallel G2-Structures determined by these geometries, and our geometric flow
ansatz. We then study the behavior of the Laplacian coflow in §3, the
Laplacian flow in §4, and the Ricci flow in §5. To do this, we reduce the
study of each rescaled flow to the analysis of a nonlinear ODE system for two
functions.

##  2\. $\mathrm{G}_{2}$-structures on 3-Sasakian 7-manifolds

In this section we recall some of the basics of 3-Sasakian geometry in 7
dimensions and outline its relationship to $\mathrm{G}_{2}$ geometry. Further
details on 3-Sasakian geometry can be found in [3, 4]. For information about
$\mathrm{G}_{2}$-structures, we refer the reader to [10] or [11, pp. 3–50].

###  2.1. 3-Sasakian 7-manifolds

We first recall the definition of a 3-Sasakian 7-manifold.

######  Definition 2.1.

A complete Riemannian 7-manifold $(M^{7},g_{M})$ is _3-Sasakian_ if it has an
orthonormal triple of Killing fields $\\{E_{1},E_{2},E_{3}\\}$ satisfying
$[E_{i},E_{j}]=2E_{k}$ for a cyclic permutation $(i,j,k)$ of $(1,2,3)$, such
that each $E_{i}$ defines a Sasakian structure on $(M,g_{M})$.

If $(M,g_{M})$ is 3-Sasakian then $g_{M}$ is Einstein with positive scalar
curvature equal to 42 (so $M$ is compact) and there is a locally free action
of $\mathrm{SU}(2)$ on $M$ whose leaf space $N$ is a 4-dimensional orbifold.
Moreover, there is a canonical metric $g_{N}$ on $N$, which is anti-self-dual
Einstein with positive scalar curvature equal to 48, such that $(M,g_{M})$ and
$(N,g_{N})$ are related by an orbifold Riemannian submersion:

$\pi:M\rightarrow N.$ (2.1)

######  Remark 2.2.

The simplest example of a 3-Sasakian 7-manifold is the 7-sphere with its
constant curvature $1$ metric. In this setting, (2.1) just becomes the usual
Hopf fibration with $M=S^{7}$ and $N=S^{4}$, and $N=S^{4}$ has its constant
curvature $4$ metric.

The Levi-Civita connection of $(N,g_{N})$ lifts to a connection on the bundle
(2.1), and so may be viewed as an $\mathfrak{su}(2)$-valued 1-form $\eta$ on
$M$, which can be written as

$\eta=\sum_{i=1}^{3}\eta_{i}\otimes T_{i},$ (2.2)

where $\eta_{1},\eta_{2},\eta_{3}$ are 1-forms on $M$ and
$\\{T_{1},T_{2},T_{3}\\}$ is a basis for $\mathfrak{su}(2)$ satisfying
$[T_{i},T_{j}]=2T_{k}$ for cyclic permutations $(i,j,k)$ of $(1,2,3)$. The
curvature of $\eta$ is then an $\mathfrak{su}$(2)-valued 2-form $\omega$ which
may be written as

$\omega=-2\sum_{i=1}^{3}\omega_{i}\otimes T_{i}$ (2.3)

for 2-forms $\omega_{1},\omega_{2},\omega_{3}$ on $M$ which are, in fact,
pullbacks of orthogonal self-dual 2-forms on $N$ since $g_{N}$ is anti-self-
dual Einstein. (The factor of $2$ and sign are chosen for convenience.)
Moreover, we have that the forms $\omega_{1},\omega_{2},\omega_{3}$ are
normalized such that

$\omega_{i}\wedge\omega_{j}=2\delta_{ij}\pi^{*}\mathrm{vol}_{N}.$ (2.4)

For later use, we record the following equations satisfied by $\eta$ and
$\omega$ where, in each case, $(i,j,k)$ are taken to be a cyclic permutation
of $(1,2,3)$:

$\displaystyle\mathrm{d}\eta_{i}$
$\displaystyle=-2\eta_{j}\wedge\eta_{k}-2\omega_{i},$ (2.5)
$\displaystyle\mathrm{d}\omega_{i}$
$\displaystyle=-2\eta_{j}\wedge\omega_{k}+2\eta_{k}\wedge\omega_{j}.$ (2.6)

The 3-Sasakian metric $g_{M}$ on $M$ may then be given in terms of the
$\eta_{i}$ and $g_{N}$ as follows:

$g_{M}=\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}+\pi^{*}g_{N},$ (2.7)

We can scale $g_{M}$ by any positive constant $c$ and then $c^{2}g_{M}$ will
still be Einstein with positive scalar curvature. We may also observe the
following well-known fact.

######  Lemma 2.3.

The metric

$\tilde{g}_{M}=\frac{1}{5}(\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2})+\pi^{*}g_{N}$
(2.8)

is Einstein with positive scalar curvature and is known as the squashed
Einstein metric on the 3-Sasakian $M^{7}$ [7, 8].

######  Remark 2.4.

The metric cone on $(M,g_{M})$ has holonomy contained in $\mathrm{Sp}(2)$,
whereas the metric cone on $(M,\tilde{g}_{M})$ (once one scales
$\tilde{g}_{M}$ appropriately) has holonomy $\mathrm{Spin}(7)$. In the first
case, the metric cone has the full holonomy $\mathrm{Sp}(2)$ if it is not
flat.

###  2.2. Natural $\mathrm{G}_{2}$-structures

We recall that a $\mathrm{G}_{2}$-structure on a 7-manifold is determined by a
3-form $\varphi$ on the manifold satisfying a certain nondegeneracy condition.
Such a 3-form determines a metric $g_{\varphi}$ and volume form
$\mathrm{vol}_{\varphi}$, and hence a dual 4-form $\psi=*_{\varphi}\varphi$,
where $*_{\varphi}$ is the Hodge star determined by $\varphi$.

Given the data in (2.1), (2.2) and (2.3) above, we may now write down a
natural family of $\mathrm{G}_{2}$-structures on a 3-Sasakian 7-manifold
$(M^{7},g_{M})$ as follows.

######  Lemma 2.5.

Given $a_{1},a_{2},a_{3},c>0$ and $\epsilon\in\\{\pm 1\\}$, if we let
$\mathbf{a}=(a_{1},a_{2},a_{3})$ then the $3$-form

$\varphi_{\mathbf{a},c,\epsilon}=\epsilon
a_{1}a_{2}a_{3}\eta_{1}\wedge\eta_{2}\wedge\eta_{3}-c^{2}(a_{1}\eta_{1}\wedge\omega_{1}+a_{2}\eta_{2}\wedge\omega_{2}+\epsilon
a_{3}\eta_{3}\wedge\omega_{3})$ (2.9)

defines a $\mathrm{G}_{2}$-structure on $M$. Moreover, this
$\mathrm{G}_{2}$-structure induces the following metric, volume form and dual
$4$-form:

$\displaystyle g_{\mathbf{a},c}$
$\displaystyle=a_{1}^{2}\eta_{1}^{2}+a_{2}^{2}\eta_{2}^{2}+a_{3}^{2}\eta_{3}^{2}+c^{2}\pi^{*}g_{N};$
(2.10) $\displaystyle\mathrm{vol}_{\mathbf{a},c,\epsilon}$
$\displaystyle=\epsilon
a_{1}a_{2}a_{3}c^{4}\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\pi^{*}\mathrm{vol}_{N};$
(2.11) $\displaystyle\psi_{\mathbf{a},c,\epsilon}$
$\displaystyle=c^{4}\pi^{*}\mathrm{vol}_{N}-c^{2}(\epsilon
a_{2}a_{3}\eta_{2}\wedge\eta_{3}\wedge\omega_{1}+\epsilon
a_{3}a_{1}\eta_{3}\wedge\eta_{1}\wedge\omega_{2}+a_{1}a_{2}\eta_{1}\wedge\eta_{2}\wedge\omega_{3}).$
(2.12)

Note that $g_{\mathbf{a},c}$ is independent of $\epsilon$.

This result is an elementary consequence of the fact that
$\omega_{1},\omega_{2},\omega_{3}$ are the pullbacks of self-dual 2-forms on
$N$ satisfying (2.4).

######  Remark 2.6.

Initially, one may allow for $a_{1},a_{2},a_{3}\in\mathbb{R}\setminus\\{0\\}$.
However, $\varphi$ and $-\varphi$ are the same $\mathrm{G}_{2}$-structure up
to a change of orientation. Moreover, there are only two possibilities: either
$a_{1},a_{2},a_{3}$ all have the same sign, or just two have the same sign.
Therefore, we can take $a_{1},a_{2},a_{3}$ to be all positive and use
$\epsilon$ to account for the two choices.

We now compute the exterior derivatives of $\varphi_{\mathbf{a},c,\epsilon}$
and $\psi_{\mathbf{a},c,\epsilon}$, which together encode all of the
information about the torsion of the $\mathrm{G}_{2}$-structure.

