Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=|∇T(x,t)|g(t)2+|Rm(x,t)|g(t)212\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$\end{document}will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda(x,t)}$$\end{document} will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda(x,t)}$$\end{document} remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.


Introduction
In this article we analyse the Laplacian flow for closed G 2 structures, which provides a potential tool for studying the challenging problem of existence of torsion-free G 2 structures, and thus Ricci-flat metrics with exceptional holonomy G 2 , on a 7dimensional manifold. We develop foundational results for the flow, both in terms of analytic and geometric aspects.
Although the existence theory of Joyce is powerful, it is a perturbative result and one has to work hard to find suitable initial data for the theory. In all known examples such data is always close to "degenerate", arising from a gluing procedure, and thus gives little sense of the general problem of existence of torsion-free G 2 structures. In fact, aside from some basic topological constraints, we have a primitive understanding of when a given compact 7-manifold could admit a torsion-free G 2 structure, and this seems far out of reach of current understanding. However, inspired by Joyce's work, it is natural to study the problem of deforming a closed G 2 structure, not necessarily with any smallness assumption on its torsion, to a torsion-free one, and to see if any obstructions arise to this procedure. A proposal to tackle this problem, due to Bryant (c.f. [Bry05]), is to use a geometric flow.
Geometric flows are important and useful tools in geometry and topology. For example, Ricci flow was instrumental in proving the Poincaré conjecture and the 1 4 -pinched differentiable sphere theorem, and Kähler-Ricci flow has proved to be a useful tool in Kähler geometry, particularly in low dimensions. In 1992, in order to study 7-manifolds admitting closed G 2 structures, Bryant (see [Bry05]) introduced the Laplacian flow for closed G 2 structures: where Δ ϕ ϕ = dd * ϕ + d * dϕ is the Hodge Laplacian of ϕ with respect to the metric g determined by ϕ and ϕ 0 is an initial closed G 2 structure. The stationary points of the flow are harmonic ϕ, which on a compact manifold are the torsion-free G 2 structures. The goal is to understand the long time behaviour of the flow; specifically, to find conditions under which the flow converges to a torsion-free G 2 structure. A reasonable conjecture (see [Bry05]), based on the work of Joyce described above, is that if the initial G 2 structure ϕ 0 on a compact manifold is closed and has sufficiently small torsion, then the flow will exist for all time and converge to a torsion-free G 2 structure.
Another motivation for studying the Laplacian flow comes from work of Hitchin [Hit00] (see also [BX]), which demonstrates its relationship to a natural volume functional. Letφ be a closed G 2 structure on a compact 7-manifold M and let [ will imply bounds on all the derivatives of the Riemann curvature: this was proved by Bando [Ban87] and comprehensively by Shi [Shi89] independently. The techniques used in [Ban87,Shi89] were introduced by Bernstein (in the early twentieth century) for proving gradient estimates via the maximum principle, and will also be used here in proving Theorem 1.2.
A key motivation for defining Λ(x, t) as in (1.2) is that the evolution equations of |∇T (x, t)| 2 and |Rm(x, t)| 2 both have some bad terms, but the chosen combination kills these terms and yields an effective evolution equation for Λ(x, t). We can then use the maximum principle to show that (1.4) satisfies a doubling-time estimate (see Proposition 4.1), i.e. Λ(t) ≤ 2Λ(0) for all time t ≤ 1 CΛ(0) for which the flow exists, where C is a uniform constant. This shows that Λ has similar properties to Riemann curvature under Ricci flow. Moreover, it implies that the assumption Λ(x, t) ≤ K in Theorem 1.2 is reasonable as Λ(x, t) cannot blow up quickly. We conclude Sect. 4 by giving a local version of Theorem 1.2.
