Local curvature estimates for the Laplacian flow

In this paper we give local curvature estimates for the Laplacian flow on closed G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{2}$$\end{document}-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{2}$$\end{document}-structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) 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 G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{2}$$\end{document}-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature Rg(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{g(t)}$$\end{document} is equal to the Laplacian of Rg(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{g(t)}$$\end{document}, plus an extra term which can be written as the difference of two nonnegative quantities.


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 2-tensors. For any 2-tensor A = A i j dx i⊗ j , define A := 1 2 ( A i j + A ji )dx i⊗ j ≡ A i j dx i⊗ j and A ∧ := 1 2 ( A i j − A ji )dx i⊗ j ≡ A ∧ i j dx i⊗ j . 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 i j dx i∧ j . Define α A := 1 2 α A i j dx i∧ j with α A i j := A i j . 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 i j dx j ≡ X j dx j and g (α) = α i g i j ∂ ∂ 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 k-forms 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 i j dx i⊗ j and B = B i j dx i⊗ j . Define A, B g := A i j B i j . There are two special cases which will be used later: 1 In our convention, for any 2-form α = 1 2 α i j dx i j , we have which justifies the notion α k as α(∂/∂ x k , ∂/∂x ). In general, for any k-form α = 1 k! α i 1 ···i k dx i 1 ∧···∧i k we have α i 1 ···i k = α(∂/∂ x i 1 , · · · , ∂/∂x i k ), because dx i 1 ∧···∧i k = σ ∈S k sgn(σ )dx i σ (1) ⊗···⊗i σ (k) .
(1) α = 1 2 α i j dx i∧ j ∈ ∧ 2 (M) and B = B i j dx i⊗ j ∈ ⊗ 2 (M). In this case, α can be written as a 2-tensor A α = A α i j dx i⊗ j with A α i j = α i j . Then α, B g := A α , B g = α i j B i j . (2) α = 1 2 α i j dx i∧ j and β = 1 2 β i j dx i∧ j ∈ ∧ 2 (M). In this case, α, β can be both written as and ||α|| 2 g = 2|α| 2 g , for any vector field X ∈ X(M) and 2-form α. 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 Ricci curvature of g is given by R jk := R i jk g i . 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.

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). [2]) For a compact 7-manifold M, the initial value problem (1.1) has a unique solution for a short time interval [0, T max ) with the maximal time T max ∈ (0, ∞] depending on ϕ.

Theorem 1.1 (Bryant-Xu
As in the Ricci flow, we can prove following results on the long time existence for the Laplacian flow (1.1). [32]) 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
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 [25] and the structure of the Eq. (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 Then sup where the bound depends only on n, K , T and .
When M is compact, the theorem immediately implies the part (a) in Theorem 1.2. Indeed, we shall show that [see (3.10) and (3.29) In the compact case, Theorem 1.3 shows that, if the conclusion in part (a) does not hold, then (3.58)]. However, by part (b) in Theorem 1.2, it is impossible. Therefore, the conclusion in part (a) is true.
As remarked in [25], 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 B g(0) (x 0 , A/ √ K ) is compactly contained in M and that 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 [25] by noting that (1.5) To verify (1.5), we use (2.26), (3.56) and (3.60) to deduce that and . Then, by (3.23) 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 [17] (for part (b)) and Sesum [37] (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 [40] or type-I Ricci flow [11], 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 [32] (see p. 171, line -6 to -3, or Open Problem (3) in p. 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 | · | g(t) in (1.6). The standard formula for the scalar curvature R g (t) gives [see (3.15)] Now the above mentioned open problem states that Is it ture that lim sup The "minus infinity" comes from the fact that along the Laplacian flow on closed G 2structures 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 Eq. (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 (3.15)], we can prove the following interesting evolution equation for R g(t) .