######  Lemma 2.7.

Let $\varphi_{\mathbf{a},c,\epsilon}$ and $\psi_{\mathbf{a},c,\epsilon}$ be as
in Lemma 2.5. Then:

$\displaystyle\mathrm{d}\varphi_{\mathbf{a},c,\epsilon}$
$\displaystyle=4c^{2}(a_{1}+a_{2}+\epsilon a_{3})\pi^{*}\mathrm{vol}_{N}$
$\displaystyle\quad-2(\epsilon a_{1}a_{2}a_{3}-c^{2}a_{1}+c^{2}a_{2}+\epsilon
c^{2}a_{3})\eta_{2}\wedge\eta_{3}\wedge\omega_{1}$
$\displaystyle\quad-2(\epsilon a_{1}a_{2}a_{3}+c^{2}a_{1}-c^{2}a_{2}+\epsilon
c^{2}a_{3})\eta_{3}\wedge\eta_{1}\wedge\omega_{2}$
$\displaystyle\quad-2(\epsilon a_{1}a_{2}a_{3}+c^{2}a_{1}+c^{2}a_{2}-\epsilon
c^{2}a_{3})\eta_{1}\wedge\eta_{2}\wedge\omega_{3};$
$\displaystyle\mathrm{d}\psi_{\mathbf{a},c,\epsilon}$ $\displaystyle=0.$

This result follows quickly from (2.5) and (2.6). Notice in particular that
the $\mathrm{G}_{2}$-structures are all _coclosed_.

######  Remark 2.8.

We note the following special cases of our family of
$\mathrm{G}_{2}$-structures.

  * •

We can always make an overall rescaling so that $c=1$. (However, we shall see
that we will require the freedom to vary the scale $c$ along our flows.)

  * •

Taking $a_{1}=a_{2}=a_{3}=c=1$ and $\epsilon=1$ gives a coclosed
$\mathrm{G}_{2}$-structure inducing the 3-Sasakian metric. It has been
referred to as the “canonical” $\mathrm{G}_{2}$-structure on a 3-Sasakian
7-manifold (see e.g. [1]).

  * •

Taking $a_{1}=a_{2}=a_{3}=a$ and $c=1$ gives the family of
$\mathrm{G}_{2}$-structures considered in [15]. The subfamily where
$\epsilon=1$ was also studied earlier in [7].

###  2.3. Nearly parallel G2-structures

We recall the definition of the distinguished class of
$\mathrm{G}_{2}$-structures that will be the focus of this paper.

######  Definition 2.9.

A $\mathrm{G}_{2}$-structure on a 7-manifold $M$ defined by a 3-form $\varphi$
with dual 4-form $\psi$ is _nearly parallel_ if

$\mathrm{d}\varphi=\lambda\psi$

for some non-zero constant $\lambda$. (A priori $\lambda$ could be a function
on $M$, but a short argument using $\mathrm{d}\psi=0$ and some representation
theory shows that it must in fact be constant.)

A nearly parallel $\mathrm{G}_{2}$-structure $\varphi$ induces an Einstein
metric $g_{\varphi}$ with positive scalar curvature. If $\lambda$ is chosen so
that the scalar curvature of $g_{\varphi}$ is 42, then the cone metric
$\mathrm{d}r^{2}+r^{2}g_{\varphi}$ on $\mathbb{R}^{+}\times M$ is Ricci-flat
and has holonomy contained in $\mathrm{Spin}(7)$, and $\varphi$ is _strictly
nearly parallel_ if the holonomy of this cone metric is $\mathrm{Spin}(7)$.
(One should compare this to Remark 2.4.)

We now record the following facts, which follow immediately from (2.5) and
(2.6), that show that our family of $\mathrm{G}_{2}$-structures contains two
nearly parallel $\mathrm{G}_{2}$-structures (up to scale).

######  Lemma 2.10.

Take $a_{1}=a_{2}=a_{3}=a$ in $\varphi_{\mathbf{a},c,\epsilon}$.

  * •

If $a=\frac{1}{\sqrt{5}}c$ and $\epsilon=1$, then the resulting
$\mathrm{G}_{2}$-structure, which we may write $c^{3}\varphi^{np}$ with
$\varphi^{np}$ independent of $c$, is (strictly) nearly parallel and its
induced metric is $c^{2}\tilde{g}_{M}$.

  * •

If $a=c$ and $\epsilon=-1$, then the resulting $\mathrm{G}_{2}$-structure,
which we may write $c^{3}\varphi^{ts}$ where $\varphi^{ts}$ is independent of
$c$, is nearly parallel and its induced metric is $c^{2}g_{M}$.

Hence, within each branch (determined by $\epsilon\in\\{\pm 1\\}$) of our
family of $\mathrm{G}_{2}$-structures, there is one natural critical point (up
to scale) for our geometric flows. For $\epsilon=1$, this is the strictly
nearly parallel $\mathrm{G}_{2}$-structure $\varphi^{np}$ inducing the
squashed Einstein metric $\tilde{g}_{M}$ on $M$, and for $\epsilon=-1$ this is
the nearly parallel $\mathrm{G}_{2}$-structure $\varphi^{ts}$ (where “ts”
stands for 3-Sasakian) inducing the 3-Sasakian metric $g_{M}$.

######  Remark 2.11.

If we take $a=\frac{1}{\sqrt{5}}c$ and $\epsilon=-1$ in Lemma 2.10 then we
obtain a coclosed $\mathrm{G}_{2}$-structure which induces the Einstein metric
$c^{2}\tilde{g}_{M}$, but is not nearly parallel. The same occurs when we
$a=c$ and $\epsilon=1$, but now for the Einstein metric $c^{2}g_{M}$: this
gives a multiple of the “canonical” $\mathrm{G}_{2}$-structure we saw earlier
(cf. Remark 2.8).

######  Remark 2.12.

It is worth noting that, by Lemma 2.10 and [6, Examples 1.14 and 1.15], the
$\mathrm{G}_{2}$-structures defined by $\varphi_{\mathbf{a},c,+1}$ and
$\varphi_{\mathbf{a},c,-1}$ cannot be homotopic through
$\mathrm{G}_{2}$-structures.

###  2.4. The ansatz

Motivated by Lemma 2.10, we will take our ansatz to be a special case of that
of Lemma 2.5 where

$a_{1}=a_{2}=a(t),\qquad a_{3}=b(t)\quad\text{and}\quad c=c(t),$ (2.13)

for positive time-dependent functions $a(t),b(t),c(t)$. These then define
1-parameter families of $\mathrm{G}_{2}$ 3-forms $\varphi_{\epsilon}(t)$
depending on $t$, with induced metric $g(t)$, volume form
$\mathrm{vol}_{\epsilon}(t)$ and dual 4-form $\psi_{\epsilon}(t)$ as follows:

$\displaystyle\varphi_{\epsilon}(t)$ $\displaystyle=\epsilon
a(t)^{2}b(t)\eta_{1}\wedge\eta_{2}\wedge\eta_{3}-a(t)c(t)^{2}(\eta_{1}\wedge\omega_{1}+\eta_{2}\wedge\omega_{2})-\epsilon
b(t)c(t)^{2}\eta_{3}\wedge\omega_{3};$ (2.14) $\displaystyle g(t)$
$\displaystyle=a(t)^{2}(\eta_{1}^{2}+\eta_{2}^{2})+b(t)^{2}\eta_{3}^{2}+c(t)^{2}\pi^{*}g_{N};$
(2.15) $\displaystyle\mathrm{vol}_{\epsilon}(t)$ $\displaystyle=\epsilon
a(t)^{2}b(t)c(t)^{4}\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\pi^{*}\mathrm{vol}_{N};$
(2.16) $\displaystyle\psi_{\epsilon}(t)$
$\displaystyle=c(t)^{4}\pi^{*}\mathrm{vol}_{N}-\epsilon
a(t)b(t)c(t)^{2}(\eta_{2}\wedge\eta_{3}\wedge\omega_{1}+\eta_{3}\wedge\eta_{1}\wedge\omega_{2})-a(t)^{2}c(t)^{2}\eta_{1}\wedge\eta_{2}\wedge\omega_{3}.$
(2.17)

We include the subscript $\epsilon$ to emphasise the choice of branch given by
$\epsilon\in\\{\pm 1\\}$, as we shall see different behaviour for distinct
choices of $\epsilon$, but drop the subscript for $g(t)$ since it is
independent of $\epsilon$. We shall make the restriction in (2.13) henceforth
in this article.

######  Remark 2.13.

The reader way wonder why we do not simply choose $a=b$ in (2.13) given that
this holds for the nearly parallel $\mathrm{G}_{2}$-structures in Lemma 2.10.
We shall see that the simpler ansatz when $a=b$ is not necessarily preserved
along the geometric flows we consider, and so we broaden our study to consider
the larger class of 1-parameter families of $\mathrm{G}_{2}$-structures given
by the condition (2.13). One could also consider curves in the full family of
$\mathrm{G}_{2}$-structures in Lemma 2.5, but this would be much more
challenging and we already exhibit interesting behaviour within the framework
provided by (2.13).