In Sect. 5 we use our Shi-type estimates to study finite-time singularities of the Laplacian flow. Given an initial closed G 2 structure ϕ 0 on a compact 7-manifold, Theorem 1.1 tells us there exists a solution ϕ(t) of the Laplacian flow on a maximal time interval [0, T 0 ). If T 0 is finite, we call T 0 the singular time. Using our global derivative estimates (1.3) for Rm and ∇T , we can obtain the following long time existence result on the Laplacian flow.
where Λ(t) is given in (1.4). Moreover, we have a lower bound on the blow-up rate: Theorem 1.3 shows that the solution ϕ(t) of the Laplacian flow for closed G 2 structures will exist as long as the quantity Λ(x, t) in (1.2) remains bounded. We significantly strengthen this first long-time existence result in Theorem 1.6 below as a consequence of our compactness theory for the flow.
1.3 Uniqueness. In Sect. 6 we study uniqueness of the Laplacian flow, including both forward and backward uniqueness.
In Ricci flow, there are two standard arguments to prove forward uniqueness. One relies on the Nash-Moser inverse function theorem [Ham82] and another relies on DeTurck's trick and the harmonic map flow (see [Ham95]). Recently, Kotschwar [Kot14] provided a new approach to prove forward uniqueness. The idea in [Kot14] is to define an energy quantity E(t) in terms of the differences of the metrics, connections and Riemann curvatures of two Ricci flows, which vanishes if and only if the flows coincide. By deriving a differential inequality for E(t), it can be shown that E(t) = 0 if E(0) = 0, which gives the forward uniqueness.
In [Kot10], Kotschwar proved backward uniqueness for complete solutions to the Ricci flow by deriving a general backward uniqueness theorem for time-dependent sections of vector bundles satisfying certain differential inequalities. The method in [Kot10] is using Carleman-type estimates inspired by [Ale,WY12]. Recently, Kotschwar [Kot16] gave a simpler proof of the general backward uniqueness theorem in [Kot10].
Here we will use the ideas in [Kot10,Kot14] to give a new proof of forward uniqueness (given in [BX]) and prove backward uniqueness of the Laplacian flow for closed G 2 structures, as stated below. As an application of Theorem 1.4, we show that on a compact manifold M 7 , the subgroup I ϕ(t) of diffeomorphisms of M isotopic to the identity and fixing ϕ(t) is unchanged along the Laplacian flow. Since I ϕ is strongly constrained for a torsionfree G 2 structure ϕ on M , this gives a test for when the Laplacian flow with a given initial condition could converge.

Compactness.
In the study of Ricci flow, Hamilton's compactness theorem [Ham95] is an essential tool to study the behavior of the flow near a singularity. In Sect. 7, we prove an analogous compactness theorem for the Laplacian flow for closed G 2 structures.
Suppose we have a sequence (M i , ϕ i (t)) of compact solutions to the Laplacian flow and let where g i (t) is the associated metric to ϕ i (t), and let inj(M i , g i (0), p i ) denote the injectivity radius of (M i , g i (0)) at the point p i . Our compactness theorem then states that under uniform bounds on Λ ϕi and inj(M i , g i (0), p i ) we can extract a subsequence of (M i , ϕ i (t)) converging to a limit flow (M, ϕ(t)).
There exists a 7-manifold M , a point p ∈ M and a solution ϕ(t) of the Laplacian flow on M for t ∈ (a, b) such that, after passing to a subsequence, We refer to Sect. 7 for a definition of the notion of convergence in Theorem 1.5.
To prove Theorem 1.5, we first prove a Cheeger-Gromov-type compactness theorem for the space of G 2 structures (see Theorem 7.1). Given this, Theorem 1.5 follows from a similar argument for the analogous compactness theorem in Ricci flow as in [Ham95].
As we indicated, Theorem 1.5 could be used to study the singularities of the Laplacian flow, especially if we can show some non-collapsing estimate as in Ricci flow (c.f. [Per]) to obtain the injectivity radius estimate (1.6). Even without such an estimate, we can use Theorem 1.5 to greatly strengthen Theorem 1.3 to the following desirable result, which states that the Laplacian flow will exist as long as the velocity of the flow remains bounded. Theorem 1.6. Let M be a compact 7-manifold and ϕ(t), t ∈ [0, T 0 ), where T 0 < ∞, be a solution to the Laplacian flow (1.1) for closed G 2 structures with associated metric g(t) for each t. If the velocity of the flow satisfies then the solution ϕ(t) can be extended past time T 0 .