Proposition 1.5
Let M be a smooth 7-manifold and ϕ(t), t ∈ [0, T ), where T ∈ (0, ∞], be a solution to the flow (1.1) for closed G 2 -structures with associated metric g(t) = g ϕ(t) for each t. Then the scalar curvature R g(t) satisfies Here The evolution Eq. (1.8) can be written simply as for some suitable time-dependent nonnegative functions A(t) and B(t). By the maximum principle we obtain Here R max (0) := max M R g(0) and R min (0) := min M R g(0) . Observe that the above wellarranged evolution equation can give us a weakly lower bound for R g(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 Sect. 2 about G 2structures, G 2 -decompositions of 2-forms and 3-forms, and general flows on G 2 -structures. In Sect. 3, we rewrite results in Sect. 2 for closed G 2 -structures, and the local curvature estimates will be given in the last subsection.

Basic theory of G -structures
In this section, we view some basic theory of G 2 -structures, following [1,[20][21][22][23]32]. Let {e 1 , . . . , e 7 } denote the standard basis of R 7 and let {e 1 , . . . , e 7 } be its dual basis. Define the 3-form where e i∧ j∧k := e i ∧ e j ∧ e k . The subgroup G 2 , which fixes φ, of GL(7, R) is the 14dimensional Lie subgroup of SO(7), acts irreducibly on R 7 , and preserves the metric and orientation for which {e 1 , · · · , e 7 } is an oriented orthonormal basis. Note that G 2 also preserves the 4-form where the Hodge star operator * φ is determined by the metric and orientation.
When M is compact, the above theorem says that a G 2 -structure ϕ is torsion-free if and only if ϕ is harmonic with respect to the induces metric g ϕ .
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.

G 2 -decompositions of ∧ 2 (M) and ∧ 3 (M)
A G 2 -structure ϕ induces splittings 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 We mainly focus on the G 2 -decompositions of ∧ 2 (M) and ∧ 3 (M). Recall that Here each component is determined by For any 2-form β = 1 2 β i j dx i∧ j ∈ ∧ 2 (M), its two components π 2 7 (β) and π 2 14 (β) are determined by To decompose 3-forms, recall two maps introduce by Bryant [1] Then i ϕ is injective and is isomorphic onto for some symmetric 2-tensor h ∈ 2 (M) and vector field X ∈ X(M). Then . (2.11)

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] It can be proved that τ 1 = τ 1 (see [23]). 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 τ ϕ : 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, Consequently, there exists a 2-tensor T = T i j dx i⊗ j , called the full torsion tensor, such that ∇ ϕ = T n ψ nabc . (2.14) Equivalently, The associated 2-tensor τ 3 := (τ 3 ) i j dx i⊗ j of τ 3 lies in the space 2 0 (M). With this convenience, the full torsion tensor T m is determined by or as 2-tensors, 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 where Then, according to (2.5) and (2.6) Karigiannis [23] (see also the equivalent formula obtained by Bryant in [1]) proved that the Ricci curvature is given by Cleyton and Ivanov [6] also derived a formula for the Ricci tensor for closed G 2 -structures in terms of d * ϕ ϕ. Taking the trace of (2.23), we obtain Btyant's formula [1] for the scalar curvature (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 [23].

General flows on G 2 -structures
For any family ϕ(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 (2.36) locally can be written as We write for g(t) and dV g(t) the metric and volume form associated to ϕ(t), respectively.

Theorem 2.3 Under the general flow (2.36), we have
These evolution equations can be found in [23].

Laplacian flows on closed G 2 -structures
We now consider the Laplacian flow for closed G 2 -structures is the Hodge Laplacian of g ϕ(t) and ϕ is an initial closed G 2 -structure. The short time existence for (3.1) on compact manifolds 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 manifolds was given in Theorem 1.2. In this section, we give a new elementary proof of Lotay-Wei's result in compact case.

