Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior

0.21 MB·English

by  Robert Bryant

LAPLACIAN FLOW FOR CLOSED G -STRUCTURES:
2
SHORT TIME BEHAVIOR
1
1 ROBERTBRYANT& FENG XU
0
2
Abstract. We prove short time existence and uniqueness of solutions
n to the Laplacian flow for closed G2 structures on a compact manifold
a M7. The result was claimed in [2], but its proof has neverappeared.
J

1
1

] Introduction
G
D Since R. Hamilton introduced Ricci flow to study Riemannian structures
. on manifolds [5], extensive work has been done to extend the ideas and
h
techniques to other geometric structures. For example, in the category of
t
a
Ka¨hler structures, Ka¨hler Ricci flow was studied in [3]. In the symplec-
m
tic background, a flow called anti-complexified Ricci flow stimulates much
[
interest and its s hort time existence has been proved in [11].

1 Many attempts have also been tried on G structures (e.g., see [9],[10]).
2
v
In this article, we are interested in the Laplacian flow
4
0 d
0 (1) σ = ∆σσ,
dt
2
1. where σ is the defining 3-form of a G2-structure, and ∆σ is the Hodge
0 Laplacian of the metric determined by σ. This flow was studied by Steven
1 Altschuler and the first author between 1992 and 1994 (see [2]). In par-
1
ticular, when the initial 3-form σ is closed and we evolve inside a fixed
: 0
v cohomology class, a natural question is under which conditions it will con-
i
X verge to a structure with holonomy in G2. Later on, when M is compact, it
is found to be the gradient flow of Hitchin’s volume functional V : [σ ] → R
r 0
a defined by
(2) V(σ) = σ∧∗ σ.
σ
Z
M
However,sinceboththeflow(1)andthefunctional(2)arediffeomorphism
invariant and since [σ ]/Diff is still infinite dimensional, it is not clear if a
0 0
short time solution to
dσ = ∆ σ
dt σ
(3)  dσ = 0
 σ(0) = σ
0

1991 Mathematics Subject Classification. Primary 53C38.
Key words and phrases. G2 structures, Geometric flow, Parabolic equation.
1
2 ROBERTBRYANT&FENGXU
exists at all. We prove this result in this paper.
Theorem 0.1. Assume M is compact. Then the initial value problem (3)
has a unique solution for a short time 0 ≤ t ≤ ǫ with ǫ depending on σ .
0
We give an outline of its proof. First, when restricted to closed forms,
the flow equation takes the form
dσ = −d∗ d∗ σ
dt σ σ
(4) 
 σ(0) =σ
0
where dσ = 0. 
0
It follows that a solution σ(t) must lie in the cohomology class [σ ]. By
0
letting σ(t) = σ +θ(t) with θ taking values in exact forms, we may rewrite
0
the flow equation in terms of θ
dθ = −d∗ d∗ (θ+σ ),
dt (θ+σ0) (θ+σ0) 0
(5) 
 θ(0)= 0.
Clearly, (3)is equivalenttotheinitialvalueproblem(5)forafamilyof exact
forms θ(t).
Of course, this flow is still diffeomorphism-invariant and thus is not par-
abolic. As in Deturck’s Trick for the Ricci Flow [4], we modify the flow by
an operator of the form L (σ) = d(Vyσ)+Vydσ = d(Vyσ):
V(σ)
dθ = −d∗ d∗ (σ +θ)+d(V(θ+σ )y(θ+σ )),
dt (σ0+θ) (σ0+θ) 0 0 0
(6) 
 θ = 0.
0
For wisely chosen V, this new flow is elliptic in the direction of closed
forms. We hope to show that the new flow has short time existence and
then,byapplyingsuitabletime-dependentdiffeomorphismstogetasolution
of the flow (5).
However, no existing theory of parabolic equations seems to be directly
applicable. Thus, it seems to us that an argument along the line of inverse
function theorems is necessary. In fact, we will use Nash Moser inverse
functiontheoremfortameF´echetspaces(see[6],pp. 171-172). Weintroduce
(7) U = θ ∈ dC∞ [0,T]×M,Λ2(M) : σ +θ is definite .