In Ricci flow, the analogue of Theorem 1.6 was proved in [Ses05], namely that the flow exists as long as the Ricci tensor remains bounded. It is an open question whether just the scalar curvature (the trace of the Ricci tensor) can control the Ricci flow, although it is known for Type-I Ricci flow [EMT11] and Kähler-Ricci flow [Zha10]. In Sect. 2.2, we see that for a closed G 2 structure ϕ, we have Δ ϕ ϕ = i ϕ (h), where i ϕ : S 2 T * M → Λ 3 T * M is an injective map defined in (2.2) and h is a symmetric 2-tensor with trace equal to 2 3 |T | 2 . Moreover, the scalar curvature of the metric induced by ϕ is −|T | 2 . Thus, comparing with Ricci flow, one may ask whether the Laplacian flow for closed G 2 structures will exist as long as the torsion tensor remains bounded. This is also the natural question to ask from the point of view of G 2 geometry. However, even though −|T | 2 is the scalar curvature, it is only first order in ϕ, rather than second order like Δ ϕ ϕ, so it would be a major step forward to control the Laplacian flow using just a bound on the torsion tensor.
1.5 Solitons. In Sect. 9, we study soliton solutions of the Laplacian flow for closed G 2 structures, which are expected to play a role in understanding the behavior of the flow near singularities, particularly given our compactness theory for the flow. Given a 7-manifold M , a Laplacian soliton of the Laplacian flow (1.1) for closed G 2 structures on M is a triple (ϕ, X, λ) satisfying where dϕ = 0, λ ∈ R, X is a vector field on M and L X ϕ is the Lie derivative of ϕ in the direction of X. Laplacian solitons give self-similar solutions to the Laplacian flow. Specifically, suppose (ϕ 0 , X, λ) satisfies (1.8). Define and let φ t be the family of diffeomorphisms generated by the vector fields X(t) such that φ 0 is the identity. Then ϕ(t) defined by is a solution of the Laplacian flow (1.1), which only differs by a scaling factor ρ(t) and pull-back by a diffeomorphism φ t for different times t. We say a Laplacian soliton (ϕ, X, λ) is expanding if λ > 0; steady if λ = 0; and shrinking if λ < 0.
Recently, there are several papers considering soliton solutions to flows of G 2 structures, e.g. [KMT12,Lin13,WeW12]. In particular, Lin [Lin13] studied Laplacian solitons as in (1.8) and proved there are no compact shrinking solitons, and that the only compact steady solitons are given by torsion-free G 2 structures.
A closed G 2 structure on a compact manifold which is stationary under the Laplacian flow must be torsion-free since here, unlike in the general non-compact setting, harmonic forms are always closed and coclosed. We show that stationary points for the flow are torsion-free on any 7-manifold and also give non-existence results for Laplacian solitons as follows. Combining Lin's [Lin13] result and the above proposition, any Laplacian soliton on a compact manifold M which is not torsion-free (if it exists) must satisfy (1.8) for λ > 0 and X = 0. This phenomenon is somewhat surprising, since it is very different from Ricci solitons Ric + L X g = λg: when X = 0, the Ricci soliton equation is just the Einstein equation Ric = λg and there are many examples of compact Einstein metrics.
Since a G 2 structure ϕ determines a unique metric g, it is natural to ask what condition the Laplacian soliton equation on ϕ will impose on g. We show that for a closed G 2 structure ϕ and any vector field X on M , we have Thus the symmetries of ϕ, namely the vector fields X such that L X ϕ = 0, are precisely given by the Killing vector fields X of g with d * (X ϕ) = 0 on M . Moreover, using (1.9) we can derive an equation for the metric g from the Laplacian soliton equation (1.8), which we expect to be of further use (see Proposition 9.4). In particular, we deduce that any Laplacian soliton (ϕ, X, λ) must satisfy 7λ + 3 div(X) = 2|T | 2 ≥ 0, which leads to a new short proof of the main result in [Lin13].
