Local curvature estimates for the Laplacian flow

In this paper we give local curvature estimates for the Laplacian flow on closed G_2-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar-Munteanu-Wang who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum's result, and the particular structure of the Laplacian flow on closed G_2-structures. As an immediate consequence, this estimates give a new proof of Lotay-Wei's result which is an analogue of Sesum's theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed G_2-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature R_t is equal to the Laplacian of R_t, plus an extra term which can be written as the difference of two nonnegative quantities.


INTRODUCTION
Let M be a smooth 7-manifold. The Laplacian flow for closed G 2 -structures on M introduced by Bryant [1] is to study the torsion-free G 2 -structures where ∆ ϕ t ϕ t = dd * ϕ t ϕ t + d * ϕ t dϕ t is the Hodge Laplacian of g ϕ t and ϕ is an initial closed G 2 -structure. Since d∂ t ϕ t = ∂ t d∆ ϕ t ϕ t = 0, we see that the flow (1.1) preserves the closedness of ϕ t . For more background on G 2 -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)

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 by C ∞ (M) and X(M). The space of k-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 any k-tensor field T ∈ ⊗ k (M), we locally have the expression T = T i 1 ···i k dx i 1 ⊗ · · · ⊗ dx i k =: T i 1 ···i k dx i 1 ⊗···⊗i k . A k-form α on M can be written in the standard form as α = 1 k! α i 1 ···i k dx i 1 ∧ · · · ∧ dx i k =: 1 k! α i 1 ···i k dx i 1 ∧···∧i k , where α i 1 ···i k is fully skew-symmetric in its indices. Using the standard forms, if we take the interior product X α of a k-form α ∈ ∧ k (M) with a vector field X ∈ X(M), we obtain the (k − 1)-form X α = 1 (k−1)! X m α mi 1 ···i k−1 dx i 1 ∧···∧i k−1 which is also in the standard form. In particular, consider the vector space ⊗ 2 (M) of 2tensors. For any 2-tensor A = A ij dx i⊗j , define A ⊙ := 1 2 Then A ⊙ is an element of ⊙ 2 (M), the space of symmetric 2-tensors. Since 1 dx i∧j = dx i⊗j − dx j⊗i , it follows that A ∧ = 1 2 A ij dx i∧j . Define α A := 1 2 α A ij dx i∧j with α A ij := 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 field X = X i ∂ ∂x i and 1-form α = α i dx i , ♭ g (X) = X i g ij dx j ≡ X j dx j and ♯ g (α) = α i g ij ∂ ∂x j ≡ α j ∂ ∂x j . Using these two natural maps, we can frequently raise or lower indices on tensors. The metric g also induces a metric on kforms g(dx i 1 ∧···∧i k , dx j 1 ∧···∧j k ) = det(g(dx i a , dx j b )) = ∑ σ∈S 7 sgn(σ)g i 1 j σ(1) · · · g i k j σ(k) where S 7 is the group of permutations of seven letters and sgn(σ) denotes the sign (±1) of an element σ of S 7 . The inner product ·, · g of two k-forms α, β ∈ ∧ k (M) now is given by α, β g = 1 k! α i 1 ···i k β i 1 ···i k = 1 k! α i 1 ···i k β j 1 ···j k g i 1 j 1 · · · g i k j k . Given two 2-tensors A, B ∈ ⊗ 2 (M), with the forms A = A ij dx i⊗j and B = B ij dx i⊗j . Define A, B g := A ij B ij . There are two special cases which will be used later: (1) α = 1 2 α ij dx i∧j ∈ ∧ 2 (M) and B = B ij dx i⊗j ∈ ⊗ 2 (M). In this case, α can be written as a 2-tensor A α = A α ij dx i⊗j with A α ij = α ij . Then α, B g := A α , B g = α ij B ij .
The Levi-Civita connection associated to a given Riemannian metric g is denoted by g ∇ or simply ∇. Our convention on Riemann curvature tensor is R m ijk ∂ ∂x m Ricci curvature of g is given by R jk := R ijkℓ g iℓ . We use dV g and * g to denote the volume form and Hodge star operator, respectively, on M associated to a metric g and an orientation.
We use the standard notion A * B to denote some linear combination of contractions of the tensor product A ⊗ B relative to the metric g t associated the ϕ t . In Theorem 1.4 and its proof, all universal constants c, C below depend only on the given real number p.
As in the Ricci flow, we can prove following results on the long time existence for the Laplacian flow (1.1). [33]) Let M be a compact 7-manifold and ϕ t , t ∈ [0, T), where T < ∞, be a solution to the flow (1.1) for closed G 2 -structures with associated metric g t = g ϕ t for each t.