0
(cid:8) (cid:0) (cid:1) (cid:9)
This is an open set of the Fr´echet space dC∞ [0,T]×M,Λ2(M) . We
consider a mapping F : U → C∞ [0,T]×M,Λ2(cid:0)(M) ×dC∞ M,Λ(cid:1)2(M)
defined by (cid:0) (cid:1) (cid:0) (cid:1)
d
(8) θ 7→ θ−∆ σ−L σ,θ| .
(cid:18)dt σ V(σ) t=0(cid:19)
whereσ = σ +θandV(σ)isavectorfielddependenton1storderderivatives
0
of σ. We will prove the following lemma:
LAPLACIAN FLOW FOR CLOSED G2-STRUCTURES: SHORT TIME BEHAVIOR 3
Lemma 0.2. Suppose θ ∈ U is a solution to
dθ−∆ σ−L σ = χ
dt σ V(σ)
(9) 
 θ(0) = θ .
0

Then for (χ,θ ) sufficiently close to (χ,θ ), there is a unique solution θ(t)
0 0
to
dθ−∆ σ−L σ = χ
dt σ V(σ)


θ(0) = θ .
0

It is for the proof of Lemma 0.2 that we use Nash Moser inverse function
theorem. To use the theorem, we need to show:
(1) The linearized map F at σ is injective;
∗
(2) F is surjective;
∗
(3) The inverse map F−1 is a smooth tame map.
∗
Roughly speaking, (3) means that the solutions to linearized equations sat-
isfy certain a priori estimates.
Below is thestructureof therest of this paper. In§1, we review G struc-
2
tures. We discuss G connections and the Levi-Civita connections. These
2
will beusefulin constructing vector fields used in Deturck’s Trick. Most im-
portantly, differential identities of G holonomy are discussed. Theseidenti-
2
ties are discovered in [2] and are basis for our later computations. Hitchin’s
functional will also be discussed in more detail. In §3, we give a detailed
proof of Theorem 0.1 assuming Lemma 0.2. In §4, we prove Lemma 0.2.
The work was done when the second named author was an MSRI post-
doctoral fellow. He would like to thank MSRI for hospitality.
1. Manifolds with G -structures
2
Inthissection,wecollectbasicfactsaboutG -structuresona7-dimensional
2
manifold M. For details, see [2].
1.1. Lie group G . LetV = R7 bethe7-dimensional Euclideanspace. Let
2
{e ,e ,··· ,e } be the standard basis of V and let {e1,··· ,e7} be the dual
1 2 7
basis.
Define a 3-form
(10) φ= e123 +e145+e167 +e246 −e257−e347 −e256
where e123 = e1∧e2∧e3, etc.
We have the following characterization of G :
2
Lemma 1.1 ([12],[1]).
(11) G = {g ∈ GL(V)|g∗φ= φ}.
2
4 ROBERTBRYANT&FENGXU
The Lie group G is compact, connected, simply connected and simple.
2
It acts irreducibly on V and preserves the metric and orientation for which
{e ,··· ,e } is an oriented orthonormal basis. We use ∗ (or ∗ when there
1 7 φ
is no danger of confusion) to denote the Hodge star operator determined by
the orientation and the metric. It particular, G also preserves the 4-form
2
(12) ∗ φ= e4567 +e2367 +e2345 +e1357 −e1346 −e1256−e1247.
φ
Using the metric, we identify V and its dual V∗ through musical isomor-
phisms
♭ :V → V∗
and
: V∗ → V.
#
Under the action of G , the spaces Λ1V∗ and Λ6V∗ are irreducible while
2
the other wedge products Λ∗(V∗) decomposes into irreducible components:
(13) Λ2V∗ = Λ2⊕Λ2
7 14
(14) Λ3V∗ = Λ3⊕Λ3⊕Λ3
1 7 27
(15) Λ4V∗ = Λ4⊕Λ4⊕Λ4
1 7 27
(16) Λ5V∗ = Λ5⊕Λ5 .