To conclude the paper in Sect. 10, we provide a list of open problems that are inspired by our work and which we intend to study in the future.

Closed G Structures
We collect some facts on closed G 2 structures, mainly based on [Bry05,Kar09]. The subgroup of GL(7, R) fixing φ is the exceptional Lie group G 2 , which is a compact, connected, simple Lie subgroup of SO(7) of dimension 14. Note that G 2 acts irreducibly on R 7 and preserves the metric and orientation for which {e 1 , e 2 , . . . , e 7 } is an oriented orthonormal basis. If * φ denotes the Hodge star determined by the metric and orientation, then G 2 also preserves the 4-form * φ φ = e 4567 + e 2367 + e 2345 + e 1357 − e 1346 − e 1256 − e 1247 .
Let M be a 7-manifold. For x ∈ M we let which is isomorphic to GL(7, R)/G 2 since φ has stabilizer G 2 . The bundle Λ 3 We call a section ϕ of Λ 3 + (M ) a positive 3-form on M and denote the space of positive 3-forms by Ω 3 + (M ). There is a 1-1 correspondence between G 2 structures (in the sense of subbundles of the frame bundle) and positive 3-forms, because given ϕ ∈ Ω 3 + (M ), the subbundle of the frame bundle whose fibre at x consists of invertible u ∈ Hom(T x M, R 7 ) such that u * φ = ϕ x defines a principal subbundle with fibre G 2 . Thus we usually call a positive 3-form ϕ on M a G 2 structure on M . The existence of G 2 structures is equivalent to the property that M is oriented and spin. GAFA We now see that a positive 3-form induces a unique metric and orientation. For a 3-form ϕ, we define a Ω 7 (M )-valued bilinear form B ϕ by where u, v are tangent vectors on M . Then ϕ is positive if and only if B ϕ is positive definite, i.e. if B ϕ is the tensor product of a positive definite bilinear form and a nowhere vanishing 7-form which defines a unique metric g with volume form vol g as follows: (2.1) The metric and orientation determines the Hodge star operator * ϕ , and we define ψ = * ϕ ϕ, which is sometimes called a positive 4-form. Notice that the relationship between g and ϕ, and hence between ψ and ϕ, is nonlinear. The group G 2 acts irreducibly on R 7 (and hence on Λ 1 (R 7 ) * and Λ 6 (R 7 ) * ), but it acts reducibly on Λ k (R 7 ) * for 2 ≤ k ≤ 5. Hence a G 2 structure ϕ induces splittings of the bundles Λ k T * M (2 ≤ k ≤ 5) into direct summands, which we denote by Λ k l (T * M, ϕ) so that l indicates the rank of the bundle. We let the space of sections Hodge duality gives corresponding decompositions of Ω 4 (M ) and Ω 5 (M ).
To study the Laplacian flow, it is convenient to write key quantities in local coordinates using summation convention. We write a k-form α as ..ik is totally skew-symmetric in its indices. In particular, we write ϕ, ψ locally as Note that the metric g on M induces an inner product of two k-forms α, β, given locally by As in [Bry05] (up to a constant factor), we define an operator i ϕ : where S 2 0 T * M denotes the bundle of trace-free symmetric 2-tensors on M . Clearly, i ϕ (g) = 3ϕ. We also have the inverse map j ϕ of i ϕ , which is an isomorphism between Λ 3 1 (T * M, ϕ) ⊕ Λ 3 27 (T * M, ϕ) and S 2 T * M . Then we have j ϕ (i ϕ (h)) = 4h + 2tr g (h)g for any h ∈ S 2 T * M and j ϕ (ϕ) = 6g.

Hodge Laplacian of ϕ.