Theorem 1.2. (Lotay-Wei
(a) If the velocity of the flow satisfies then the solution ϕ t can be extended past time T.
Here T t is the torsion of ϕ t (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 compact 7-manifold and ϕ t , t ∈ [0, T), where T < ∞, be a solution to the flow (1.1) for closed G 2 -structures with associated metric g t = g ϕ t for each t. Suppose that where the bound depends only on n, K, T and Λ.
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 In the compact case, Theorem 1.3 shows that, if the conclusion in part (a) does not hold, then T = T max and sup (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 smooth 7-manifold and ϕ t , t ∈ [0, T), where T < ∞, be a solution to the flow (1.1) for closed G 2 -structures with associated metric g t = g ϕ t for each t. Assume that there exist constants A, K > 0 and a point x 0 ∈ M such that the geodesic ball Then, for any p ≥ 5, there exists c = c(p) > 0 so that 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 where d 1 , d 2 are constants depending on K, T, A, and Actually, this follows from the same argument in [24] by noting that To verify (1.5), we use (2.26), (3.61) and (3.65) to deduce that ||∇ t T t || ≤ c||Rm t || t and Then, by (3.31) and the Cauchy inequality 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 Kähler-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].
We can ask the same question for the Laplacian flow on closed G 2 -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 closed G 2 -structures will exist as long as the torsion tensor or scalar curvature remains bounded. Let g t be the associated metric of ϕ t . Then the evolution equation for g t is given by For the Laplacian flow on closed G 2 -structures, the torsion T t is actually a 2-form for each t, hence we use the norm | · | t in (1.6). The standard formula for the scalar curvature R t gives (see (3.23))

Now the above mentioned open problem states that
Is it ture that lim The "minus infinity" comes from the fact that along the Laplacian flow on closed G 2 -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 evolution equation (1.7) where the last two terms on the right-hand side do not have good signature. However, using the closedness of ϕ t (in particular, the identity (3.23)), we can prove the following interesting evolution equation for R t . .
Observe that the above well-arranged evolution equation can give us a weakly lower bound for R t , 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 Section 2 about G 2 -structures, G 2 -decompositions of 2-forms and 3-forms, and general flows on G 2 -structures. In Section 3, we rewrite results in Section 2 for closed G 2 -structures, and the local curvature estimates will be given in the last subsection. The author, together with other six friends, thanks Yunhui Wu who personally provided us 14, the dimension of G 2 , 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 version of this paper. He also thanks Jason Lotay for his interested in this paper.
where the Hodge star operator * φ is determined by the metric and orientation.
For a smooth 7-manifold M and a point x ∈ M, define as in [33] and the bundle We call a section ϕ of ∧ 3 + (T * M) a positive 3-form on M or a G 2 -structure on M, and denote the space of positive 3-forms by ∧ 3 + (M). The existence of G 2 -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 G 2 -structures, we see that any ϕ ∈ ∧ 3 + (M) uniquely determines a Riemannian metric g ϕ and an orientation dV ϕ , hence the Hodge star operator * ϕ and the associated 4-form We also have the isomorphisms ♭ ϕ := ♭ g ϕ and ♯ ϕ := ♯ g ϕ . For a given G 2 -structure ϕ ∈ ∧ 3 + (M), we denote by ·, · ϕ , ·, · , | · | ϕ , || · || ϕ , the corresponding inner products ·, · g ϕ , ·, · g ϕ and norms | · | g ϕ , || · || g ϕ .
Given a G 2 -structure ϕ ∈ ∧ 3 + (M). We say that ϕ is torsion-free if ϕ is parallel with respect to the metric g ϕ . Equivalently, ϕ is torsion-free if and only if ϕ ∇ϕ = 0, where ϕ ∇ is the Levi-Civita connection of g ϕ . [11]) The G 2 -structure ϕ is torsion-free if and only if ϕ is both closed (i.e., dϕ = 0) and co-closed (i.e., d * ϕ ϕ = dψ = 0). When M is compact, the above theorem says that a G 2 -structure ϕ is torsionfree if and only if ϕ is harmonic with respect to the induces metric g ϕ .

Theorem 2.1. (Fernández-Gray
We say that a G 2 -structure ϕ is closed (resp., co-closed) if dϕ = 0 (resp., dψ = 0). Theorem 2.1 can be restated as that a G 2 -structure is torsion-free if and only if it is both closed and co-closed.

