Local curvature estimates for the Laplacian flow

YI LI

ABSTRACT. In this paper we give local curvature estimates for the Laplacian flow on closedG2-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar-Munteanu- Wang [24] who gave local curvature estimates for the Ricci flow on complete man- ifolds and then provided a new elementary proof of Sesum’s result [36], and the particular structure of the Laplacian flow on closedG2-structures. As an immedi- ate consequence, this estimates give a new proof of Lotay-Wei’s [33] result which is an analogue of Sesum’s theorem.

The second result is about an interesting evolution equation for the scalar cur- vature of the Laplacian flow of closedG₂-structures. Roughly speaking, we can prove that the time derivative of the scalar curvatureRtis equal to the Laplacian ofRt, plus an extra term which can be written as the difference of two nonnegative quantities.

1. INTRODUCTION

Let

M

be a smooth 7-manifold. The Laplacian flow for closedG₂-structures on

M

introduced by Bryant [1] is to study the torsion-freeG₂-structures

(1.1) ∂_tϕ_t

=

∆ϕtϕ_t, ϕ₀

=

ϕ,

where∆ϕtϕt

=

dd^∗_ϕtϕt

+

d^∗_ϕtdϕtis the Hodge Laplacian ofgϕt and ϕis an initial closedG2-structure. Sinced∂tϕt

=

∂td∆ϕtϕt

=

0, we see that the flow (1.1) pre- serves the closedness of ϕt. For more background on G2-structures, see Section 2. When

M

is compact, the flow (1.1) can be viewed as the gradient flow for the Hitchin functional introduced by Hitchin [17]

(1.2) H :

[

ϕ

]

+

−→

R⁺, ϕ

7−→

¹ 7 Z

Mϕ

∧

ψ

=

Z

M

∗

_ϕ1.

Hereϕis a closedG2-structure on

M

and

[

ϕ

]

₊is the open subset of the cohomol- ogy class

[

ϕ

]

consisting ofG2-structures. Any critical point ofH gives a torsion- freeG2-structure.

The study of Laplacian flows on some special 7-manifolds, Laplacian solitons, and other flows onG2-structures can be found in [12, 13, 14, 15, 18, 23, 28, 34, 35, 37, 38].

Recently, Donaldson [6, 7, 8, 9] studied the co-associative Kovalev-Lefschetz fibrationsG₂-manifolds andG₂-manifolds with boundary.

2010Mathematics Subject Classification. Primary 53C44, 53C10.

Key words and phrases.Laplacian flow,G2-structures, local curvature estimates.

1

1.1. Notions and conventions. To state the main results, we fix our notions used throughout this paper. Let

M

be as before a smooth 7-manifold. The space of smooth functions and the space of smooth vector fields are denoted respectively byC^∞

(M)

andX

(M)

. The space ofk-tenors (i.e.,

(

0,k

)

-covariant tensor fields) and k-forms on

M

are denoted, respectively, by

⊗

^k

(M) =

C^∞

(⊗

^k

(

T^∗

M))

and

∧

^k

(M) =

C^∞

(∧

^k

(

T^∗

M))

. For anyk-tensor fieldT

∈ ⊗

^k

(M)

, we locally have the expressionT

=

T_i1···i_kdxⁱ¹

⊗ · · · ⊗

dx^ik

=

: T_i1···i_kdx^{i1⊗···⊗ik}. Ak-formαon

M

can be written in thestandard formasα

=

_k!¹α_i1···i_kdxⁱ¹

∧ · · · ∧

dx^ik

=

: _k!¹α_i1···i_kdx^{i1∧···∧ik}, where α_i1···i_k is fully skew-symmetric in its indices. Using the standard forms, if we take the interior product X

y

^α of ak-form α

∈ ∧

^k

(M)

with a vector field X

∈

X

(M)

, we obtain the

(

k

−

1

)

-formX

y

α

=

_(k−1)!¹ X^mα_mi1···i_k−1dx^{i1∧···∧ik−1}which is also in the standard form. In particular, consider the vector space

⊗

²

(M)

of 2- tensors. For any 2-tensorA

=

Aijdx^i⊗j, defineA :

=

¹₂

(

Aij

+

Aji

)

dx^i⊗j

≡

A_ijdx^i⊗j and A^∧ :

=

¹₂

(

A_ij

−

A_ji

)

dx^i⊗j

≡

A^∧_ijdx^i⊗j. Then A is an element of

²

(M)

, the space of symmetric 2-tensors. Since¹dx^i∧j

=

dx^i⊗j

−

dx^j⊗i, it follows that A^∧

=

¹₂A_ijdx^i∧j. Define α^A :

=

₂¹α_ij^Adx^i∧j with α_ij^A :

=

A_ij. Then we see that α^A

=

A^∧

∈ ∧

²

(M)

and

⊗

²

(M) =

²

(M) ⊕ ∧

²

(M)

.