Since dϕ = 0, from (2.8) and (2.9) we have that the Hodge Laplacian of ϕ is equal to where in the third equality we used τ ∧ ϕ = − * ϕ τ since τ ∈ Ω 2 14 (M ). In local coordinates, we write (2.17) as We can decompose Δ ϕ ϕ into three parts: where π k l : Ω k (M ) → Ω k l (M ) denotes the projection onto Ω k l (M ), a is a function, X is a vector field andh is a trace-free symmetric 2-tensor. We now calculate the values of a, X,h.
For a, we take the inner product of ϕ and Δ ϕ ϕ, and using the identity (2.16) (since τ ∈ Ω 2 14 (M )), where in the last equality we used |τ | 2 = 1 2 τ ij τ kl g ik g jl . For X, we use the contraction identities (2.4), (2.6), (2.7) and the definition of i ϕ : where the index of tensors are raised using the metric g. The last equality follows from the fact thath im is symmetric in i, m, but ϕ mil is skew-symmetric in i, m. Using (2.18), we have where in the above calculation we used (2.12), (2.15), (2.16) and the totally skewsymmetry in ϕ ijk and ψ ijkl . So X = 0 and thus the Ω 3 7 (M ) part of Δ ϕ ϕ is zero. To find h, using the decomposition (2.19), X = 0 and the contraction identities (2.4) and (2.5), we have (as in [GY09]) The left-hand side of the above equation can be calculated using (2.18): where we used (2.16) and that for closed We conclude that (2.21)

Ricci curvature and torsion.
Since ϕ determines a unique metric g on M , we then have the Riemann curvature tensor Rm of g on M . Our convention is the following: Recall that Rm satisfies the first Bianchi identity: We also have the following Ricci identities when we commute covariant derivatives of a (0, k)-tensor α: Karigiannis [Kar09] derived the following second Bianchi-type identity for the full torsion tensor.
Lemma 2.1. (2.24) Proof. The proof of (2.24) in [Kar09] is indirect, but as remarked there, (2.24) can also be established directly using (2.10)-(2.12) and the Ricci identity. We provide the detail here for completeness.
where in the third equality we used (2.10), (2.12) and (2.23), and in the fourth equality we used the contraction identity (2.4).
We now consider the Ricci tensor, given locally as R ik = R ijkl g jl , which has been calculated for closed G 2 structures (and more generally) in [Bry05,CI07,Kar09]. We give the general result from [Kar09] here.
Proposition 2.2. The Ricci tensor of the associated metric g of the G 2 structure ϕ is given locally as (2.25) In particular, for a closed G 2 structure ϕ, we have Proof. We multiply (2.24) by −ϕ jp k : where the last equality is due to (2.22). The formula (2.25) follows. For a closed G 2 structure, we have and Then we obtain which is (2.26).
We noted earlier that Rm and ∇T are second order in ϕ, and T is essentially ∇ϕ, so we would expect Rm and ∇T to be related. We show explicitly using Proposition 2.2 that, for closed G 2 structures, this is the case.
We can also deduce a useful, already known, formula for the scalar curvature of the metric given by a closed G 2 structure.
Corollary 2.5. The scalar curvature of a metric associated to a closed G 2 structure satisfies Proof. By taking trace in (2.26), using T ij = − 1 2 τ ij and (2.16), we obtain the scalar curvature This result is rather striking since it shows that the scalar curvature, which is a priori second order in the metric and hence in ϕ, is given by a first order quantity in ϕ when dϕ = 0.

J. D. LOTAY AND Y. WEI GAFA
Recall that the Laplacian flow for a closed G 2 structure is From (2.20) and (2.21), the flow (3.1) is equivalent to where h is the symmetric 2-tensor given in (2.21). We may write h in terms of the full torsion tensor T ij as follows: For closed ϕ, the Ricci curvature is equal to so we can also write h as Notice that T k i = T il g kl and T il = −T li . Throughout this section and the remainder of the article we will use the symbol Δ to denote the "analyst's Laplacian" which is a non-positive operator given in local coordinates as ∇ i ∇ i . This is in contrast to Δ ϕ , which is the Hodge Laplacian and is instead a non-negative operator.