The torsion tensors of a G 2 -structure. By Hodge duality we obtain the
Since τ 2 ∈ ∧ 2 14 (M, ϕ), it follows that τ 2 ∧ ϕ = − * ϕ τ 2 . Then (2.12) can be written as in the sense of Bryant [1] (2.13) It can be proved that τ 1 = τ 1 (see [22]). We call τ 0 the scalar torsion, τ 1 the vector torsion, τ 2 the Lie algebra torsion, and τ 3 the symmetric traceless torsion. We also call Recall that a G 2 -structure ϕ is torsion-free if and only if dϕ = dψ = 0 by Theorem 2.1. From (2.12) we see that ϕ is torsion-free if and only if the intrinsic torsion forms τ ϕ ≡= 0; that is, Equivalently, Write The associated 2-tensor τ 3 With this convenience, the full torsion tensor T ℓm is determined by Here the 2-form ♯ ϕ (τ 1 ) ϕ is defined by As an application, this gives another proof of Theorem 2.1.
For fixed indices i and j, set Then, according to (2.5) and (2.6) Karigiannis [22] (see also the equivalent formula obtained by Bryant in [1]) proved that the Ricci curvature is given by Cleyton and Ivanov [5] also derived a formula for the Ricci tensor for closed G 2structures in terms of d * ϕ ϕ. Taking the trace of (2.23), we obtain Btyant's formula [1] for the scalar curvature For a closed G 2 -structure, we have τ 0 = τ 1 = τ 3 = 0 and then R = − 1 4 ||τ 2 || 2 ϕ ≤ 0. On the other hand, we have (τ 2 ) ij = −2T ij by (2.20). Thus the full torsion tensor T is actually a 2-form and the scalar curvature can be written in terms of T Hence, for closed G 2 -structures, scalar curvatures are always non-positive.
Finally, we mention a Bianchi type identity The proof can be found in [22].

General flows on G 2 -structures.
For any family (ϕ t ) t of G 2 -structures, according to the decomposition (2.10), we can consider the general flow where h t ∈ ⊙ 2 (M) and X t ∈ X(M). The general flow (2.28) locally can be written as We write for g t and dV t the metric and volume form associated to ϕ t , respectively.
These evolution equations can be found in [22].

LAPLACIAN FLOWS ON CLOSED G 2 -STRUCTURES
We now consider the Laplacian flow for closed G 2 -structures where ∆ ϕ t ϕ t = dd * ϕ t ϕ t + d * ϕ t dϕ t is the Hodge Laplacian of g ϕ t and ϕ is an initial closed G 2 -structure. The short time existence for (3.1) was proved by Bryant and Xu [2], see also Theorem 1.