A given Riemannian metric g on

M

determines two isomorphisms between vector fields and 1-forms:

[

_g : X

(M) −→ ∧

¹

(M)

and

]

_g :

∧

¹

(M) −→

X

(M)

, where, for every vector fieldX

=

Xⁱ_∂x^∂_i and 1-formα

=

α_idxⁱ,

[

_g

(

X

) =

Xⁱg_ijdx^j

≡

Xjdx^j and

]

_g

(

α

) =

α_ig^ij_∂x^∂_j

≡

α^{j ∂}

∂x^j. Using these two natural maps, we can fre- quently raise or lower indices on tensors. The metricgalso induces a metric onk- formsg

(

dx^{i1∧···∧ik},dx^{j1∧···∧jk}

) =

_det

(

g

(

dx^ia,dx^jb

)) =

_∑σ∈S

7sgn

(

σ

)

g^i1jσ(1)

· · ·

g^ikjσ(k) whereS₇is the group of permutations of seven letters and sgn

(

σ

)

denotes the sign

(±

1

)

of an elementσofS₇. The inner product

h·

,

·i

_gof twok-formsα,β

∈ ∧

^k

(M)

now is given by

h

α,β

i

g

=

_k!¹α_i1···i_kβ^i1···ik

=

_k!¹α_i1···i_kβ_j1···j_kg^i1j1

· · ·

g^ikjk.

Given two 2-tensors A,B

∈ ⊗

²

(M)

, with the forms A

=

Aijdx^i⊗j and B

=

B_ijdx^i⊗j. Define

hh

A,B

ii

_g :

=

A_ijB^ij. There are two special cases which will be used later:

(1) α

=

¹₂α_ijdx^i∧j

∈ ∧

²

(M)

andB

=

B_ijdx^i⊗j

∈ ⊗

²

(M)

. In this case,αcan be written as a 2-tensor A^α

=

A^α_ijdx^i⊗jwith A_ij^α

=

α_ij. Then

hh

α,B

ii

_g :

= hh

A^α,B

ii

g

=

α_ijB^ij.

(2) α

=

¹₂α_ijdx^i∧j and β

=

¹₂β_ijdx^i∧j

∈ ∧

²

(M)

. In this case,α,βcan be both written as 2-tensors A^α

=

A^α_ijdx^i⊗j andB^β

=

B^β_ijdx^i⊗j with A^α_ij

=

α_ij and B^β_ij

=

β_ij. Then

hh

α,β

ii

_g:

= hh

A^α,B^β

ii

_g

=

α_ijβ^ij

=

2

h

α,β

i

_g.

1In our convention, for any 2-formα=¹₂αijdx^ij, we have α

∂

∂x^k, ∂

∂x^`

=¹ 2α_ij

dx^i⊗j−dx^j⊗i

∂

∂x^k, ∂

∂x^`

=¹ 2α_ij

δⁱ_kδ_{`</sub><sup>j</sup>−δ<sub>k</sub><sup>j</sup>δ<sup>i</sup><sub>`}

=¹

2(α_k`−α`k) =α_k` which justifies the notionα<sub>k</sub>`asα(∂/∂x^k,∂/∂x^`). In general, for anyk-formα= _k!¹α_i1···ikdx^{i1∧···∧ik}we haveα_i1···ik =α(∂/∂xⁱ¹,· · ·,∂/∂x^ik), becausedx^{i1∧···∧ik}=_∑σ∈S

ksgn(σ)dx^{iσ(1)⊗···⊗iσ(k)}.

The norm of A

∈ ⊗

²

(M)

is defined by

||

A

||

²_g :

= hh

A,A

ii

_g

=

A_ijA^ij, while the norm ofα

∈ ∧

^k

(M)

is

|

α

|

_g² :

= h

α,α

i

_g

=

_k!¹α_i1···i_kα^i1···ik. In particular,

||

X

||

²_g

=

X_iXⁱ

= |[

_g

(

X

)|

²_gand

||

α

||

²_g

=

2

|

α

|

²_g, for any vector fieldX

∈

X

(M)

and 2-formα.

The Levi-Civita connection associated to a given Riemannian metric g is de- noted by^g

∇

or simply

∇

. Our convention on Riemann curvature tensor isR^m_ijk∂x^∂m

:

=

Rm

(

_∂x^∂_i,_∂x^∂_j

)

^∂

∂x^k

= (∇

_i

∇

_j

− ∇

_j

∇

_i

)

^∂

∂x^k andR_ijk`:

=

R^m_ijkg_m`. The Ricci curvature ofgis given byRjk :

=

Rijk`g<sup>i`</sup>. We usedVgand

∗

_gto denote the volume form and Hodge star operator, respectively, on

M

associated to a metric gand an orienta- tion.

We use the standard notion A

∗

Bto denote some linear combination of con- tractions of the tensor productA

⊗

Brelative to the metricgtassociated theϕt. In Theorem 1.4 and its proof, all universal constantsc,Cbelow depend only on the given real numberp.

1.2. Main results. Applying De Turck’s trick and Hamilton’s Nash-Moser inverse function theorem, Bryant and Xu [2] proved the following local time existence for (1.1).