For the ansatz, we have the following simplification and slight extension of
Lemma 2.7.

######  Lemma 2.14.

Let $\varphi_{\epsilon}=\varphi_{\epsilon}(t)$ and
$\psi_{\epsilon}=\psi_{\epsilon}(t)$ be given by Lemma 2.5 with the conditions
in (2.13). Then:

$\displaystyle\mathrm{d}\varphi_{\epsilon}$ $\displaystyle=4c^{2}(2a+\epsilon
b)\pi^{*}\mathrm{vol}_{N}-2\epsilon
b(a^{2}+c^{2})\eta_{2}\wedge\eta_{3}\wedge\omega_{1}$
$\displaystyle\quad-2\epsilon
b(a^{2}+c^{2})\eta_{3}\wedge\eta_{1}\wedge\omega_{2}-2\epsilon(a^{2}b+2\epsilon
ac^{2}-bc^{2})\eta_{1}\wedge\eta_{2}\wedge\omega_{3};$
$\displaystyle\mathrm{d}\psi_{\epsilon}$ $\displaystyle=0.$

Moreover, we may write
$\mathrm{d}\varphi_{\epsilon}=\tau_{0}\psi_{\epsilon}+*\tau_{3}$ where

$\displaystyle\tau_{0}$
$\displaystyle=\frac{4}{7a^{2}c^{2}}\left(4a(a^{2}+c^{2})+\epsilon
b(2a^{2}-c^{2})\right)$

and $\tau_{3}\wedge\varphi_{\epsilon}=0=\tau_{3}\wedge\psi_{\epsilon}$.

###### Proof.

The formulas for $\mathrm{d}\varphi_{\epsilon}$ and
$\mathrm{d}\psi_{\epsilon}$ are immediate from Lemma 2.7. We then compute that

$\displaystyle\mathrm{d}\varphi_{\epsilon}\wedge\varphi_{\epsilon}$
$\displaystyle=4\big{(}\epsilon a^{2}bc^{2}(2a+\\!\epsilon b)+2\epsilon
abc^{2}(a^{2}+c^{2})+bc^{2}(a^{2}b+2\epsilon
ac^{2}-\\!bc^{2})\big{)}\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\pi^{*}\mathrm{vol}_{N}$
$\displaystyle=4\epsilon bc^{2}\big{(}4a(a^{2}+c^{2})+\epsilon
b(2a^{2}-c^{2})\big{)}\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\pi^{*}\mathrm{vol}_{N}.$

The formula for $\tau_{0}$ follows. ∎

######  Remark 2.15.

Lemma 2.14 shows that, regardless of the choice of $\epsilon\in\\{\pm 1\\}$,
we can always choose initial conditions for our flows of
$\mathrm{G}_{2}$-structures such that $\tau_{0}=0$ (and necessarily
$\tau_{3}\neq 0$), even though we are trying to flow to nearly parallel
$\mathrm{G}_{2}$-structures, which must have $\tau_{0}\neq 0$ and
$\tau_{3}=0$.

##  3\. Laplacian coflow

We start by studying the Laplacian coflow, which is arguably the natural flow
for our ansatz of coclosed $\mathrm{G}_{2}$-structures since it manifestly
preserves the coclosed condition. We recall that this flow, if it is well-
posed and stays within the ansatz, is given by

$\frac{\partial}{\partial
t}\psi_{\epsilon}(t)=\Delta_{\psi_{\epsilon}(t)}\psi_{\epsilon}(t)=\mathrm{d}\mathrm{d}^{*}_{\psi_{\epsilon}(t)}\psi_{\epsilon}(t),$
(3.1)

for the closed 4-forms $\psi_{\epsilon}(t)$ in (2.17).

###  3.1. The flow equations

Since we have that

$\mathrm{d}\mathrm{d}^{*}_{\psi_{\epsilon}}\psi_{\epsilon}=\mathrm{d}*\mathrm{d}\varphi_{\epsilon},$

it is straightforward to compute the right-hand side of (3.1) from Lemma 2.14
as follows.

######  Lemma 3.1.

The Hodge Laplacian of $\psi_{\epsilon}$ in (2.17) is given by:

$\begin{split}\Delta_{\psi_{\epsilon}}\psi_{\epsilon}&=8\left(2a^{2}+b^{2}+2c^{2}+\frac{2\epsilon
bc^{2}}{a}-\frac{b^{2}c^{2}}{a^{2}}\right)\pi^{\ast}\textnormal{vol}_{N}\\\
&\qquad-4\Big{(}b^{2}+\frac{4\epsilon
a^{3}b}{c^{2}}+\frac{2a^{2}b^{2}}{c^{2}}+\frac{2\epsilon
bc^{2}}{a}-\frac{b^{2}c^{2}}{a^{2}}\Big{)}(\eta_{2}\wedge\eta_{3}\wedge\omega_{1}+\eta_{3}\wedge\eta_{1}\wedge\omega_{2})\\\
&\qquad-4\Big{(}2a^{2}-b^{2}+2c^{2}+\frac{4\epsilon
a^{3}b}{c^{2}}+\frac{2a^{2}b^{2}}{c^{2}}-\frac{2\epsilon
bc^{2}}{a}+\frac{b^{2}c^{2}}{a^{2}}\Big{)}\eta_{1}\wedge\eta_{2}\wedge\omega_{3}\end{split}$
(3.2)

In particular, (3.2) is in the same form as (2.17) and so the Laplacian coflow
(3.1) is well-defined.

Given this result and (2.17), we may write down the Laplacian coflow (3.1) as
the following system of ordinary differential equations for the coefficient
functions $a(t)$, $b(t)$, $c(t)$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(c^{4})$
$\displaystyle=8\left(2a^{2}+b^{2}+2c^{2}+\frac{2\epsilon
bc^{2}}{a}-\frac{b^{2}c^{2}}{a^{2}}\right);$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(a^{2}c^{2})$
$\displaystyle=4\left(2a^{2}-b^{2}+2c^{2}+\frac{4\epsilon
a^{3}b}{c^{2}}+\frac{2a^{2}b^{2}}{c^{2}}-\frac{2\epsilon
bc^{2}}{a}+\frac{b^{2}c^{2}}{a^{2}}\right);$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(abc^{2})$
$\displaystyle=4\left(\epsilon b^{2}+\frac{4a^{3}b}{c^{2}}+\frac{2\epsilon
a^{2}b^{2}}{c^{2}}+\frac{2bc^{2}}{a}-\frac{\epsilon
b^{2}c^{2}}{a^{2}}\right).$

We can simplify the analysis of these equations by introducing new variables
as follows.

######  Lemma 3.2.

Define

$X=\frac{a^{2}}{c^{2}}\quad\text{and}\quad Y=\frac{ab}{c^{2}}$

and introduce a new variable $s$ by

$\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{1}{c^{2}}.$

If we let $\dot{X}=\frac{\mathrm{d}X}{\mathrm{d}s}$ and
$\dot{Y}=\frac{\mathrm{d}Y}{\mathrm{d}s}$, then the Laplacian coflow equations
for (2.17) imply that

$\displaystyle\dot{X}$
$\displaystyle=\frac{4}{X^{2}}\left((X+1)Y^{2}+2\epsilon(2X^{2}-2X-1)XY-2X^{2}(2X-1)(X+1)\right);$
(3.3) $\displaystyle\dot{Y}$
$\displaystyle=\frac{4Y}{X^{2}}\left(2(1-X)Y^{2}+\epsilon(2X^{2}-3X-1)Y+2X(1-2X)\right).$
(3.4)

We note that $X$ and $Y$ are scale-invariant quantities and that solutions to
(3.3)–(3.4) give the solutions to the Laplacian coflow (3.1) up to rescaling.

###  3.2. Critical points and dynamics

To understand the dynamics of the flow (3.3)–(3.4), we need to study its
critical points. Some straightforward calculations show the following.

######  Lemma 3.3.

The only critical points for $X,Y>0$ to the system (3.3)–(3.4) are:

$X=Y=\frac{1}{5}\quad\text{and}\quad\epsilon=1$ (3.5)

and

$X=Y=1\quad\text{and}\quad\epsilon=-1.$ (3.6)

Moreover, if $\epsilon=1$ the condition $X=Y$ is preserved, but if
$\epsilon=-1$ the condition $X=Y$ is not preserved except when $X=Y=1$.

######  Remark 3.4.

By Lemma 2.10, the critical points (3.5)–(3.6) correspond to the 4-forms
$\psi^{np}$ and $\psi^{ts}$ dual to the nearly parallel
$\mathrm{G}_{2}$-structures $\varphi^{np}$ and $\varphi^{ts}$ respectively.
Thus, Lemma 3.3 shows that the only critical points for (3.1) up to rescaling
are $\psi^{np}$ and $\psi^{ts}$.

Before considering the general ansatz, we note that if we set $X=Y$ and
$\epsilon=1$ in (3.3)–(3.4) then we obtain:

$\dot{X}=4(1-5X).$

Hence, $\dot{X}$ is positive for $X<1/5$ and negative for $X>1/5$, which
clearly shows the stability along the line $X=Y$ of the critical point (3.5).
Thus $\psi^{np}$ is stable within the restricted ansatz (2.13) with $a=b$.