Evolution of the metric. Under a general flow for G
where h(t), X(t) are a time-dependent symmetric 2-tensor and vector field on M respectively, it is well known that (see [Bry05,Joy00] and explicitly [Kar09]) the associated metric tensor g(t) evolves by Substituting (3.4) into this equation, we have that under the Laplacian flow (3.1) (also given by (3.2)), the associated metric g(t) of the G 2 structure ϕ(t) evolves by Thus the leading term of the metric flow (3.6) corresponds to the Ricci flow, as already observed in [Bry05].
From (3.6) we have that the inverse of the metric evolves by and the volume form vol g(t) evolves by where we used the fact that the scalar curvature R = −|T | 2 . Hence, along the Laplacian flow, the volume of M with respect to the associated metric g(t) will non-decrease; in fact, the volume form is pointwise non-decreasing (again as already noted in [Bry05]).

Evolution of torsion.
By [Kar09, Lemma 3.7], the evolution of the full torsion tensor T under the flow (3.2) is given by 2 (3.10) Using the contraction identity (2.5) and Ricci identity (2.23), the first term on the right hand side of (3.10) is equal to where we used ∇ m T mi = 0 in the fourth equality and the Bianchi identity (2.22) in the last equality. Using the contraction identity (2.6) and (2.10), we can calculate the second term on the right hand side of (3.10) as follows: where in the last equality we used T k m δ m k = 0 and T k m ϕ mq We can further simplify the above equations by noting that where we used the expression of Ricci tensor in (2.26). Therefore, we have The above evolution equation of the torsion tensor can be expressed schematically as where * indicates a contraction using the metric g(t) determined by ϕ(t).

Evolution of curvature.
To calculate the evolution of the Riemann curvature tensor we will use well-known general evolution equations. Recall that for any smooth one-parameter family of metrics g(t) on a manifold evolving by (3.14) for some time-dependent symmetric 2-tensor η(t), the Riemann curvature tensor, Ricci tensor and scalar curvature evolve by (see e.g. [CK04, Lemma 6.5]) where Δ L denotes the Lichnerowicz Laplacian The first six terms in the evolution equation come from the −2Ric term in (3.6). Then, as in Ricci flow, by applying Bianchi identities and commuting covariant GAFA derivatives, we can obtain We write the above equation schematically as in (3.13): Then from (3.7) and (3.18), noting that |T | 2 = −R ≤ C|Rm| for some universal constant C, we have Similarly, substituting (3.6) into (3.16) and (3.17), we obtain the evolution equation of the Ricci tensor and the evolution equation of the scalar curvature Remark 3.1. We shall only require the schematic evolution equations (3.13) and (3.18) for T and Rm to derive our Shi-type estimates. To obtain these equations we used the fact that ϕ remains closed under the evolution, which is a particular property of the Laplacian flow. If one is able to obtain the same schematic evolution equations for T and Rm for another flow of G 2 structures, then the methods of this article will apply more generally to give Shi-type estimates for that flow.

Derivative Estimates of Curvature and Torsion
In this section, we use the evolution equations derived in Sect. 3 to obtain global derivative estimates for the curvature tensor Rm and torsion tensor T . Throughout, we use * to denote some contraction between tensors and often use the same symbol C for a finite number of constants for convenience. First, we show a doubling-time estimate for Λ(t) defined in (1.4), which roughly says that Λ(t) behaves well and cannot blow up quickly.  (1.1) on a compact 7-manifold for t ∈ [0, ]. There exists a constant C such that Proof. We will calculate a differential inequality for Λ(x, t) given in (1.2), and thus for Λ(t) = sup x∈M Λ(x, t). Since we already have an evolution equation for |Rm| 2 in (3.19), it suffices to compute the evolution of |∇T | 2 .
Recall that for any smooth family of metrics g(t) evolving by (3.14), the Christoffel symbols of the Levi-Civita connection of g(t) evolve by Thus, for any time-dependent tensor A(t), we have the commutation formula (see The fact that the metric g is parallel gives that for any two tensors A, B, Then using (3.6), (3.13) and (4.1), we see that where in the last equality we used (2.10) and (2.12) in the form ∇ϕ = T * ψ, ∇ψ = T * ϕ, and we commuted covariant derivatives using the Ricci identity, i.e.