Theorem 1.1. (Bryant-Xu [2]) For a compact 7-manifold

M

, the initial value prob- lem (1.1) has a unique solution for a short time interval

[

0,Tmax

)

with the maximal time Tmax

∈ (

0,∞

]

depending onϕ.

As in the Ricci flow, we can prove following results on the long time existence for the Laplacian flow (1.1).

Theorem 1.2. (Lotay-Wei[33]) Let

M

be a compact7-manifold and ϕt, t

∈ [

0,T

)

, where T

<

∞, be a solution to the flow (1.1) for closed G2-structures with associated metric gt

=

g_ϕtfor each t.

(a) If the velocity of the flow satisfies sup

M×[0,T)

||

_∆tϕt

||

_t

<

_∞,

then the solutionϕtcan be extended past time T.

(b) If T

=

Tmax, then

t→Tlim_maxsup

M

||

Rmt

||

²_t

+ ||∇

_tTt

||

²_t

=

_∞.

HereTtis the torsion ofϕt(see (2.14)).

In this paper, we give a new elementary proof of Theorem 1.2, based on the idea of [24] and the structure of the equation (1.1).

Theorem 1.3. Let

M

be a compact7-manifold and ϕt, t

∈ [

0,T

)

, where T

<

_{∞, be a} solution to the flow (1.1) for closed G2-structures with associated metric gt

=

g_ϕtfor each t. Suppose that

K:

=

sup

M×[0,T)

||

Rict

||

_t

<

_{∞, Λ}:

=

sup

M

||

Rm0

||

₀.

Then

sup

M×[0,T)

||

Rmt

||

_t

<

∞, where the bound depends only on n,K,T andΛ.

When

M

is compact, the theorem immediately implies the part (a) in Theorem 1.2. Indeed, we shall show that (see (3.18) and (3.37))

sup

M×[0,T)

||

∆tϕ_t

||

_t

<

∞

⇐⇒

sup

M×[0,T)

||

Rict

||

_t

<

∞.

In the compact case, Theorem 1.3 shows that, if the conclusion in part (a) does not hold, thenT

=

Tmaxand sup_M×[0,Tmax)

||

Rmt

||

_t

<

_∞which implies sup_M×[0,Tmax)

(||

Rmt

||

²_t

+ ||∇

_tTt

||

²_t

) <

∞, since the norm

||∇

_tTt

||

²_t can be controlled by

||

Rmt

||

²_t (see (3.63)). However, by part (b) in Theorem 1.2, it is impossible. Therefore, the conclusion in part (a) is true.

As remarked in [24], to prove Theorem 1.3, it suffices to establish the following integral estimate.

Theorem 1.4. Let

M

be a smooth7-manifold and ϕt, t

∈ [

0,T

)

, where T

<

_{∞, be a} solution to the flow (1.1) for closed G2-structures with associated metric gt

=

g_ϕtfor each t. Assume that there exist constants A,K

>

0and a point x0

∈ M

such that the geodesic ball Bg₀

(

x0,A/

√

K

)

is compactly contained in

M

and that

|

Rict

|

_t

≤

K on Bg₀

x0, A

√

K

× [

0,T

]

. Then, for any p

≥

5, there exists c

=

c

(

p

) >

0so that

Z

B_g0(x₀,A/2√

K)

||

Rmt

||

_t^pdVt

≤

c

(

1

+

K

)

e^cKT Z

B_g0(x₀,A/√

K)

||

Rm0

||

₀^pdV0

+

cK^p

1

+

A^−2p

e^cKTvolt

Bg₀

x₀, A

√

K (1.3)

for all t

∈ [

0,T

]

.

Now by the standard De Giorgi-Nash-Moser iteration (our manifold is compact and the Ricci curvature is uniformly bounded), under the condition in Theorem 1.4, we can prove

(1.4)

||

RmT

||

_T

(

x₀

) ≤

d₁

(

d₂

+

_Λ0

)

_, whered₁,d₂are constants depending onK,T,A, and

Λ0:

=

sup

B_g0(x₀,A/√ K)

||

Rm0

||

₀.

Actually, this follows from the same argument in [24] by noting that (1.5)

(

∆t

−

∂_t

)||

Rmt

||

_t

≥ −

c

||

Rmt

||

²_t.

To verify (1.5), we use (2.26), (3.61) and (3.65) to deduce that

||∇

tTt

|| ≤

c

||

Rmt

||

t

and

||∇

²_tTt

||

_t

≤

c

||∇

_tRmt

||

_t

+

c

||

Rmt

||

^3/2_t .

Then, by (3.31) and the Cauchy inequality

||∇

_tRmt

||

²_t

≤ −

¹

2

(

∂t

−

_∆t

)||

Rmt

||

²_t

+

c

||

Rmt

||

³_t

+

c

||

Rmt

||

^3/2_t

||∇

_tRmt

||

_t

≤ −

¹

2

(

∂_t

−

∆t

)||

Rmt

||

²_t

+

c

||

Rmt

||

³_t

+ ||∇

_tRmt

||

²_t which implies (1.5). Now the estimate (1.4) yields Theorem 1.3.