######  Remark 3.5.

Lemma 3.3 shows that the coclosed $\mathrm{G}_{2}$-structure with $\epsilon=1$
inducing the 3-Sasakian metric (up to scale), as well as the one with
$\epsilon=-1$ inducing the squashed Einstein metric, have no significance for
the Laplacian coflow. It also shows that we need to use the ansatz (2.13) with
$a$ and $b$ distinct (i.e. allowing $X\neq Y$) to understand the Laplacian
coflow for $\epsilon=-1$.

We provide dynamic plots of the equations (3.3)–(3.4) in Figure 3.1 for
$\epsilon=\pm 1$. In the plots, the curves $\gamma_{X}$ and $\gamma_{Y}$
across which $\dot{X}$ and $\dot{Y}$ change sign respectively are also shown,
along with the line $X=Y$.

Figure 3.1. Dynamic plots for Laplacian coflow for $\epsilon=1$ and
$\epsilon=-1$

Figure 3.1 indicates that the critical points (3.5) and (3.6), which
correspond to $\psi^{np}$ and $\psi^{ts}$ as in Remark 3.4, are both _stable_.
We now show that this is indeed true.

######  Proposition 3.6.

The $4$-forms $\psi^{np}$ and $\psi^{ts}$ dual to the nearly parallel
$\mathrm{G}_{2}$-structures $\varphi^{np}$ and $\varphi^{ts}$ are stable sinks
under the Laplacian coflow (3.1), after rescaling.

###### Proof.

We study the linearization of the flow equations (3.3)–(3.4) at the critical
points (3.5) and (3.6) to determine their stability.

At $X=Y=1/5$ and $\epsilon=1$, the linearized equations are

$\displaystyle\dot{X}=-\frac{76}{5}X-\frac{24}{5}Y\quad\text{and}\quad\dot{Y}=-\frac{12}{5}X-\frac{88}{5}Y.$

(Note that $X=Y$ is preserved by the above system as expected.) The associated
$2\times 2$ matrix of coefficients of $X,Y$ in the above equations has two
negative eigenvalues ($-64/5$ and $-20$) and so (3.5) is a stable critical
point.

Similarly, at $X=Y=1$ and $\epsilon=-1$, the linearized equations are

$\displaystyle\dot{X}=-76X+24Y\quad\text{and}\quad\dot{Y}=-36X+8Y,$

noting that this time $X=Y$ is not preserved. Here, the matrix one obtains
again has two negative eigenvalues, which are $-4$ and $-64$, so the critical
point (3.6) is stable. ∎

######  Remark 3.7.

We see from (3.3)–(3.4) that if we allow $X=0$ or $Y=0$ then there are
additional critical points:

$(X,Y)=\left(\frac{1}{2},0\right)\quad\text{for $\epsilon=\pm 1$.}$

We can see these critical points in Figure 3.1. We can also consider
$(X,Y)=(0,0)$ to be a degenerate critical point, even though the equations
(3.3)–(3.4) are not defined there. We can understand these additional critical
points geometrically as follows.

Recall the fibration (2.1) of $M^{7}$ over $N^{4}$. The point $(0,0)$
corresponds to sending the 3-dimensional fibres of (2.1) to zero size (since
$a=b=0$), and so $M$ has collapsed to $N$ (or a point). In this setting, the
4-form $\psi_{\epsilon}$ reduces to simply the volume form of $N$ (or zero if
the collapse is to a point).

If we instead view the fibres of (2.1) as circle bundles over $S^{2}$ (where
$E_{3}$ is tangent to the circle direction in the notation of Definition 2.1),
then at $(1/2,0)$ the circle fibres have now collapsed (as $b=0$ in (2.13)).
Since $a\neq 0$ there $M^{7}$ has collapsed to a 6-manifold $Z$ which is a
2-sphere bundle over $N$. This 6-manifold $Z$ is the _twistor space_ of $N$,
and at this critical point it will be endowed with its _nearly Kähler_ metric
$g_{Z}$, which is an Einstein metric on $Z$ with positive scalar curvature.
This Einstein metric $g_{Z}$ is not Kähler (unlike the standard choice of
metric on the twistor space), but instead is related to $\mathrm{G}_{2}$
geometry as the metric cone over a 6-dimensional nearly Kähler manifold will
have holonomy $\mathrm{G}_{2}$.

###  3.3. Long-time behaviour

The plots in Figure 3.1 suggest that, within our ansatz, any initial condition
flows to the unique (up to scale) nearly parallel $\mathrm{G}_{2}$-structure
in the family. We now show that this is indeed the case. For the statement, as
in Remark 3.4, we denote by $\psi^{np}$ and $\psi^{ts}$ the duals of the
nearly parallel $\mathrm{G}_{2}$-structures $\varphi^{np}$ and $\varphi^{ts}$
defined in Lemma 2.10, and recall that they induce the squashed Einstein
metric and 3-Sasakian metric respectively.

######  Theorem 3.8.

Let $\psi_{\epsilon}=\psi_{\epsilon}(0)$ be a closed $4$-form as in (2.17)
dual to a $\mathrm{G}_{2}$-structure. The solution to the Laplacian coflow
(3.1) starting at $\psi_{\epsilon}$ converges, after rescaling, to $\psi^{np}$
if $\epsilon=1$ and to $\psi^{ts}$ if $\epsilon=-1$, which are the only
critical points of the rescaled flow. In particular, the nearly parallel
$\mathrm{G}_{2}$-structures given by $\psi^{np}$ and $\psi^{ts}$ are stable
for (3.1) after rescaling.

Theorem 3.8 gives Theorem 1.1 in the introduction. The proof of Theorem 3.8 is
quite lengthy, so we break it up into several smaller results.

####  3.3.1. Strategy

Our aim is to prove that, given any initial condition, the flow (3.3)–(3.4)
for $(X(s),Y(s))$ exists for all $s$ and converges to the critical point with
$X>0$ and $Y>0$. From (3.3)–(3.4), long time existence will be guaranteed as
long as $X,Y$ remain bounded and $X$ remains bounded away from $0$. Moreover,
periodic orbits are not possible as we know the Laplacian coflow is a gradient
flow. Hence, long time existence will imply convergence to the critical point
with $X,Y>0$ if we additionally know that $Y$ remains bounded away from $0$.

To deduce the result for the Laplacian coflow we observe that the evolution
equation for $c^{2}$, which determines the parameter $s$ by Lemma 3.2, is:

$\frac{\mathrm{d}}{\mathrm{d}t}(c^{2})=8\left(2X+\frac{Y^{2}}{X}+2+2\epsilon\frac{Y}{X}-\frac{Y^{2}}{X^{2}}\right),$

so $c^{2}$ cannot blow up in finite time because $X,Y$ are bounded and bounded
away from zero. In fact, $c^{2}$ grows at most linearly, so we can integrate
$\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{1}{c^{2}}$ to find $s$ as a function of
the Laplacian coflow parameter $t$.

Throughout the proof, we denote the curves where $\dot{X}$ and $\dot{Y}$
change sign, respectively, by $\gamma_{X}$ and $\gamma_{Y}$ as in Figure 3.1.

####  3.3.2. Bounds on $X$

We start by looking at the behaviour of $X$.

######  Lemma 3.9.

The function $X(s)$ is bounded away from zero and can only diverge in the
region to the left of $\gamma_{X}$.

###### Proof.

We first see that $X$ is decreasing whenever it is to the right of
$\gamma_{X}$ and so $X$ will remain bounded by its initial condition in this
region. We then see that $X$ is increasing in the region to the left of
$\gamma_{X}$, which is the region containing $X=0$, and so $X$ is bounded away
from zero in this region by its initial condition. ∎

Since $\gamma_{X}$ meets $Y=X$ (at the critical point), the unbounded part of
$\gamma_{X}$ lies above the line $Y=X$, i.e. where $Y>X$. Therefore, it
suffices to show that $Y$ remains bounded in the region to the left of
$\gamma_{X}$ where additionally $Y>X$ to deduce that $X$ is bounded
everywhere.

####  3.3.3. Bounds on $Y$: $\epsilon=1$

Given our earlier discussion, we now turn to showing that $Y$ remains bounded
and bounded away from zero. We start with the case $\epsilon=1$.

######  Lemma 3.10.

When $\epsilon=1$, $Y(s)$ can only diverge in the region above the upper part
of $\gamma_{Y}$ and can only tend to zero in the region above the lower part
of $\gamma_{Y}$ but below the line $Y=X$.

