GEOMETRIC FLOWS OF G 2-STRUCTURES ON 3-SASAKIAN 7-MANIFOLDS AARON KENNON AND JASON D. LOTAY

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GEOMETRIC FLOWS OF G2-STRUCTURES ON 3-SASAKIAN

7-MANIFOLDS

AARON KENNON AND JASON D. LOTAY

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 G2-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 G2-structures provide natural critical points of the (rescaled)

geometric ﬂows of G2-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 G2-structures. We also

compare the behaviour of the ﬂows of G2-structures with the (rescaled) Ricci ﬂow.

Contents

1. Introduction 1

2. G2-structures on 3-Sasakian 7-manifolds 4

3. Laplacian coﬂow 8

4. Laplacian ﬂow 15

5. Ricci ﬂow 17

References 22

1. Introduction

1.1. Nearly parallel G2-structures. A G2-structure on a 7-manifold is encoded by a

3-form ϕsatisfying a certain nondegeneracy condition, and such a 3-form determines a Rie-

mannian metric and orientation. One of the most important types of G2-structure is a nearly

parallel G2-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

G2-structure admits a conical metric with exceptional holonomy Spin(7) (and so is Ricci-

ﬂat), and thus nearly parallel G2-structures are also important in the study of asymptotically

conical and conically-singular 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 G2-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

Date: February 24, 2023.

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).

arXiv:2210.12962v2 [math.DG] 22 Feb 2023

2 AARON KENNON AND JASON D. LOTAY

[7]. The largest class of 7-manifolds that are known to admit nearly parallel G2-structures

are the 3-Sasakian 7-manifolds, which are the focus of this paper.

1.2. Geometric ﬂows. Nearly parallel G2-structures are natural to study from the per-

spective of several geometric ﬂows. Since a nearly parallel G2-structure induces a positive

Einstein metric, it is natural to evolve its induced metric gby the Ricci ﬂow:

∂g

∂t =−2Ric(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 G2-structure contains more information than

the metric (since the same metric is induced by a whole family of G2-structures), so it is

worthwhile to examine ﬂows of G2-structures relevant to nearly parallel G2-structures, and

compare and contrast its behaviour to the Ricci ﬂow.

Two such geometric ﬂows of G2-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]1).

1.2.1. Laplacian ﬂow. The Laplacian ﬂow evolves the 3-form ϕdeﬁning the G2-structure by

its Hodge Laplacian:

∂ϕ

∂t = ∆ϕϕ= (dd∗

ϕϕ+ d∗

ϕd)ϕ. (1.2)

(Here, we emphasise the nonlinearity in the formal adjoint d∗

ϕof the exterior derivative,

since the metric and orientation depend on ϕ.) The Laplacian ﬂow has received particular

attention in the context of closed G2-structures (when dϕ= 0), where it has many attractive

features, particularly with regards to torsion-free G2-structures (when dϕ= 0 and d∗

ϕϕ= 0),

which deﬁne Ricci-ﬂat metrics with holonomy contained in G2. For foundational results and

a survey of recent developments in the Laplacian ﬂow for closed G2-structures see e.g. [11,

17, 18].

A nearly parallel G2-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

G2-structure is, however, not closed but coclosed: the deﬁning 3-form ϕsatisﬁes d∗

ϕϕ= 0.

Whilst it may seem potentially plausible to study coclosed G2-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 G2-structure. An example situation where it has proved instructive to

use the Laplacian ﬂow to study coclosed G2-structures can be found in [16].

1.2.2. Laplacian coﬂow. Currently the best candidate2for studying coclosed G2-structures is

the Laplacian coﬂow, which evolves the closed 4-form ψ=∗ϕϕdual to the 3-form ϕdeﬁning

the G2-structure by its Hodge Laplacian:

∂ψ

∂t = ∆ψψ= (d∗

ψd + dd∗

ψ)ψ= dd∗

ψψ, (1.3)

1It should be noted that in [12] the opposite sign for the velocity of the Laplacian coﬂow is used.

2The Laplacian coﬂow for coclosed G2-structures has many attractive features analogous to the Lapla-

cian ﬂow for closed G2-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.

GEOMETRIC FLOWS OF G2-STRUCTURES ON 3-SASAKIAN 7-MANIFOLDS 3

using the fact that ψis closed. (The 4-form ψinduces the metric just like ϕ, but not the

orientation, though an orientation can be ﬁxed by the initial choice of G2-structure.) The

Laplacian coﬂow preserves the cohomology class [ψ] of ψ, where it may be viewed as the

gradient ﬂow of the Hitchin volume functional, and the induced ﬂow of the metric gdeﬁned

by ψis

∂g

∂t =−2Ric(g) + Q(dϕ),(1.4)

where Qis a quadratic expression in dϕ: see [9, 11] for details. Since Qonly depends on

ﬁrst order information on ψ, 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 G2-structure deﬁnes a self-

similarly expanding solution to the Laplacian coﬂow (1.3), whereas its induced metric de-

ﬁnes 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 G2-structures satisfy a parametric h-principle (see [6]).