The analogue of Theorem 1.2 in the Ricci flow was proved by Hamilton [16] (for part (b)) and Sesum [36] (for part (a)). It is an open question (due to Hamilton, see [3]) that the Ricci flow will exist as long as the scalar curvature remains bounded.

For the K¨ahler-Ricci flow [39] or type-I Ricci flow [10], this question was settled.

For the general case, some partial result on Hamilton’s conjecture was carried out in [3].

For the Ricci-harmonic flow introduce by List [29, 30] (see also, [31, 32]), the analogue of Theorem 1.2 was proved in [29, 30] (see also, [31, 32]) and [4] (see [27] for another proof). The author [25, 26] extended Cao’s result [3] to the Ricci- harmonic flow. The same Hamilton’s conjecture was asked by the author in [25, 26].

We can ask the same question for the Laplacian flow on closedG2-structures. In [33] (see Page 171, line -6 to -3, or Open Problem (3) in Page 230), Lotay and Wei asked that whether the Laplacian flow on closedG2-structures will exist as long as the torsion tensor or scalar curvature remains bounded. Let gt be the associated metric ofϕ_t. Then the evolution equation forgtis given by

(1.6) ∂tgij

= −

2Rij

−

⁴

3

|

Tt

|

²_tgij

−

4T_i^kT_kj.

For the Laplacian flow on closedG2-structures, the torsionTtis actually a 2-form for eacht, hence we use the norm

| · |

_tin (1.6). The standard formula for the scalar curvatureRtgives (see (3.23))

(1.7) ∂tRt

=

_∆tRt

+

2

||

Rict

||

²_t

−

²

3R²_t

+

4R<sub>ijk`</sub>T<sup>ik</sup>T<sup>j`</sup>

+

4

(∇

^jT^ik

)(∇

_iT_jk

)

. Now the above mentioned open problem states that

Is it ture that lim

t→T_maxRt

= −

_∞?

The “minus infinity” comes from the fact that along the Laplacian flow on closed G₂-structures the scalar curvature is always nonpositive (see (2.26)). The following Proposition 1.5 is motivate to solve this problem, and starts from the basic evolu- tion equation (1.7) where the last two terms on the right-hand side do not have good signature. However, using the closedness of ϕt (in particular, the identity (3.23)), we can prove the following interesting evolution equation forRt.

Proposition 1.5. Let

M

be a smooth7-manifold andϕt, t

∈ [

0,T

)

, where T

∈ (

0,∞

]

, be a solution to the flow (1.1) for closed G2-structures with associated metric gt

=

g_ϕtfor

each t. Then the scalar curvature Rtsatisfies

∂tRt

=

_∆tRt

+

2

R_ij

+

² 3

|

Tt

|

²_tg_ij

2 t

+

¹ 2

R_ijabR^ijmn

−

ψ_abmn

2 t

+

¹ 2

2TiaTjbR^ijmn

−

ψ_abmn

2 t

+

¹

2

2TbamTbbn

−

ψ_abmn

2 t

+

₂

||

Tbt

||

²_t

+

₄

||∇

tTt

||

²_t

−

||

Rmt

||

²_t

+

²⁶ 9 R²_t

+

¹

2

R_ijabR^ijmn

2

(1.8) t

+

2

TiaT_jbR^ijmn

2

t

+

2

||

bTt

||

⁴_t

+

210

. HereTb_ij

=

T_i^kT_kj.

Observe that the above well-arranged evolution equation can give us a weakly lower bound forRt, which can not prove or disprove the conjecture of Lotay and Wei.

We give an outline of the current paper. We review the basic theory in Sec- tion 2 aboutG2-structures, G2-decompositions of 2-forms and 3-forms, and gen- eral flows onG2-structures. In Section 3, we rewrite results in Section 2 for closed G2-structures, and the local curvature estimates will be given in the last subsection.

1.3. Acknowledgments. The author is supported in part by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746).

The main result was carried out during the Young Geometric Analysts Forum 2018, 29th January – 2th February, in Tsinghua Sanya International Mathematics Forum.

The author, together with other six friends, thanks Yunhui Wu who personally provided us 14, the dimension ofG2, very fresh coconuts during the forum.

The author thanks Joel Fine, Brett Kotschwar, Chengjian Yao, Yong Wei, and Anton Thalmaier for useful discussion on the Laplacian flows and the earlier ver- sion of this paper. He also thanks Jason Lotay for his interested in this paper.