###### Proof.

In this case, $\dot{Y}<0$ in the region above the lower part of $\gamma_{Y}$
and below or to the right of the upper part of $\gamma_{Y}$, so $Y$ remains
bounded by its initial value in this region. Note that $\gamma_{Y}$ meets the
line $X=0$ at $Y=0$ and $Y=1/2$. Hence, in the same region we just considered,
we are either to the left of $\gamma_{X}$ and so $\dot{X}>0$, which means we
cannot reach $(0,0)$, or we are to the right of $\gamma_{X}$ but also above
$\gamma_{Y}$. In this latter region, if we are above the line $Y=X$ we cannot
cross it by Lemma 3.3, and so $Y$ remains bounded away from zero here.

We then notice that $\dot{Y}>0$ in the region bounded by the lower part of
$\gamma_{Y}$, in which $Y$ is bounded, and so $Y$ is additionally bounded away
from $0$ here. ∎

We now have our crucial observation for the case $\epsilon=1$.

######  Lemma 3.11.

When $\epsilon=1$, $Y(s)$ is decreasing and hence bounded in the region where
$Y>X$ and $X\geq 1$.

###### Proof.

In this setting, we can rewrite (3.4) in the following manner:

$\dot{Y}=\frac{4Y}{X^{2}}\left(2(1-X)(Y-X)Y-XY-Y+2X(1-2X)\right).$

Hence, $\dot{Y}<0$ when $Y>X$ and $X\geq 1$. Therefore, $Y$ will be bounded in
this region. ∎

Since the only part of the quadrant with $X,Y>0$ where $Y$ can become
unbounded is where $Y>X$ by Lemma 3.10, we deduce that $Y$ can only become
unbounded, when $\epsilon=1$, if $X$ remains in $(0,1)$. We now show that this
is impossible.

######  Proposition 3.12.

For $\epsilon=1$, there are no solutions $(X(s),Y(s))$ above the upper part of
$\gamma_{Y}$ with $X(s)\in(0,1)$ and $Y(s)$ unbounded.

###### Proof.

Suppose not and that we have such a solution. Note that there is a finite
$\bar{Y}>0$ such that for all $X\in(0,1)$,

$Y^{2}+2(2X^{2}-2X-1)XY-2X^{2}(2X-1)(X+1)>0\quad\text{and}\quad(2X^{2}-3X-1)Y+2X(1-2X)<0$
(3.7)

for all $Y>\bar{Y}$. Since $Y$ is increasing and unbounded in the region under
consideration, we may assume that $Y(s)>\bar{Y}$. Then, comparing (3.7) and
(3.3)–(3.4), we see that

$\dot{X}\geq\frac{4}{X}Y^{2}\quad\text{and}\quad\dot{Y}\leq\frac{8(1-X)}{X^{2}}Y^{3}.$

Hence, we see that

$\frac{\mathrm{d}Y}{\mathrm{d}X}\leq\frac{2(1-X)}{X}Y.$

Grönwall’s inequality then shows that there are constants $C_{0},C_{1}$
depending only on the initial conditions so that $Y$ is bounded by
$C_{0}X^{2}e^{-2X}+C_{1}$. Since we assumed $X(s)\in(0,1)$, this is a
contradiction. ∎

Our results so far show that, when $\epsilon=1$, both $X$ and $Y$ are bounded
and that $X$ is bounded away from zero. To complete the proof of Theorem 3.8
in the $\epsilon=1$ case we therefore only need the following.

######  Lemma 3.13.

When $\epsilon=1$, $Y(s)$ is bounded away from zero.

###### Proof.

By Remark 3.7 and Lemma 3.10, we need only consider points near
$(X,Y)=(1/2,0)$ above $\gamma_{Y}$. Linearizing around the critical point
$(1/2,0)$, so writing $X=\frac{1}{2}+\delta X_{1}$ and $Y=\delta Y_{1}$ for
$\delta$ small, we find that the linear term in $\delta$ gives

$\dot{X_{1}}+\dot{Y_{1}}=-24(X_{1}+Y_{1})\quad\text{and}\quad\dot{Y_{1}}=0.$

These equations are degenerate along the line $X_{1}+Y_{1}=0$, which is
tangent to $\gamma_{Y}$ at $(1/2,0)$. However, $\gamma_{Y}$ lies above this
line for all points near $(1/2,0)$, so we may restrict to the region where
$X_{1}+Y_{1}>0$. Therefore, to leading order, $Y_{1}$ remains a non-zero
constant and $X_{1}+Y_{1}$ decreases with an exponential rate. Hence, $Y$ will
be bounded away from $0$, as claimed. ∎

We can now put our results together so far.

###### Proof of Theorem 3.8 for $\epsilon=1$.

Lemma 3.9 shows that $X$ is bounded away from $0$ and bounded above if $Y$ is
bounded. Lemma 3.10, Lemma 3.11 and Proposition 3.12 show that $Y$ is bounded
and hence $X$ is bounded. Finally, Lemma 3.13 shows that $Y$ is bounded away
from $0$. The observations in §3.3.1 then give the result. ∎

####  3.3.4. Bounds on $Y$: $\epsilon=-1$

Having proved Theorem 3.8 for $\epsilon=1$ now move on to the case where
$\epsilon=-1$. The arguments here are similar to the $\epsilon=1$ case, but
often easier.

######  Lemma 3.14.

When $\epsilon=-1$, $Y(s)$ can only diverge in the region to the left of
$\gamma_{Y}$ and above the line $Y=X$, and can only tend to $0$ to the right
of $\gamma_{Y}$ but below the line $Y=X$.

###### Proof.

These are elementary observations given that $Y$ is decreasing to the right of
$\gamma_{Y}$ and increasing to the left of $\gamma_{Y}$. ∎

We now see that the evolution equation (3.4) for $Y$ has a useful feature when
$\epsilon=-1$.

######  Lemma 3.15.

When $\epsilon=-1$, there exists a least $\bar{X}\in(1,2)$ such that
$X\leq\bar{X}$ whenever $(X,Y)$ is on $\gamma_{Y}$. Hence, $\dot{Y}\leq 0$ and
thus $Y$ is bounded whenever $X\geq\bar{X}$.

###### Proof.

This is an elementary calculation, showing that $X$ takes a maximum value on
the curve $\gamma_{Y}$. ∎

We deduce from Lemma 3.14 and Lemma 3.15 that the only way $Y$ can become
unbounded when $\epsilon=-1$ is if $X$ remains in the interval $(0,\bar{X})$.
We now show that this is not possible.

######  Proposition 3.16.

Recall $\bar{X}$ from Lemma 3.15. For $\epsilon=-1$, there are no solutions
$(X(s),Y(s))$ to the left of $\gamma_{Y}$ and above $Y=X$ with
$X(s)\in(0,\bar{X})$ and $Y(s)$ unbounded.

###### Proof.

We suppose, for a contradiction, that there is such a solution. Suppose that
the solution enters the part of the region where $X\geq 1$. Then $X$ is
strictly increasing, so $X>1$ for all subsequent times. However, the line
$\gamma_{Y}$ asymptotes to $X=1$ as $Y\to\infty$, so we must eventually have
that $X$ is decreasing, which is a contradiction.

We deduce that $X(s)\in(0,1)$ for all $s$. We note that there is a finite
$\bar{Y}>0$ such that

$\begin{split}Y^{2}-2(2X^{2}-2X-1)XY-2X^{2}(2X-1)(X+1)>0\\\
\text{and}\quad(2X^{2}-3X-1)Y-2X(1-2X)>0\end{split}$ (3.8)

for all $X\in(0,1)$ and $Y>\bar{Y}$. Since $Y$ is increasing in the region we
are studying and we are assuming it is unbounded, we may restrict to the case
where $Y(s)>\bar{Y}$. Comparing (3.8) to (3.3)–(3.4) yields the differential
inequalities

$\dot{X}\geq\frac{4}{X}Y^{2}\quad\text{and}\quad\dot{Y}\leq\frac{8(2-X)Y^{3}}{X^{2}}.$

We deduce that

$\frac{\mathrm{d}Y}{\mathrm{d}X}\leq\frac{2(2-X)}{X}Y$

and so, by Grönwall’s inequality, we have that $Y$ is bounded by
$X^{4}e^{-2X}$ (up to multiplicative factors and additive constants depending
only on the initial conditions). Since $X\in(0,1)$, this forces our required
contradiction. ∎

To complete the proof in the $\epsilon=-1$ we now only need to show that $Y$
stays away from $0$.

######  Lemma 3.17.

When $\epsilon=-1$, $Y(s)$ is bounded away from zero.

###### Proof.

The proof is entirely analogous to that of Lemma 3.13. Remark 3.7 and Lemma
3.14 show that we may restrict attention to points near $(1/2,0)$ to the right
of $\gamma_{Y}$. Writing $X=\frac{1}{2}+\delta X_{1}$ and $Y=\delta Y_{1}$ for
$\delta$ small, we find that the linear term in $\delta$ gives

$\dot{X_{1}}-\dot{Y_{1}}=-24(X_{1}-Y_{1})\quad\text{and}\quad\dot{Y_{1}}=0.$

Note that the line $X_{1}=Y_{1}$ is tangent to $\gamma_{Y}$ at $(1/2,0)$ and
that $\gamma_{Y}$ lies above this line. Therefore, we need only consider
$X_{1}-Y_{1}>0$ and see that, to leading order, $Y_{1}$ remains a non-zero
constant whilst $X_{1}-Y_{1}$ exponentially decreases. Hence, $Y$ will be
bounded away from $0$. ∎

We may now complete the proof of Theorem 3.8.