Evolution equations along the Laplacian flow
Since the Laplacian flow (3.1) preserves the closedness of ϕ(t), it follows from (3.10) that we have where From Theorem 2.3, we see that the associated metric tensor g(t) evolves by and the volume form dV g(t) evolves by Hence, along the flow (3.1), the volume of g(t) is nondecreasing. Introduce the following notions where g(t) := g i j ∇ i ∇ j is the usual Laplacian of g(t) and g(t) is the Hodge Laplacian of g(t), and also the 2-tenor Sic g(t) with components Then the evolution Eq. (3.4) can be written as The trace of Sic g(t) is exactly the scalar curvature, up to a multiplying constant, It was proved in [32] that This identity together with (2.26) shows that the boundedness of g(t) ϕ(t) is equivalent to the boundedness of Ric g(t) . The evolution Eq. (2.41) implies that for the Laplacian flow on closed G 2 -structures, the torsion T i j evolves by evolves Furthermore, we can prove
For a geometric flow ∂ t g i j = η i j , where η i j is a family of symmetric 2-tensors, we have (e.g. see formula (2.66), (2.29), and (2.30) in [5]) where (div g(t) η(t)) j = ∇ i η i j . Applying those evolution equations to where the symmetric 2-tensor T (t) is given by Plugging those identities into the above evolution equation for R g(t) , we get (3.14) 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 (2.29) and (2.30), to simplify those two terms. Using the identity (3.15), the term ∇ j ∇ i T i j can be simplified as follows.
On the other hand, from the Ricci identity we see that the evolution Eq. (3.14) is equivalent to From (3.7) and (3.13) we can rewrite the term ||Ric g(t) || 2 g(t) in (3.16) in terms of Sic g(t) according to the following relation: where we used the identities tr g(t) Replacing R g(t) by S g(t) according to the identity (3.9), we can rewrite (3.16) as

Similarly, replacing Ric g(t) , T (t) g(t) by Sic g(t) , T (t) g(t) with respect to the identity
we obtain the following evolution equation for S g(t) , (3.17) Next, we try to deal with the last bracket in (3.17), which contains two terms R i jk T ik T j and (∇ j T ik )(∇ i T jk ). Using (2.27) and (2.33), 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 i jk ϕ k ab = g ia g jb − g ib g ja + ψ i jab (see [23]), we arrive at Since, by our convention, and it follows that and (3.17) can be written as Finally, we deal with the last term J on the right-hand side of (3.18). From the identity ψ i jk ψ i jk = 168, we find that − 168 Plugging the expression for J into (3.18), we obtain

Proposition 3.2 The scalar curvature R g(t) or S g(t) evolves by
. (3.19) Since S g(t) = 2 3 R g(t) , it follows from the above theorem that (1.8) holds true. Before giving local curvature estimates for Laplacian flow in the next subsection, we derive evolution equations for Ric g(t) , Rm g(t) , and T (t) in different forms. Using the Lichnerowicz Laplacian we see that the evolution equation for R i j can be written as But the first term is equal to we have . (3.20) Consequently, the norm of Ric g(t) satisfies 28 Page 18 of 37 (3.21) The general formula (e.g. formula (2.66) in [5]) for R i jk gives Hence, the evolution equation for ||Rm g(t) || 2 g(t) is given by Moreover, it was proved in [32] that where C 1 is some universal constant, and Squaring (3.25) gives for another universal constant C 2 which may differs from C 1 . The Cauchy-Schwartz inequality shows , so that the evolution inequality (3.26) becomes

27)
Here C 3 is a universal constant.

Main idea of proving Theorem 1.4
In this section, we consider the Laplacian flow (3.1) on M × [0, T ], where T ∈ (0, T max ). From now on we always omit the time subscripts from all considered quantities. From (3.7), (3.21), (3.23), (3.24), and (3.27) we have Choose an open domain of M and assume that on × [0, T ], Then the torsion T satisfies 2 ||T || K 1/2 and metrics g(t) are all equivalent to g(0). We also observe from (2.25) and (3.11) that ||Ric|| 1 ⇐⇒ | ϕ| 1 (3.29) and the following simple fact for any tensor A. Choose a Lipschitz function η with support in (and independent of time t) and consider the quantity d dt where p ≥ 5. As in [28], we introduce the following "good" quantities and also "bad" quantities We split the proof of Theorem 1.4 into four steps.