2. BASIC THEORY OFG₂-STRUCTURES

In this section, we view some basic theory ofG₂-structures, following [1, 19, 20, 21, 22, 33]. Let

{

e₁,

· · ·

,e₇

}

denote the standard basis ofR⁷and let

{

e¹,

· · ·

,e⁷

}

be its dual basis. Define the 3-form

φ:

=

e^1∧2∧3

+

e^1∧4∧5

+

e^1∧6∧7

+

e^2∧4∧6

−

e^2∧5∧7

−

e^3∧4∧7

−

e^3∧5∧6,

wheree^i∧j∧k :

=

eⁱ

∧

e^j

∧

e^k. The subgroup G2, which fixesφ, of GL

(

7,R

)

is the 14-dimensional Lie subgroup ofSO

(

7

)

, acts irreducibly onR⁷, and preserves the metric and orientation for which

{

e1,

· · ·

,e7

}

is an oriented orthonormal basis.

Note thatG2also preserves the 4-form

∗

_φφ

=

e^{4∧5∧6∧7}

+

e^{2∧3∧6∧7}

+

e^{2∧3∧4∧5}

+

e^{1∧3∧5∧7}

−

e^{1∧3∧4∧6}

−

e^{1∧2∧5∧6}

−

e^{1∧2∧4∧7}. where the Hodge star operator

∗

_φis determined by the metric and orientation.

For a smooth 7-manifold

M

and a pointx

∈ M

, define as in [33]

∧

³₊

(

T_x^∗

M)

:

=

ϕx

∈ ∧

³

(

T_x^∗

M)

: u^∗φ

=

ϕxfor some invertible mapu

∈

Hom_R

(

Tx

M

,R⁷

)

and the bundle

∧

³₊

(

T^∗

M)

:

=

^G

x∈M

∧

³₊

(

T_x^∗

M)

.

We call a sectionϕof

∧

³₊

(

T^∗

M)

apositive3-formon

M

or aG2-structureon

M

, and denote the space of positive 3-forms by

∧

³₊

(M)

. The existence ofG2-structures is equivalent to the property that

M

is oriented and spin, which is equivalent to the vanishing of the first and second Stiefel-Whitney classes. From the definition of G₂-structures, we see that any ϕ

∈ ∧

³₊

(M)

uniquely determines a Riemann- ian metric g_ϕand an orientationdV_ϕ, hence the Hodge star operator

∗

_ϕand the associated 4-form

(2.1) ψ:

= ∗

_ϕϕ.

We also have the isomorphisms

[

_ϕ:

= [

_gϕ and

]

_ϕ:

= ]

_gϕ. For a givenG2-structure ϕ

∈ ∧

³₊

(M)

, we denote by

h·

,

·i

_ϕ,

hh·

,

·ii

,

| · |

_ϕ,

|| · ||

_ϕ, the corresponding inner products

h·

,

·i

_gϕ,

hh·

,

·ii

_gϕand norms

| · |

_gϕ,

|| · ||

_gϕ.

Given aG2-structure ϕ

∈ ∧

³₊

(M)

. We say that ϕistorsion-freeif ϕis parallel with respect to the metricgϕ. Equivalently,ϕis torsion-free if and only if^ϕ

∇

ϕ

=

0, where^ϕ

∇

is the Levi-Civita connection ofg_ϕ.

Theorem 2.1. (Fern´andez-Gray[11])The G2-structureϕis torsion-free if and only if ϕis both closed (i.e., dϕ

=

0) and co-closed (i.e., d

∗

_ϕϕ

=

dψ

=

0).

When

M

is compact, the above theorem says that aG2-structure ϕis torsion- free if and only ifϕis harmonic with respect to the induces metricgϕ.

We say that aG₂-structureϕisclosed(resp.,co-closed) ifdϕ

=

_{0 (resp.,}dψ

=

_0).

Theorem 2.1 can be restated as that aG₂-structure is torsion-free if and only if it is both closed and co-closed.

2.1. G2-decompositions of

∧

²

(M)

and

∧

³

(M)

. AG2-structure ϕinduces split- tings of the bundles

∧

^k

(

T^∗

M)

, 2

≤

k

≤

5, into direct summands, which we denote by

∧

^k_`

(

T^∗

M

,ϕ

)

with

`

being the rank of the bundle. We let the space of sections of

∧

^k_`

(

T^∗

M

,ϕ

)

by

∧

^k_`

(M

,ϕ

)

. Define the natural projections (2.2) π_`^k:

∧

^k

(M) −→ ∧

^k_`

(M

,ϕ

)

, α

7−→

π_`^k

(

α

)

.

We mainly focus on theG2–decompositions of

∧

²

(M)

and

∧

³

(M)

. Recall that

∧

²

(M) = ∧

²₇

(M

,ϕ

) ⊕ ∧

²₁₄

(M

,ϕ

)

, (2.3)

∧

³

(M) = ∧

³₁

(M

,ϕ

) ⊕ ∧

₇³

(M

,ϕ

) ⊕ ∧

³₂₇

(M

,ϕ

)

. (2.4)

Here each component is determined by

∧

²₇

(M

,ϕ

) = {

X

y

ϕ:X

∈

X

(M)} = {

β

∈ ∧

²

(M)

:

∗

_ϕ

(

ϕ

∧

β

) =

2β

}

,

∧

²₁₄

(M

,ϕ

) = {

β

∈ ∧

²

(M)

:ψ

∧

β

=

0

} = {

β

∈ ∧

²

(M)

:

∗

_ϕ

(

ϕ

∧

β

) = −

β

}

,

∧

³₁

(M

,ϕ

) = {

fϕ: f

∈

C^∞

(M)}

,

∧

³₇

(M

,ϕ

) =

ⁿ

∗

_ϕ

(

ϕ

∧

α

)

_:α

∈ ∧

¹

(M)

^o

= {

X

y

ψ:X

∈

X

(M)}

,

∧

³₂₇

(M

,ϕ

) = {

η

∈ ∧

³

(M)

:η

∧

ϕ

=

η

∧

ψ

=

0

}

.