###### Proof of Theorem 3.8 for $\epsilon=-1$.

We first see that Lemma 3.9 shows that $X$ is bounded away from $0$ and
bounded if $Y$ is bounded. Lemmas 3.14 and 3.15, together with Proposition
3.16, show that $Y$ is bounded, and thus $X$ is also bounded. Finally, Lemma
3.17 shows that $Y$ is bounded away from zero, which completes the proof by
the discussion in §3.3.1. ∎

##  4\. Laplacian flow

We now consider the Laplacian flow for our family of
$\mathrm{G}_{2}$-structures in (2.14). We recall that this flow, if it is
well-posed and stays within the ansatz, is given by

$\frac{\partial}{\partial
t}\varphi_{\epsilon}(t)=\Delta_{\varphi_{\epsilon}(t)}\varphi_{\epsilon}(t)=\mathrm{d}^{*}_{\varphi(t)}\mathrm{d}\varphi_{\epsilon}(t),$
(4.1)

for the coclosed 3-forms $\varphi_{\epsilon}(t)$ in (2.14). Here we have to be
particularly mindful that the coclosed condition may not be preserved by the
flow, let alone the ansatz.

###  4.1. The flow equations

We first observe that the Hodge Laplacian of the 3-form defining the
$\mathrm{G}_{2}$-structure follows immediately from that of the 4-form by
taking the Hodge star:

$\mathrm{d}\mathrm{d}^{*}_{\varphi_{\epsilon}}\varphi_{\epsilon}=*\Delta_{\psi_{\epsilon}}\psi_{\epsilon}.$

To compute the Hodge star we use the following relations:

$\displaystyle*_{\varphi_{\epsilon}}\pi^{*}\mathrm{vol}_{N}$
$\displaystyle=\frac{\epsilon
a^{2}b}{c^{4}}\eta_{1}\wedge\eta_{2}\wedge\eta_{3};$
$\displaystyle*_{\varphi_{\epsilon}}(\eta_{2}\wedge\eta_{3}\wedge\omega_{1}+\eta_{3}\wedge\eta_{1}\wedge\omega_{2})$
$\displaystyle=\frac{\epsilon}{b}(\eta_{1}\wedge\omega_{1}+\eta_{2}\wedge\omega_{2});$
$\displaystyle*_{\varphi_{\epsilon}}(\eta_{1}\wedge\eta_{2}\wedge\omega_{1})$
$\displaystyle=\frac{\epsilon b}{a^{2}}\eta_{3}\wedge\omega_{3}.$

We can now use Lemma 3.1 to find an expression for the Hodge Laplacian of
$\varphi_{\epsilon}$, which we need to consider the Laplacian flow (4.1).

######  Lemma 4.1.

The Hodge Laplacian of $\varphi_{\epsilon}$ in (2.14) is given by:

$\begin{split}\Delta_{\varphi_{\epsilon}}\varphi_{\epsilon}&=\frac{8\epsilon
a^{2}b}{c^{2}}\left(\frac{2a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}+2+\frac{2\epsilon
b}{a}-\frac{b^{2}}{a^{2}}\right)\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\\\
&\qquad-4\Big{(}\epsilon b+\frac{4a^{3}}{c^{2}}+\frac{2\epsilon
a^{2}b}{c^{2}}+\frac{2c^{2}}{a}-\frac{\epsilon
bc^{2}}{a^{2}}\Big{)}(\eta_{1}\wedge\omega_{1}+\eta_{2}\wedge\omega_{2})\\\
&\qquad-4\epsilon
b\Big{(}2-\frac{b^{2}}{a^{2}}+\frac{2c^{2}}{a^{2}}+\frac{4\epsilon
ab}{c^{2}}+\frac{2b^{2}}{c^{2}}-\frac{2\epsilon
bc^{2}}{a^{3}}+\frac{b^{2}c^{2}}{a^{4}}\Big{)}\eta_{3}\wedge\omega_{3}\end{split}$
(4.2)

In particular, (4.2) is coclosed and in the same form as (2.14), so the
Laplacian flow (4.1) is well-defined.

######  Remark 4.2.

We observe that Lemma 4.1 shows that the coclosed condition on the
$\mathrm{G}_{2}$-structure is preserved along the Laplacian flow in this
situation.

As a consequence of Lemma 4.1 and (2.14), we can write down the Laplacian flow
(4.1) as a system of ODEs for $a(t)$, $b(t)$, $c(t)$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(a^{2}b)$
$\displaystyle=\frac{8a^{2}b}{c^{2}}\left(\frac{2a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}+2+\frac{2\epsilon
b}{a}-\frac{b^{2}}{a^{2}}\right);$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(ac^{2})$
$\displaystyle=4\Big{(}\epsilon b+\frac{4a^{3}}{c^{2}}+\frac{2\epsilon
a^{2}b}{c^{2}}+\frac{2c^{2}}{a}-\frac{\epsilon bc^{2}}{a^{2}}\Big{)};$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(bc^{2})$
$\displaystyle=4b\Big{(}2-\frac{b^{2}}{a^{2}}+\frac{2c^{2}}{a^{2}}+\frac{4\epsilon
ab}{c^{2}}+\frac{2b^{2}}{c^{2}}-\frac{2\epsilon
bc^{2}}{a^{3}}+\frac{b^{2}c^{2}}{a^{4}}\Big{)}.$

Just as for the Laplacian coflow, it turns out that the Laplacian flow is
easier to analyze if we introduce scale-invariant quantities and a new time
parameter. The resulting equations we obtain then describe the rescaled
Laplacian flow.

######  Lemma 4.3.

Define

$X=\frac{a^{2}}{c^{2}}\quad\text{and}\quad Y=\frac{ab}{c^{2}}$

and introduce a new variable $s$ by

$\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{1}{c^{2}}.$

If we let $\dot{X}=\frac{\mathrm{d}X}{\mathrm{d}s}$ and
$\dot{Y}=\frac{\mathrm{d}Y}{\mathrm{d}s}$, then the Laplacian flow equations
(4.1) imply that

$\displaystyle\dot{X}$
$\displaystyle=\frac{4}{X^{2}}\left(2X^{2}(2X-1)(X+1)+2\epsilon(1+2X-2X^{2})XY-(1+X)Y^{2}\right);$
(4.3) $\displaystyle\dot{Y}$
$\displaystyle=\frac{4Y}{X^{2}}\left(2X(2X-1)+\epsilon(1+3X-2X^{2})Y+2(X-1)Y^{2}\right).$
(4.4)

###  4.2. Critical points and dynamics

We observe that (4.3)–(4.4) are just the negative of the equations (3.3)–(3.4)
arising from the Laplacian coflow. We therefore have the following.

######  Lemma 4.4.

The only critical points for $X,Y>0$ to (4.3)–(4.4) are:

$X=Y=\frac{1}{5}\quad\text{and}\quad\epsilon=1$

and

$X=Y=1\quad\text{and}\quad\epsilon=-1,$

and the condition $X=Y$ is preserved for $\epsilon=1$, but not preserved for
$\epsilon=-1$ except when $X=Y=1$.

The observation that the Laplacian flow equations (4.3)–(4.4) are the negative
of the Laplacian coflow equations also implies the following result, based on
our stability analysis for the Laplacian coflow, which gives Theorem 1.2 in
the Introduction.

######  Theorem 4.5.

The only critical points of the Laplacian flow (4.1), after rescaling, are the
nearly parallel $\mathrm{G}_{2}$ structures $\varphi^{ts}$ and $\varphi^{np}$
inducing the 3-Sasakian and squashed Einstein metrics. Both critical points
are unstable sources under the rescaled Laplacian flow.

For completeness, we provide the dynamic plots in Figure 4.1 for the Laplacian
flow (4.1) for our ansatz with $\epsilon=1$ and $\epsilon=-1$. We again
indicate the curves $\gamma_{X}$ and $\gamma_{Y}$ where $\dot{X}$ and
$\dot{Y}$ change sign, respectively. Of course, the dynamics are simply the
opposite of those which appear in the Laplacian coflow plot in Figure 3.1

Figure 4.1. Dynamic plots for Laplacian flow for $\epsilon=1$ and
$\epsilon=-1$

######  Remark 4.6.

Figure 4.1 shows that several different behaviours are possible for the
rescaled Laplacian flow when the initial condition is close to a nearly
parallel $\mathrm{G}_{2}$-structure. One possibility is flowing to the origin,
which corresponds again to the 7-manifold $M$ collapsing to the 4-manifold $N$
through the fibration (2.1). Notice that the 3-dimensional fibres of (2.1) are
calibrated by the $\mathrm{G}_{2}$-structure $\varphi$, i.e. $\varphi$
restricts to be the volume form on the fibres, and so are associative by
definition. The fact that $\mathrm{d}\varphi\neq 0$ means that the volume of
any compact associative 3-fold is not necessarily topologically determined,
unlike for closed $\mathrm{G}_{2}$-structures.