7 14
p
Here Λ denotes an irreducible G representation of dimension d. Different
d 2
spaces in this list with the same dimension are naturally isomorphic. For
example, ∗ : Λ4 ≃ Λ3 . The space Λ2 is naturally isomorphic to the Lie
27 27 14
algebra g through musical isomorphisms.
2
ThespaceΛ3 deservesspecialattention. Anexplicitalgebraicdescription
27
of this space is [2]
Λ3 = {γ ∈ Λ3|γ ∧φ= 0,γ ∧∗φ = 0}.
27
As an irreducible representation of G , it has the highest weight (2,0). The
2
space of traceless symmetric bilinear forms on V is also 27-dimensional and
hasthehighestweight(2,0). SotheremustbeaG -equivariantisomorphism
2
between them. One such is given in [2]: Define j :Λ3 → Sym2(V) by
φ 27 0
(17) j (γ)(v,w) = ∗ ((vyφ)∧(wyφ)∧γ).
φ φ
1.2. G -structures. Suppose now M is a 7-dimensional smooth manifold.
2
For each x ∈ M, we define the set definite 3-forms
(Λ3+)x = {σx ∈ Λ3Tx∗M|∃u∈ HomR(TxM,V),u∗φ= σx}.
We call
Λ3(M) = (Λ3)
+ + x
ax
the bundle of definite 3-forms. It is a subbunle of Λ3(M). In fact, (Λ3)
+ x
contains a single open orbit of GL(T M). Thus the bundle Λ3(M) is open
x +
in Λ3.
LAPLACIAN FLOW FOR CLOSED G2-STRUCTURES: SHORT TIME BEHAVIOR 5
Definition 1.2. A section σ of the bundle Λ3(M) is called a G -structure
+ 2
on M.
A G -structure on M reduces the total coframe bundle F to a principal
2
G -subbundle
2
Fσ = {(x,u)|u ∈HomR(TxM,V),u∗φ =σx}.
Because G acts reducibly on ΛpV∗ for 2 ≤ p ≤ 5, the vectors bundles of
2
formsΛp(M)onM decomposesasdirectsumscorrespondingly. Forinstance
Λ2(M) = Λ2(M)⊕Λ2 (M)
7 14
for Λ2(M) = F × Λ2V∗ and for Λ2 (M) = F × Λ2 V∗.
7 σ G2 7 14 σ G2 14
1.2.1. Connectionsandtorsions. Wediscussvariousconnectionsdetermined
by a G -structure σ.
2
First, associated with σ, there is a canonical metric, denoted by gσ. On
each tangent space T M, gσ is simply the pull-back by u of the standard
x
metric on V for any u ∈ F | . This does not depend on the choice of u
σ x
because u only differs by an element of G .
2
The oriented orthonormal coframe bundle of gσ is obtained from F by
σ
extension of the structure group to SO(7), i.e., F ·SO(7) ⊂ F.
σ
We denote by ω = (ω )the Levi-Civita connection on F ·SO(7). It is an
ij σ
so(7)-valued one-form. If {ω } is a local orthonormal coframe, i.e., a local
i
section of F ·SO(7), we have
σ
dω = −ω ∧ω .
i ij j
The dual frame field e has covariant derivatives
i
∇σ e = e ω (X).
X i j ji
When we restrict ω to the subbundle F , it decomposes as
σ
ω = θ⊕τ
where θ takes value in the Lie algebra g ⊂ so(7) and τ takes value in
2
so(7)/g . It is easy to show that θ defines a G -connection on F . It
2 2 σ
follow that τ is semibasic with respect to F → M. Thus τ is a section of
σ
F × (so(7)/g ⊗V∗).
σ G2 2
Definition 1.3. We call τ the torsion tensor of the connection θ.