For any 2-formβ

=

¹₂β_ijdx^i∧j

∈ ∧

²

(M)

, its two componentsπ²₇

(

β

)

andπ₁₄²

(

β

)

are determined by

π₇²

(

β

) =

^β

+ ∗

_ϕ

(

ϕ

∧

β

)

3

=

¹

2 1

3β_ab

+

¹

6β<sup>`m</sup>ψ<sub>`mab</sub>

dx^ab, (2.5)

π₁₄²

(

β

) =

^2β

− ∗

_ϕ

(

ϕ

∧

β

)

3

=

¹

2 2

3β_ab

−

¹

6β<sup>`m</sup>ψ<sub>`mab</sub>

dx^ab. (2.6)

To decompose 3-forms, recall two maps introduce by Bryant [1]

(2.7) i_ϕ:

²

(M) −→ ∧

³

(M)

, j_ϕ:

∧

³

(M) −→

²

(M)

, where

i_ϕ

(

h

)

:

=

h_ijg^j`dxⁱ

∧

∂

∂x^`

y

ϕ

=

¹

2h<sub>i`</sub>ϕ<sup>`</sup>_jkdx^ijk

=

¹ 6

hi`ϕ<sup>`</sup>_jk

+

hj`ϕ<sub>i</sub><sup>`</sup>_k

+

hk`ϕ<sub>ij</sub><sup>`</sup>

dx^ijk, h

=

hijdx^ij

∈

²

(M)

, (2.8)

and

(2.9)

j_ϕ

(

η

)

(

X,Y

)

:

= ∗

_ϕ

((

X

y

ϕ

) ∧ (

Y

y

ϕ

) ∧

η

)

.

Theni_ϕ is injective and is isomorphic onto

∧

³₁

(M

,ϕ

) ⊕ ∧

³₂₇

(M

,ϕ

)

, andj_ϕ is an isomorphism between

∧

³₁

(M

,ϕ

) ⊕ ∧

₂₇³

(M

,ϕ

)

_and

²

(M)

. Moreover, for any 3- formη

∈ ∧

³

(M)

, we have

(2.10) η

=

i_ϕ

(

h

) +

X

y

^ψ

for some symmetric 2-tensorh

∈

²

(M)

and vector fieldX

∈

X

(M)

. Then η

=

h_i^`dxⁱ

∧

∂

∂x^`

y

ϕ

+

X^` ∂

∂x^`

y

ψ

=

¹

2h_i<sup>`</sup>ϕ<sub>`jk</sub>dx^ijk

+

¹

6X<sup>`</sup>ψ<sub>`ijk</sub>dx^ijk

=

¹ 6

3hi`

ϕ_`jk

+

X<sup>`</sup>ψ<sub>`ijk</sub>

dx^ijk

=

¹

6η_ijkdx^ijk.

Writehash_ij

=

h^˚_ij

+

¹₇tr_ϕ

(

h

)

g_ϕ, where ˚h

∈

²₀

(M)

is the trace-free part ofh, one has

(2.11) η

=

³

7 trϕ

(

h

)

ϕ

| {z }

π³₁(η)

+

¹ 2h˚i`

ϕ_`jkdx^ijk

| {z }

π₂₇³(η)

+

¹

6X<sup>`</sup>ψ<sub>`ijk</sub>dx^ijk

| {z }

π³₇(η)

.

2.2. The torsion tensors of aG₂-structure. By Hodge duality we obtain theG₂- decompositions of 4-forms

∧

⁴

(M) = ∧

⁴₁

(M

,ϕ

) ⊕ ∧

₇⁴

(M

,ϕ

) ⊕ ∧

⁴₂₇

(M

,ϕ

)

and 5- forms

∧

⁵

(M) = ∧

⁵₇

(M

,ϕ

) ⊕ ∧

⁵₁₄

(M

,ϕ

)

, respectively. By definition, we can find formsτ0

∈

C^∞

(M)

,τ1,eτ1

∈ ∧

¹

(M)

,τ2

∈ ∧

²₁₄

(M

,ϕ

)

, andτ3

∈ ∧

³₂₇

(M

,ϕ

)

such that

(2.12) dϕ

=

τ₀ψ

+

3τ1

∧

ϕ

+ ∗

_ϕτ₃, dψ

=

4eτ₁

∧

ψ

− ∗

_ϕτ₂.