##  5\. Ricci flow

In this section we compare the results we have obtained for the Laplacian flow
and coflow of our $\mathrm{G}_{2}$-structures with the behaviour of the
induced metric of the $\mathrm{G}_{2}$-structures under the Ricci flow. We
recall that this comparison is useful because the nearly parallel
$\mathrm{G}_{2}$-structures, which are critical points of the rescaled
Laplacian flow and coflow that we studied earlier, induce Einstein metrics
which are then critical points of the Ricci flow up to scale. It is also of
interest because if we use the Laplacian coflow for coclosed
$\mathrm{G}_{2}$-structures (1.3), then the induced flow (1.4) on the induced
metric of the $\mathrm{G}_{2}$-structures is the Ricci flow plus lower order
terms determined algebraically by the torsion of the
$\mathrm{G}_{2}$-structure.

###  5.1. The flow equations

We wish to study the Ricci flow for our ansatz

$\frac{\partial}{\partial t}g(t)=-2\mathrm{Ric}(t),$ (5.1)

if it exists. From general theory the Ricci flow will have short time
existence starting from our metric ansatz (2.15), though it is not immediately
obvious that the ansatz will be preserved.

We begin by computing the Ricci curvature of $g(t)$ from (2.15). (Note that
$g(t)$ is independent of $\epsilon\in\\{\pm 1\\}$.)

######  Lemma 5.1.

Let $g(t)$ be as in (2.15). The Ricci curvature $\mathrm{Ric}(t)$ of $g(t)$
satisfies

$\displaystyle\mathrm{Ric}$
$\displaystyle=2\left(2-\frac{b^{2}}{a^{2}}+\frac{2a^{4}}{c^{4}}\right)(\eta_{1}^{2}+\eta_{2}^{2})+2\left(\frac{b^{4}}{a^{4}}+\frac{2b^{4}}{c^{4}}\right)\eta_{3}^{2}+2\left(6-\frac{2a^{2}+b^{2}}{c^{2}}\right)\pi^{*}g_{N}.$

In particular, $\mathrm{Ric}(t)$ is of the same form as (2.15), and so the
Ricci flow (5.1) is well-defined.

###### Proof.

Our approach is to use the O’Neill formulas for Riemannian submersions [20].

We let $\nabla$ denote the Levi-Civita connection of the 3-Sasakian metric
$g_{M}$. Recall the orthonormal Killing fields $E_{1},E_{2},E_{3}$ in
Definition 2.1 and let $B_{0},B_{1},B_{2},B_{3}$ be local orthonormal vector
fields on $M$ which are horizontal for the Riemannian submersion (2.1). It is
then straightforward to compute that

$\displaystyle\nabla_{E_{i}}E_{j}=\sum_{k=1}^{3}\epsilon_{ijk}E_{k},\quad\nabla_{E_{i}}B_{l}=0,\quad\nabla_{B_{l}}E_{i}=-\sum_{m=0}^{4}\sigma_{ilm}B_{m},\quad(\nabla_{B_{l}}B_{m})^{\mathcal{V}}=\sum_{i=1}^{3}\sigma_{ilm}E_{i},$

where $()^{\mathcal{V}}$ indicates the vertical projection with respect to
(2.1), $\epsilon_{ijk}$ is the sign of the permutation $(i,j,k)$ of $(1,2,3)$,
and $\sigma_{ilm}$ is skew-symmetric in $l,m$ satisfying $\sigma_{i0j}=1$ and
$\sigma_{ijk}=\epsilon_{ijk}$ for $i,j,k\in\\{1,2,3\\}$. In particular, we
notice that the fibres of the Riemannian submersion (2.1) are totally
geodesic, so the O’Neill tensor often denoted $T$ vanishes, and that the other
O’Neill tensor, often called $A$, is horizontally divergence-free.

We then let $\tilde{\nabla}$ denote the Levi-Civita connection of $g=g(t)$ and
let

$\displaystyle\tilde{E}_{1}=\frac{E_{1}}{a},\quad\tilde{E}_{2}=\frac{E_{2}}{a},\quad\tilde{E}_{3}=\frac{E_{3}}{b},\quad\tilde{B}_{l}=\frac{B_{l}}{c}.$

We may then compute that the quantities we need to complete our computation
are:

$\displaystyle[\tilde{E}_{2},\tilde{E}_{3}]=\frac{2}{b}\tilde{E}_{1},\quad[\tilde{E}_{3},\tilde{E}_{1}]=\frac{2}{b}\tilde{E}_{2},\quad[\tilde{E}_{1},\tilde{E}_{2}]=\frac{2b}{a^{2}}\tilde{E}_{3},$
$\displaystyle\tilde{\nabla}_{\tilde{E}_{2}}\tilde{E}_{3}=\frac{b}{a^{2}}\tilde{E}_{1},\quad\tilde{\nabla}_{\tilde{E}_{3}}\tilde{E}_{1}=\frac{2a^{2}-b^{2}}{a^{2}b}\tilde{E}_{2},\quad\tilde{\nabla}_{\tilde{E}_{1}}\tilde{E}_{2}=\frac{b}{a^{2}}\tilde{E}_{3},\quad\tilde{\nabla}_{\tilde{E}_{i}}\tilde{B}_{l}=0,$
$\displaystyle\tilde{\nabla}_{\tilde{B}_{l}}\tilde{E}_{1}=-\frac{a}{c^{2}}\sum_{m=0}^{4}\sigma_{1lm}\tilde{B}_{m},\quad\tilde{\nabla}_{\tilde{B}_{l}}\tilde{E}_{2}=-\frac{a}{c^{2}}\sum_{m=0}^{4}\sigma_{2lm}\tilde{B}_{m},\quad\tilde{\nabla}_{\tilde{B}_{l}}\tilde{E}_{3}=-\frac{b}{c^{2}}\sum_{m=0}^{4}\sigma_{3lm}\tilde{B}_{m}.$

Using the fact that the metric on $N$ is Einstein with scalar curvature $48$
and the O’Neill formulas, we see that the Ricci tensor of $g$ is diagonal and
satisfies:

$\displaystyle\mathrm{Ric}(\tilde{E}_{1},\tilde{E}_{1})=\mathrm{Ric}(\tilde{E}_{2},\tilde{E}_{2})$
$\displaystyle=\frac{4}{a^{2}}-\frac{2b^{2}}{a^{4}}+\frac{4a^{2}}{c^{4}},$
$\displaystyle\mathrm{Ric}(\tilde{E}_{3},\tilde{E}_{3})$
$\displaystyle=\frac{2b^{2}}{a^{4}}+\frac{4b^{2}}{c^{4}},$
$\displaystyle\mathrm{Ric}(\tilde{B}_{l},\tilde{B}_{l})$
$\displaystyle=\frac{12c^{2}-2(2a^{2}+b^{2})}{c^{4}}$

as desired. ∎

Given Lemma 5.1, we can now write down the Ricci flow equations for our ansatz
in (2.15):

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(a^{2})$
$\displaystyle=-4\left(2-\frac{b^{2}}{a^{2}}+2\frac{a^{4}}{c^{4}}\right);$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(b^{2})$
$\displaystyle=-4\left(\frac{b^{4}}{a^{4}}+2\frac{b^{4}}{c^{4}}\right);$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(c^{2})$
$\displaystyle=-4\left(6-2\frac{a^{2}}{c^{2}}-\frac{b^{2}}{c^{2}}\right).$

To simplify the analysis of these equations we introduce some scale-invariant
quantities and rescale the time parameter to find the following after a short
computation.

######  Lemma 5.2.

Define

$A=\frac{a^{2}}{c^{2}}\quad\text{and}\quad B=\frac{b^{2}}{c^{2}}$

and introduce a new variable $s$ by

$\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{1}{c^{2}}.$

If we let $\dot{A}=\frac{\mathrm{d}A}{\mathrm{d}s}$ and
$\dot{B}=\frac{\mathrm{d}B}{\mathrm{d}s}$, then the Ricci flow equations for
(2.15) imply that

$\displaystyle\dot{A}$
$\displaystyle=\frac{4(1-A)}{A}\left(B(1+A)-2A(1-2A)\right);$ (5.2)
$\displaystyle\dot{B}$
$\displaystyle=\frac{4B}{A^{2}}\left(2A^{2}(3-A)-B(1+3A^{2})\right).$ (5.3)

The equations (5.2)–(5.3) describe the rescaled Ricci flow.

###  5.2. Critical points and dynamics

It is straightforward to find the critical points and observe some basic facts
about the dynamics of the flow equations (5.2)–(5.3) as follows.

######  Lemma 5.3.

The only critical points for $A,B>0$ of the flow equations (5.2)–(5.3) are

$A=B=\frac{1}{5}\quad\text{and}\quad A=B=1.$

Moreover, the lines $A=B$ and $A=1$ are preserved by the flow.