For this reason, we call V = so(7)/g ⊗V∗ the torsion space of G . As
τ 2 2
an representation of G , V ≃ V∗⊗V∗ and thus decomposes into
2 τ
V = V ⊕V ⊕V ⊕V
τ (0,0) (1,0) (0,1) (2,0)
where V denotes the irreducible representation with the highest weight
(p,q)
(p,q), in particular V = V and V = g .
(1,0) (0,1) 2
Correspondingly τ has four components, τ , τ , τ , and τ . These compo-
0 1 2 3
nents show up in the differentials of σ and d∗σ:
6 ROBERTBRYANT&FENGXU
Proposition 1.4 ([2]). For any G -structure, there exist unique differential
2
forms τ ∈ Ω0(M), τ ∈ Ω1(M), τ ∈ Ω2 (M), and τ ∈ Ω3 (M) so that the
0 1 2 14 3 27
following equations hold:
(18) dσ = τ ∗σ+3τ ∧σ+∗τ ,
0 1 3
(19) d∗σ = 4τ ∧∗σ+τ ∧σ.
1 2
Various G structures with one or several of the components vanishing
2
received much interest in the literature. We are mainly interested in closed
G -structures.
2
Definition 1.5. A G -structure is closed if its definite 3-form σ is closed.
2
ByProposition1.4,theconditiondσ = 0isequivalenttoτ = τ = τ = 0.
0 1 3
Thus the only torsion component is a 2-form τ ∈ Ω2 .
2 14
Definition 1.6. A G -structure is called 1-flat if dσ = d∗σ = 0.
2
Equivalently, σ is 1-flat if and only if ∆ σ = 0 where ∆ is the Hodge
σ σ
Laplacian of gσ. In this case, the connections ω and θ coincide and the
holonomy of the Levi-Civita connection lies in G .
2
1.3. Differential identities. By composing the exterior differential oper-
ator d with various algebraic bundle isomorphisms, we can define many dif-
p
ferential operators d : Ω → Ω for all p,q ∈ {1,7,14,27}. More explicitly,
q p q
for f ∈ Ω0, α ∈Ω1, β ∈Ω2 and γ ∈ Ω3 ,
7 14 27
d1f = df
7
d7α =∗d(α∧∗σ)
7
d7 α = π2 dα
14 14
d7 α = π3 d∗(α∧∗σ)
27 27
d14β = π3 dβ
27 27
d27γ = ∗(π4 dγ)
27 27
where πk denote the projection onto Λk. and the rest are determined by
p p
(dp)∗ = dq
q p
for p 6= q as well as
d1 = d1 = d1 = d14 = 0.
14 27 1 14
1.3.1. Torsion free case. When the G structure is torsion free, the Levi-
2
Civita connection ∇σ coincides with the G connetion and its holonomy
2
lies in G . As in the case of Ka¨hler geometry, there are various differential
2
identities by abstract nonsense. These identities are listed in [2], pp. 25
(except that the minus sign in the formula d(α∧∗σ) = −∗d7α is a typo).
7
These identities are important for our computations. The proof is based on
abstract nonsense and constant checking. Besides the identities listed there,
We need one more that we now describe.
LAPLACIAN FLOW FOR CLOSED G2-STRUCTURES: SHORT TIME BEHAVIOR 7
Recall that Ω3 consists of 3-forms whose wedge products with σ and ∗σ
27
vanish. It maybe identified with Sym2–the space of trace free symmetric
0
bilinear forms. An explicit map is given in [2] by
(20) j (γ)(φ)(v,w) = ∗ ((vyσ)∧(wyσ)∧γ)
σ σ
for any γ ∈ Ω3 , v ∈ TM and w ∈ TM. By definition d27γ takes value in
27 7
Ω1. So is div(j (γ)). Here we define
7 σ
div(h) = gjlgikh e
ij,k l
for any symmetric two tensor h . They must agree up to a constant.
ij
Lemma 1.7. If the G structure is torsion free, we have
2
(21) (d27(γ)) = A·div(j (γ))
7 # σ
for some universal nonzero constant A.