Sinceτ₂

∈ ∧

²₁₄

(M

,ϕ

)

, it follows thatτ₂

∧

ϕ

= − ∗

_ϕτ₂. Then (2.12) can be written as in the sense of Bryant [1]

(2.13) dϕ

=

τ₀ψ

+

_3τ1

∧

ϕ

+ ∗

_ϕτ₃, dψ

=

_4eτ₁

∧

ψ

+

τ₂

∧

ϕ.

It can be proved thatτ₁

=

τ_e1(see [22]). We callτ₀thescalar torsion,τ₁thevector torsion,τ₂theLie algebra torsion, andτ₃thesymmetric traceless torsion. We also call τ_ϕ:

= {

τ0,τ₁,τ2,τ3

}

theintrinsic torsion formsof theG2-structureϕ.

Recall that aG2-structureϕis torsion-free if and only ifdϕ

=

dψ

=

0 by Theo- rem 2.1. From (2.12) we see thatϕis torsion-free if and only if the intrinsic torsion formsτ_ϕ

≡=

0; that is,τ0

=

τ1

=

τ2

=

τ3

=

0.

Lemma 2.2. (Fern´andez-Gray,[11])For any X

∈

X

(M)

, the3-form

∇

_Xϕlines in the space

∧

³₇

(M

,ϕ

)

. Therefore the covariant derivative

∇

ϕ

∈ ∧

¹

(M) ⊗ ∧

³₇

(M)

.

Consequently, there exists a 2-tensorT

=

T_ijdx^i⊗j, called thefull torsion tensor, such that

(2.14)

∇

_`ϕ

=

T_`ⁿψ_nabc.

Equivalently,

(2.15) T_`m

=

¹

24

(∇

_`ϕ_abc

)

ψmabc. Write

τ₁

= (

τ₁

)

_idxⁱ

∈ ∧

¹

(M)

, (2.16)

τ2

=

¹

2

(

τ2

)

_abdx^ab

∈ ∧

²₁₄

(M)

, (2.17)

τ₃

=

¹

2

(

τ₃

)

_i<sup>`</sup>ϕ<sub>`ij</sub>dx^ijk

∈ ∧

³₂₇

(M

,ϕ

)

. (2.18)

The associated 2-tensorτ3:

= (

τ3

)

_ijdx^i⊗jofτ3lies in the space

²₀

(M)

. With this convenience, the full torsion tensorT_`mis determined by

(2.19) T`m

=

^τ0

4 g`m

− (

τ₃

)

_`m

− ]

_ϕ

(

τ₁

) y

^ϕ_`m

−

¹ 2

(

τ₂

)

_`m or as 2-tensors,

(2.20) T

=

^τ0

4 gϕ

−

τ3

− ]

_ϕ

(

τ₁

) y

^ϕ

−

¹ 2τ₂. Here the 2-form

]

_ϕ

(

τ₁

) y

ϕis defined by

]

_ϕ

(

τ₁

) y

ϕ

=

¹

2

]

_ϕ

(

τ₁

) y

ϕ

dx^a∧b

=

¹ 2

(

τ₁

)

_kϕ^k_ab

dx^a∧b. As an application, this gives another proof of Theorem 2.1.

For fixed indicesiandj, set

(2.21) R_ij|k` :

=

R_ijk`is skew-symmetric inkand

`

, where

(2.22) R_ij|••:

=

¹

2R<sub>ij|k`</sub>dx<sup>k`</sup>

=

¹

2R<sub>ijk`</sub>dx<sup>k`</sup>

∈ ∧

²

(M)

. Then, according to (2.5) and (2.6)

R_ijk`

=

R_ij|k`

=

π²₇

(

R_ij|••

)

k`

+

π²₁₄

(

R_ij|••

)

k`, where

π²₇

(

R_ij|••

)

k`

=

¹

3R_ij|k`

+

¹

6R_ij|abψ^ab_k`

=

¹

3R_ijk`

+

¹

6R_ijabψ^ab_k`,

π²₁₄

(

R_ij|••

)

k`

=

²

3R_ij|k`

−

¹

6R_ij|abψ^ab_k`

=

¹

3R_ijk`

−

¹

6R_ijabψ^ab_k`. Karigiannis [22] (see also the equivalent formula obtained by Bryant in [1]) proved that the Ricci curvature is given by

R_jk

=

R<sub>ijk`</sub>g<sup>i`</sup>

=

₃π₇²

(

R_ij|••

)

k`g<sup>i`</sup>

=

³ 2

π₁₄²

(

R_ij|••

)

k`g<sup>i`</sup>

= − ∇

_iT_jm

− ∇

_jT_im

ϕ^m_kⁱ

−

T_jⁱT_ik

+

tr_ϕT

T_jk

+

T_jbT_iaψ^iab_k, (2.23)