Lemma 5.2 leads us to draw the plot in Figure 5.1 showing the dynamics of the
equations. If we denote the curve where $B(1+A)=2A(1-2A)$ for $A,B\geq 0$ by
$\gamma_{A}$ and the curve $2A^{2}(3-A)=B(1+3A^{2})$ for $A,B\geq 0$ by
$\gamma_{B}$, these are the curves which, together with the line $A=1$,
determine the sign of $\dot{A}$ and $\dot{B}$.

Figure 5.1. Dynamic plot for Ricci flow

The first plot in Figure 5.1 indicates that there is a stable critical point
at $A=B=1$, where $\gamma_{A}$ and $A=1$ intersect. In the second plot, we
focus on the critical point $A=B=1/5$ where $\gamma_{A}$ and $\gamma_{B}$
intersect, which appears to be unstable (in fact, a saddle point).

######  Remark 5.4.

The curves $\gamma_{A}$ and $\gamma_{B}$ also intersect at $A=B=0$, which is a
degenerate critical point in terms of the equations (5.2)–(5.3), since they
are not defined there. From the geometric point of view, here $M^{7}$ has
collapsed to the 4-dimensional orbifold $N$ (or a point) in the fibration
(2.1), since $a=b=0$ in (2.15), and so any flow lines tending to $(0,0)$
converge to the Einstein metric $g_{N}$ on $N$ up to scale (or $0$). We can
avoid the possibility of converging to a point along the rescaled Ricci flow,
since the diameter will stay bounded away from $0$.

######  Remark 5.5.

One may observe from (5.2)–(5.3) that there are two other degenerate critical
points if we allow $A=0$ or $B=0$:

$(A,B)=\left(\frac{1}{2},0\right)\quad\text{and}\quad(A,B)=(1,0).$

These critical points correspond to a collapsed situation where the $S^{1}$
fibres corresponding to $E_{3}$ (in the notation of Definition 2.1) now have
zero size since $b=0$ in (2.13). The 7-manifold $M$ has therefore collapsed to
the _twistor space_ $Z^{6}$, which is a 2-sphere bundle over $N^{4}$. The
metrics with $(A,B)=(1,0)$ and $(A,B)=(1/2,0)$ correspond to two Einstein
metrics on the twistor space $Z$: the canonical _Kähler–Einstein_ and _nearly
Kähler_ metric, respectively. It is natural to see these Einstein metrics
appear at the boundary of our Ricci flow ansatz as critical points. However,
it is interesting to note that the Kähler–Einstein metric on $Z$ does not play
a distinguished role in the study of the Laplacian coflow and Laplacian flow,
whereas the nearly Kähler metric does.

We now study the dynamics of the flow and show the following, recalling the
fibration (2.1) of $M^{7}$ over a 4-dimensional base $N^{4}$.

######  Theorem 5.6.

The only critical points for the Ricci flow (5.1), after rescaling, are the
3-Sasakian metric $g_{M}$ and the squashed Einstein metric $\tilde{g}_{M}$ on
$M^{7}$. The 3-Sasakian metric is a stable critical point for the rescaled
Ricci flow within the ansatz (2.15), whilst the squashed Einstein metric is an
unstable critical point which is a saddle point. Moreover, there is an open
set of initial metrics in the ansatz (2.15), which can be chosen arbitrarily
near $\tilde{g}_{M}$, such that they flow either to $g_{M}$ or to the
collapsed limit (even after rescaling) where the 3-dimensional fibres in (2.1)
shrink to $0$ and the flow converges to the Einstein 4-orbifold $(N,g_{N})$.

###### Proof.

We recall the curves $\gamma_{A}$, $\gamma_{B}$ introduced after Lemma 5.3 and
plotted in Figure 5.1.

For $A>1$ we have that $\dot{A}<0$ and hence $A$ remains bounded as long as
the flow exists. We also see that $\dot{B}<0$ whenever $B$ is above the curve
$\gamma_{B}$ (which passes through $(0,0)$ and $(3,0)$) and hence also remains
bounded as long as the flow exists. Since

$\frac{\mathrm{d}}{\mathrm{d}t}(c^{2})=-4(6-2A-B),$

the right-hand side is bounded and so $c^{2}$ cannot blow up in finite time.
Moreover, $c^{2}$ goes to zero at a linear rate in $t$ at most, and so we can
integrate $\frac{1}{c^{2}}$ with respect to $t$ to obtain $s$ in Lemma 5.2.

We linearize (5.2)–(5.3) around $(A,B)=(1,1)$ to obtain:

$\dot{A}=-16A\quad\text{and}\quad\dot{B}=-16B.$

Hence $(1,1)$ is a stable critical point, which corresponds to the 3-Sasakian
metric.

We now observe that if we start above the curve $\gamma_{A}$, which passes
through $(0,0)$ and $(\frac{1}{2},0)$, then $\dot{A}>0$ for $A<1$, so $A$ is
always bounded away from $0$ in this region, depending on its initial value.
Recalling that $\dot{A}<0$ for $A>1$, we deduce that all of the terms in
(5.2)–(5.3) are bounded and there can be no periodic orbits in this region as
$A$ cannot cross the line $A=1$ by Lemma 5.3. We also note that $\dot{B}>0$
when $A\in(1/5,1)$ and $B$ is near $0$ since then $(A,B)$ lies below
$\gamma_{B}$, and thus $B$ is bounded away from $0$ when $A\in(1/5,1)$.

Since $(1/5,1/5)$ lies on $\gamma_{A}$, we deduce that the flow will converge
to the critical point at $(1,1)$ if the initial value of $(A,B)$ lies above or
to the right of the curve $\gamma_{A}$ and $A$ is greater than $1/5$
initially. This proves the statement about initial conditions near
$\tilde{g}_{M}$ which flow to $g_{M}$, since $\tilde{g}_{M}$ corresponds to
$(1/5,1/5)$.

Now if we specialize to the line $A=B$ which, by Lemma 5.2, is preserved, we
see that

$\dot{A}=4(1-A)(5A-1).$ (5.4)

We see immediately that for $A=B<1/5$ we have $\dot{A}<0$, for $A=B\in(1/5,1)$
we have $\dot{A}>0$. (For $A=B>1$ we have $\dot{A}<0$ again, as expected by
the stability of $(1,1)$.) Thus, the critical point at $(1/5,1/5)$ is
unstable, even within the restricted ansatz when $A=B$. Furthermore, if we
linearize the system (5.2)–(5.3) around $(1/5,1/5)$ we obtain

$\dot{A}=-\frac{16}{5}A+\frac{96}{5}B\quad\text{and}\quad\dot{B}=\frac{192}{5}A-\frac{112}{5}B,$

from which it follows that $(1/5,1/5)$ is a saddle point, as the matrix
corresponding to the above dynamical system has one positive and one negative
eigenvalue: $16$ and $-208/5$.

We see from (5.4) that we have initial conditions for (5.2)–(5.3), even with
$A=B$, such that $A$ and $B$ go to $0$. In fact, suppose we choose any initial
condition in the region $\mathcal{R}$ below $\gamma_{A}$ but above
$\gamma_{B}$, which means in particular that $A,B\in(0,\frac{1}{5})$. Since
the flow lines enter $\mathcal{R}$ vertically from above along $\gamma_{A}$
and horizontally from the right along $\gamma_{B}$, no flow lines can leave
$\mathcal{R}$ so $A,B$ are bounded and can only reach $0$ when $(A,B)=(0,0)$.
In $\mathcal{R}$, $\dot{A}<0$ and $\dot{B}<0$ and there are no periodic
orbits. We conclude that all flow lines starting in $\mathcal{R}$ must
converge to $(0,0)$. We again note as in Remark 5.4 that $M$ cannot collapse
to a point along the rescaled Ricci flow. The discussion in Remark 5.4 then
implies that the point $(0,0)$ corresponds to the fibres of the fibration
(2.1) collapsing, even in the rescaled Ricci flow, so that $M^{7}$ collapses
to the base $N^{4}$ (with its Einstein metric). Since $\mathcal{R}$ is open
and has $(1/5,1/5)$ on its boundary, this complete the proof. ∎

Theorem 5.6 proves Theorem 1.3 in the Introduction.

######  Remark 5.7.

By dynamical systems theory, there is a 1-dimensional stable manifold for the
squashed Einstein metric within our rescaled Ricci flow ansatz. We can discern
this stable manifold from the plots in Figure 5.1. It might be interesting to
understand whether this stable manifold (or, equally, the corresponding
unstable manifold) has any geometric significance, e.g. any special curvature
properties.

##### Acknowledgements.

AK would like to thank Rafe Mazzeo and his advisor Dave Morrison for useful
suggestions related to this paper. JDL would like thank the Simons Laufer
Mathematical Sciences Institute (previously known as MSRI) in Berkeley,
California, for hospitality during the latter stages of this project. The
authors also thank the Simons Collaboration on Special Holonomy in Geometry,
Analysis, and Physics for support and many interesting talks which inspired
this research.

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