Evolution equations for closed G 2 -structures.
Since the Laplacian flow (3.1) preserves the closedness of ϕ t , it follows from (3.10) that we have From Theorem 2.3, we see that the associated metric tensor g t evolves by and the volume form dV t evolves by Hence, along the flow (3.1), the volume of g t is nondecreasing.
Introduce the following notions (3.14) t := ∂ t − t , | · | t := | · | ϕ t , ∆ t := ∆ ϕ t , where t := g ij ∇ i ∇ j is the usual Laplacian of g t and ∆ t is the Hodge Laplacian of g t , and also the 2-tenor Sic t with components Then the evolution equation (3.12) can be written as Moreover, the trace of Sic t is exactly the scalar curvature, up to a multiplying constant, It was proved in [33] that This identity together with (2.26) shows that the boundedness of ∆ t ϕ t is equivalent to the boundedness of Ric t .
The evolution equation (2.33) implies that for the Laplacian flow on closed G 2structures, the torsion T ij evolves by evolves Furthermore, we can prove Proof. See [33].
For a geometric flow ∂ t g ij = η ij , for some symmetric 2-tensor η ij , we have where the symmetric 2-tensor T is given by Plugging those identities into the above evolution equation for R t , we get Observe that the last two terms on the right-hand side of (3.22) are not determined of their signs. In the following, we shall use the identity follows from from (3.3) and (3.4), to simplify those two terms. Using the identity (3.23), the term ∇ j ∇ i T ij can be simplified as follows.
On the other hand, from the Ricci identity we see that the evolution equation (3.22) is equivalent to From (3.15) and (3.21) we can rewrite the term ||Ric t || 2 t in (3.24) in terms of Sic t according to the following relation: where we used tr Replacing R t by S t according to the identity (3.17), we can rewrite (3.24) as Similarly, replacing Ric t , T t t by Sic t , T t t with respect to the identity we obtain the following evolution equation for S t , Next, we try to deal with the last bracket in (3.25), which contains two terms R ijkℓ T ik T jℓ and (∇ j T ik )(∇ i T jk ). Using (2.27) and (3.7), the term ( By symmetry the term is equal to, interchanging i ↔ j and a ↔ b in the second term, which is zero. Similarly, we have, by interchanging m ↔ n and then i ↔ j, a ↔ b in the first term, Therefore, using the identity ϕ ijk ϕ k ab = g ia g jb − g ib g ja + ψ ijab (see [22]), we arrive at Since, by our convention, it follows that and (3.25) can be written as Finally, we deal with the last term J on the right-hand side of (3.26). From the identity ψ ijkℓ ψ ijkℓ = 168, we find that Plugging the expression for J into (3.26), we obtain Proposition 3.2. The scalar curvature R t or S t evolves by Since S t = 2 3 R t , it follows from the above theorem that (1.6) holds true.
Before giving local curvature estimates for Laplacian flow in the next subsection, we derive evolution equations for Ric t , Rm t , and T t in different forms. Using the Lichnerowicz Laplacian we see that the evolution equation for R ij can be written as But the first term is equal to Consequently, the norm of Ric t satisfies The general formula for R ℓ ijk gives Hence, the evolution equation for ||Rm t || 2 t is given by Moreover, it was proved in [33] that where C 1 is some universal constant, and for another universal constant C 2 which may differs from C 1 . The Cauchy-Schwartz inequality shows so that the evolution inequality (3.34) becomes Here C 3 is a universal constant.
Choose an open domain Ω of M and assume that on Ω × [0, T], Then the torsion T satisfies ||T|| K 1/2 and metrics g t are all equivalent to g 0 . We also observe from (2.25) and (3.19) that (3.37) ||Ric|| 1 ⇐⇒ |∆ϕ| 1 and the following simple fact for any tensor A.
Choose a Lipschitz function η with support in Ω and consider the quantity where p ≥ 5. As in [27], we introduce the following "good" quantities and also "bad" quantities We split the proof of Theorem 1.4 into four steps.
(a) In the first step, we can show that, see Lemma 3.3, In the second step, we can prove that the term c − ||T|| 2 ||Rm|| p−1 η 2p dV is bounded from above by (see (3.47)) Observe that the above integral is nonnegative, since the scalar curvature R is nonpositive along the Laplacian flow on closed G 2 -structures. Hence we obtain from the first step that, see Lemma 3.4, (c) In the next two steps, we estimate the bad terms B 1 and B 2 . In the third step, B 1 is estimated by (see (3.57)) Then the second step can be simplified as, see Lemma 3.5, (d) Finally, we estimate the term B 2 . In this step we shall use the assumption that p ≥ 5. Using the inequality ||∇T|| ||Rm|| and ||∇ 2 T|| ||∇Rm|| + ||Rm||||T|| + ||∇T|||T|| + ||T|| 3 , we can prove (see (3.67)) Plugging it into the third step, we arrive at, see Lemma 3.6, If we choose a geodesic ball Ω := B g 0 (x 0 , ρ/ √ K) and a cut-off function η so that ||∇φ|| ≤ √ Ke cKT /ρ, then the above inequality gives a proof of Theorem 1.4.
We are going to carry out the above mentioned four steps. From (3.39) and the above evolution equations, we have It was proved in [24] that the first integral in (3.39) is bounded by Since ||T|| 2 = −R, the same inequality holds for the integral To deal with the last term in the bracket of (3.39), we use the same argument of [24] to conclude According to the Cauchy-Schwartz inequality, the first and second integrals are bounded by Hence we obtain Using T = T * T and R = −||T|| 2 yields Hence, using (3.40), (3.41), and (3.42), we arrive at Lemma 3.3. One has In the following computations, we are mainly going to estimate or simplify the bad terms B 1 , B 2 , and also the term involving − ||T|| 2 . Integration by parts on the last integral in (3.43) and using R = −||T|| 2 , we obtain The first two integrals can be simplified by using the Cauchy-Schwarz inequality as follows: Using the identity, where p ≥ 5, Similarly, we can prove According to (3.44) we get c ||∇T|| 2 ||Rm|| p−1 η 2p dV Lemma 3.4. One has We next estimate B 1 and B 2 . Actually, we shall see that B 1 can be estimated in terms of B 2 . Hence the key step is to estimate B 2 . For B 1 , using we obtain From the estimates ∇||Ric|| 2 ||Ric||||∇Ric||, ∇||Rm|| p−1 ||Rm|| p−2 ||∇Rm||, Consider the term The three terms in the bracket can be estimated as follows. Firstly The same estimate holds for Finally, In the following, we estimate the left four terms in (3.49). We start from terms involving the scalar curvature.