Proof. Consider the following diagram
Λ3 ⊗Λ1 −→δ Λ4 −π→74 Λ4
27 7
↓ j ⊗1
σ
Sym2⊗Λ1 −→c Λ1 −∧→σ Λ4
0 7
where δ denotes the skewsymmetrization, c denotes the contraction using
the metric and π4 is the orthogonal projection. All spaces have a natural
7
G action and all linear maps are G -equivariant. By Schur’s Lemma, there
2 2
must be a nonzero constant A, so that
π4◦δ(γ ⊗α) = A′·c(j (γ)⊗α)∧σ.
7 σ
On the other hand, we have
d(γ) = δ◦∇σγ,
1
d27γ∧σ = π4d(γ) = π4◦δ◦∇σγ
4 7 7 7
and
div(j (γ))♭ = c(∇σ(j (γ))) = c(j ⊗1(∇σγ)).
σ σ σ
since ∇σ coincides with the G connection and thus j is ∇σ-parallel. Com-
2 σ
bining all these we get
d27γ = A·div(j (γ))♭
7 σ
for A= 4A′. (cid:3)
Remark 1.8 (How to determine A). Because it is universal, we might simply
assume M = R7. Since its value will not be important for us, we do not
pursue further.
8 ROBERTBRYANT&FENGXU
1.3.2. Torsion case. When the underlying G structure has torsion, the
2
identity in Lemma 1.7 and the differential identities in [2] are no longer
true in general. One needs to modify the identities by lower order deriva-
tives. However, since we will only be interested in the principal symbols
of various differential operators, these lower order terms are not essential.
Thus when we do computations, we may proceed as if the underlying struc-
ture were torsion free. This observation is very important. Hopefully this
rule will be clear once we do concrete calculations.
1.4. The Laplacian flow. The flow was introduced in [2]. The idea is to
evolve the definite 3-form σ in the direction of its Hodge Laplacian:
d
(22) σ = ∆ σ.
σ
dt
Note that, since ∆ depends on σ itself, this flow is nonlinear. It has a large
symmetry group: In fact, it is preserved by diffeomorphisms. Thus, it is not
parabolic.
Thestable solutions are, of course, given by1−flatG -structures. Then,
2
as is well-known, the underlying metric is Ricci-flat. In this sense, the flow
resembles the Ricci flow or the Ka¨hler Ricci flow.
Suppose now σ(t) is a solution. If the initial structure σ is closed, then
0
it is not hard to show that a solution σ(t) remains closed. Then the flow
equation satisfied by σ(t) simplifies to
d
(23) = −d∗d∗σ.
dt
Thus σ(t) stays in the same cohomology class [σ ].
0
1.5. Hitchin’sfunctional. Supposeσ isaclosedG -structureoncompact
0 2
M7. Define
[σ ] = {σ +dβ|β ∈ Ω2(M)}.
0 + 0
In [7], Hitchin defined a volume functional on [σ ]:
0
(24) V(σ) = σ∧∗σ.
Z
M
He computed the first variation of this functional
7
(25) δV(δσ) = δσ∧∗σ,
18 Z
M
where δσ is an exact 3-form.
The Laplacian flow may be viewed as a gradient flow of V with respect
to an unusual metric on [σ ]. We now describe this metric. First, note
0
that the tangent space of [σ ] at any point σ ∈ [σ ] is simply dΩ2(M). Let
0 0
G be the Green’s operator of the Hodge Laplacian ∆ . We define for any
σ σ
ψ,ψ′ ∈dΩ2(M),
7
(26) hψ,ψ′i = gσ(G ψ,ψ′)∗ 1.
σ σ σ
18 Z
M
LAPLACIAN FLOW FOR CLOSED G2-STRUCTURES: SHORT TIME BEHAVIOR 9
It can be shown that (26) in deed defines a metric on dΩ2(M).