Therefore, coclosed G2-structures exist on any (compact or non-compact) 7-manifold admit-

ting a G2-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 G2-structure on a compact

manifold.

1.3. 3-Sasakian 7-manifolds. A 3-Sasakian 7-manifold is a Riemannian 7-manifold Mso

that the metric cone over it is hyperk¨ahler. One can use the 3-Sasakian structure to deﬁne

two3distinct nearly parallel G2-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 G2-structures on 7-manifolds M, it was shown in [19] that

if the third Betti number b3(M)6= 0, then under the Ricci ﬂow any Einstein metric induced

by a nearly parallel G2structure is linearly unstable and therefore dynamically unstable.

As 3-Sasakian 7-manifolds Mnecessarily have b3(M) = 0, this class of examples admitting

nearly parallel G2-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.

3In fact, there are three natural nearly parallel G2-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.

4 AARON KENNON AND JASON D. LOTAY

1.5. Main results. On any 3-Sasakian 7-manifold we introduce two disjoint 3-parameter

families of coclosed G2-structures deﬁned in terms of the 3-Sasakian structure. These families

of G2-structures each include exactly one of the natural nearly parallel G2-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 G2-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 G2-structure in either

of our families converges, after rescaling, to the nearly parallel G2-structure in that family.

In particular, the nearly parallel G2-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 G2-structures are unstable sources within their families

under the rescaled Laplacian ﬂow, so coclosed G2-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. G2-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 G2geometry. Further details on 3-Sasakian geometry can be found

in [3, 4]. For information about G2-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 (M7, gM) is 3-Sasakian if it has an

orthonormal triple of Killing ﬁelds {E1, E2, E3}satisfying [Ei, Ej]=2Ekfor a cyclic permu-

tation (i, j, k) of (1,2,3), such that each Eideﬁnes a Sasakian structure on (M, gM).

If (M, gM) is 3-Sasakian then gMis Einstein with positive scalar curvature equal to 42

(so Mis compact) and there is a locally free action of SU(2) on Mwhose leaf space Nis

GEOMETRIC FLOWS OF G2-STRUCTURES ON 3-SASAKIAN 7-MANIFOLDS 5

a 4-dimensional orbifold. Moreover, there is a canonical metric gNon N, which is anti-self-

dual Einstein with positive scalar curvature equal to 48, such that (M, gM) and (N, gN) are

related by an orbifold Riemannian submersion:

π:M→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=S7and N=S4, and N=S4has its constant curvature 4 metric.

The Levi-Civita connection of (N, gN) lifts to a connection on the bundle (2.1), and so

may be viewed as an su(2)-valued 1-form ηon M, which can be written as

η=

i=1

ηi⊗Ti,(2.2)

where η1, η2, η3are 1-forms on Mand {T1, T2, T3}is a basis for su(2) satisfying [Ti, Tj]=2Tk

for cyclic permutations (i, j, k) of (1,2,3). The curvature of ηis then an su(2)-valued 2-form

ωwhich may be written as

ω=−2

i=1

ωi⊗Ti(2.3)

for 2-forms ω1, ω2, ω3on Mwhich are, in fact, pullbacks of orthogonal self-dual 2-forms on

Nsince gNis anti-self-dual Einstein. (The factor of 2 and sign are chosen for convenience.)

Moreover, we have that the forms ω1, ω2, ω3are normalized such that

ωi∧ωj= 2δij π∗volN.(2.4)

For later use, we record the following equations satisﬁed by ηand ωwhere, in each case,

(i, j, k) are taken to be a cyclic permutation of (1,2,3):

dηi=−2ηj∧ηk−2ωi,(2.5)

dωi=−2ηj∧ωk+ 2ηk∧ωj.(2.6)

The 3-Sasakian metric gMon Mmay then be given in terms of the ηiand gNas follows:

gM=η2

1+η2

2+η2

3+π∗gN,(2.7)

We can scale gMby any positive constant cand then c2gMwill still be Einstein with

positive scalar curvature. We may also observe the following well-known fact.

Lemma 2.3. The metric

˜gM=1

5(η2

1+η2

2+η2

3) + π∗gN(2.8)

is Einstein with positive scalar curvature and is known as the squashed Einstein metric on

the 3-Sasakian M7[7, 8].

Remark 2.4. The metric cone on (M, gM) has holonomy contained in Sp(2), whereas the

metric cone on (M, ˜gM) (once one scales ˜gMappropriately) has holonomy Spin(7). In the

ﬁrst case, the metric cone has the full holonomy Sp(2) if it is not ﬂat.

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GEOMETRICFLOWSOFG2-STRUCTURESON3-SASAKIAN7-MANIFOLDSAARONKENNONANDJASOND.LOTAYAbstract.A3-Sasakianstructureona7-manifoldmaybeusedtodenetwodistinctEinsteinmetrics:the3-SasakianmetricandthesquashedEinsteinmetric.BothmetricsareinducedbynearlyparallelG2-structureswhichmayalsobeexpressedintermsofthe3-Sa...

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GEOMETRIC FLOWS OF G 2-STRUCTURES ON 3-SASAKIAN 7-MANIFOLDS AARON KENNON AND JASON D. LOTAY

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