= −∇

_iTjn

ϕ_nkⁱ

+ ∇

_jTin

ϕ_nkⁱ

−

TjiTik

+

trϕT

Tjk

−

TjbTiaψ^iab_k. Cleyton and Ivanov [5] also derived a formula for the Ricci tensor for closedG2- structures in terms ofd^∗_ϕϕ. Taking the trace of (2.23), we obtain Btyant’s formula [1] for the scalar curvature

R

= −

12

∇

^`

(

τ₁

)

_`

+

²¹

8 τ₀²

− ||

τ3

||

²_ϕ

+

5

||]

_ϕ

(

τ₁

) y

^ϕ

||

²_ϕ

−

¹ 4

||

τ₂

||

²_ϕ,

= −

12

∇

^`

(

τ₁

)

_`

+

²¹

8 τ₀²

− ||

τ3

||

²_ϕ

+

30

|

τ₁

|

²_ϕ

−

¹ 2

|

τ₂

|

²_ϕ, (2.24)

For a closedG2-structure, we haveτ₀

=

τ₁

=

τ₃

=

0 and thenR

= −

¹₄

||

τ₂

||

²_ϕ

≤

0. On the other hand, we have

(

τ₂

)

_ij

= −

2Tijby (2.20). Thus the full torsion tensor Tis actually a 2-form

(2.25) T

=

¹

2Tijdx^ij

∈ ∧

²

(M)

and the scalar curvature can be written in terms ofT (2.26) R

= −||

T

||

²_ϕ

= −

2

|

T

|

²_ϕ

≤

0.

Hence, for closedG2-structures, scalar curvatures are always non-positive.

Finally, we mention a Bianchi type identity (2.27)

∇

_iT_j`

− ∇

_jT_i`

= −

¹

2R_ijabϕ^ab_`

−

T_iaT_jbϕ^ab_`

= −

1

2R_ijab

+

T_iaT_jb

ϕ^ab_`. The proof can be found in [22].

2.3. General flows on G₂-structures. For any family

(

ϕt

)

_t of G₂-structures, ac- cording to the decomposition (2.10), we can consider the general flow

(2.28) ∂tϕt

=

i_ϕt

(

ht

) +

Xt

y

ψt

whereht

∈

²

(M)

andXt

∈

X

(M)

. The general flow (2.28) locally can be written as

(2.29) ∂_tϕ_ijk

=

hi`

ϕ_`jk

+

hj`

ϕ_i`k

+

h_k<sup>`</sup>ϕ<sub>ij`</sub>

+

X<sup>`</sup>ψ<sub>`ijk</sub>.

We write forgtanddVtthe metric and volume form associated toϕ_t, respectively.

Theorem 2.3. Under the general flow (2.28), we have

∂tgij

=

2hij, (2.30)

∂tg^ij

= −

2h^ij, (2.31)

∂tdVt

= (

trtht

)

dVt, (2.32)

∂_tTpq

=

Tpmhmq

−

TpmX^kϕ_kmq

− (∇

_khip

)

ϕ^ki_q

+ ∇

_pXq. (2.33)

These evolution equations can be found in [22].

3. LAPLACIAN FLOWS ON CLOSEDG₂-STRUCTURES

We now consider the Laplacian flow for closedG2-structures (3.1) ∂tϕt

=

_∆ϕtϕt

=

_∆tϕt, ϕ₀

=

_ϕ,

where∆ϕtϕt

=

dd^∗_ϕtϕt

+

d^∗_ϕtdϕtis the Hodge Laplacian ofgϕt and ϕis an initial closedG2-structure. The short time existence for (3.1) was proved by Bryant and Xu [2], see also Theorem 1.1.

A criterion for the long time existence for the Lapalcian flow on compact man- ifolds was given in Theorem 1.2. In this section, we give a new elementary proof of Lotay-Wei’s result in compact case.

3.1. Basic theory of closedG₂-structures. Let

∧

³_+,•

(M) ⊂ ∧

³₊

(M

,ϕ

)

be the set of all closedG2-structures on

M

. Ifϕ

∈ ∧

³_+,•

(M)

is closed, i.e.,dϕ

=

0, thenτ₀,τ₁,τ₃ are all zero, so the only nonzero torsion form is

(3.2) τ

≡

τ₂

=

¹

2

(

τ₂

)

_ijdx^ij

=

¹ 2τ_ijdx^ij. According to (2.20) and (2.25), we haveT_ij

= −

¹₂τ_ijso that

(3.3) T

≡

¹

2T_ijdx^ij or equivalently T

= −

¹ 2τ,

is a 2-form. Sincedψ

=

τ

∧

ϕ

= − ∗

_ϕτ, we getd^∗_ϕτ

= ∗

_ϕd

∗

_ϕτ

= − ∗

_ϕd²ψ

=

0 which is given in local coordinates by

(3.4)

∇

ⁱτij

=

0

For a closedG₂-structure ϕ, according to (2.23), the Ricci curvature is given by (in this caseT_ijis a 2-form)

Rjk

= ∇

_jTim

− ∇

_iTjm

ϕ^m_kⁱ

−

TjiTik

+

TjbTiaψ^iab_k.