Also note ∆ G ψ = ψ for any exact form ψ. With this in mind, we now
σ σ
rewrite (25) as
δV(δσ) = 7 δσ∧∗σ
18 M
R
= 7 gσ(δσ,σ)∗ 1
18 M σ
R
= 7 gσ(∆ G δσ,σ)∗ 1
18 M σ σ σ
R
= 7 gσ(G δσ,∆ σ)∗ 1
18 M σ σ σ
R
= hδ ,∆ σi .
σ σ σ
Thus under the metric (26), the gradient vector field of V is ∆ σ.
σ
2. Deturck’s Trick
In this section, we apply Deturck’s Trick. We have to find the vector field
in L σ so that (6) is strictly parabolic in the direction of closed forms.
V(σ)
The vector field is constructed in a similar way to [4].
2.1. Deformation of G -structures. The result is due to D. Joyce [8].
2
Proposition 2.1. Let σ be a family of G structures on M. Then there
s 2
exist three differential forms f0 ∈ Ω0, f1 ∈ Ω1 and f3 ∈ Ω3 uniquely
27
characterized by
∂
(27) σ = 3f0σ+∗ (f1∧σ)+f3.
s σ
∂s
In terms of these, the variation of the metric gσ is given by
∂ 1
(28) gσ = 2f0gσ + j (f3).
σ
∂s 2
The variation of the dual 4-form ∗ σ is
σ
∂
(29) (∗ σ)= 4f0∗ σ+f1∧σ−∗ f3.
σ σ σ
∂s
The proposition will be useful in linearization. If we deform the structure
in a closed direction, we have
d(3f0σ+∗ (f1∧σ)+f3) = 0.
σ
Using differential identities to expand out we get
4
(30) 0= d7f1+l.o.t
7 1
1 1
(31) 0 = d1f0+ d7f1+ d27f3+l.o.t
7 6 7 12 7
(32) 0 = d7 f1+d27f3+l.o.t.
27 27
10 ROBERTBRYANT&FENGXU
Remark 2.2. The lower order terms are algebraic in (f0,f1,f3), acted on
by the various torsion tensors of σ. It is possible to work them out, but we
will not need them in this paper.
2.2. The vector field V(σ). Let ∇0 be a fixed torsion free connection on
M. Let ∇σ be the Levi-Civita connection of the Riemannian metric gσ
determined by σ. Then, the difference
T = ∇σ −∇0
is a well-defined tensor on M. In fact, since both connections are torsion
free, T takes valued in TM ⊗Sym2T∗M. Using gσ, we identify TM with
T∗M and decompose
TM ⊗Sym2T∗M ≃ TM ⊕TM ⊗Sym2T∗M.
0
This gives us two vector fields from T: One from TM component and
the other by taking contraction of TM ⊗ Sym2T∗M. To avoid confusing
0
constants, we define V and V in a slightly differently way. Locally, if {e }
1 2 i
is a frame field and {ωi} is the dual coframe field and T = 1Ti e ⊗ωj ◦ωk
2 jk i
with Ti = Ti , then
jk kj
1
V = gpqTi e
1 7 pq i
and
2
V = (gkjTi e +5V )
2 A ik j 1
where the repeated indices represent a summation and A is the constant
defined in Lemma 1.7.
For eachpair(λ,µ)ofrealnumbers(thatwewilldetermineinthefuture),
define
(33) V(σ) = λV +µV .
1 2
and correspondingly
(34) Q (σ) = λL σ+µL σ
λ,µ V1 V2
where L is the Lie derivative. Note that since the vector fields V and V
1 2
involves one derivatives of σ, Q is a second order differential operator on
λ,µ
σ when λ,µ are not simultaneously 0.
2.3. Linearization of Q. Now we hope to linearize Q . By Cartan’s
λ,µ
formula,
Q σ = λd(V yσ)+λV ydσ+µd(V yσ)+µV ydσ.
λ,µ 1 1 2 2
If σ is closed,
Q σ =d(q σ)
λ,µ λ,µ
with
q σ = λV yσ+µV yσ.
λ,µ 1 2