Then we can calculate the evolution of the squared norm of ∇T : where we used |T | 2 = −R ≤ C|Rm| for a constant C in the last inequality. Now, using (3.19) and (4.3), we obtain By Young's inequality, namely ab ≤ 1 2 a 2 + 2 b 2 for any > 0 and a, b ≥ 0, for all > 0 we have (4.7) The terms |Rm| 3 , |Rm||∇T | 2 and |∇T | 3 can all be bounded above by Λ 3 = (|Rm| 2 + |∇T | 2 ) 3 2 up to a multiplicative constant. Using this bound and substituting (4.5)-(4.7) into (4.4) we obtain for any > 0. Choosing so C ≤ 1 then yields The idea behind the calculations leading to (4.8) is that the negative gradient terms appearing in the evolution equations of |∇T | 2 and |Rm| 2 allow us to kill the remaining bad terms to leave us with an effective differential inequality. This is precisely the motivation for the definition Λ(x, t) in (1.2) as a combination of |∇T | and |Rm|.
Recall that Λ(t) = sup M Λ(x, t), which is a Lipschitz function of time t. Applying the maximum principle to (4.8), we deduce that in the sense of lim sup of forward difference quotients. We conclude that as long as t ≤ min{ , 2 We now derive Shi-type derivative estimates for the curvature tensor Rm and torsion tensor T along the Laplacian flow, using Λ(x, t) given in (1.2).

Theorem 4.2. Suppose that K > 0 and ϕ(t) is a solution to the Laplacian flow
Proof. The proof is by induction on k. The idea is to define a suitable function f k (x, t) for each k, in a similar way to the Ricci flow, which satisfies a parabolic differential inequality amenable to the maximum principle. For the case k = 1, we define for α to be determined later. To calculate the evolution of f , we first need to calculate the evolution of ∇Rm and ∇ 2 T . Using (3.6), (3.18) and (4.1), = ∇ΔRm + Rm * ∇Rm + ∇Rm * T * T + Rm * T * ∇T where in the last equality we used the commuting formula ∇ΔRm = Δ∇Rm + Rm * ∇Rm.
Then using (3.7), (4.12) and |T | ≤ C|Rm| Similarly, we can use (4.1) and (4.2) to obtain (4.14) where we use the symbols T 2 and T 3 here to mean contractions of two or three copies of T respectively, and again use |T | ≤ C|Rm| Using Young's inequality, we know that for all > 0 we have Substituting these bounds into (4.13) and (4.15), for suitably chosen small > 0 as before, then yields Then, from (4.8) and (4.16), we obtain By hypothesis Λ(t) = sup x∈M Λ(x, t) ≤ K and tK ≤ 1, so using the above inequality and Young's inequality to combine the middle three terms implies (4.17) We can choose α sufficiently large that C − α ≤ 0 and thus Note that f (x, 0) = α(|∇T | 2 + |Rm| 2 ) ≤ αK 2 , so applying the maximum principle to the above inequality implies that From the definition (4.11) of f , we obtain (4.10) for k = 1: Given this, we next prove k ≥ 2 by induction. It is clear that we need to obtain differential inequalities for |∇ k Rm| 2 and |∇ k+1 T | 2 , so this is how we proceed. Suppose (4.10) holds for all 1 ≤ j < k. From (4.1), for any time-dependent tensor A(t) we have (4.18) By (3.6), (3.18) and (4.18), we have where in the last equality we used the Ricci identity (4.20) J. D. LOTAY AND Y. WEI GAFA Using (4.19), the evolution of the squared norm of ∇ k Rm is: Applying (4.10) for 1 ≤ j < k to (4.21), we get where the constant C depends on the constants C j , 1 ≤ j < k in (4.10) and we used Young's inequality to estimate Similarly, we have (4.23) The second line of (4.23) can be estimated using the second line of (4.21). To estimate the third line of (4.23), for 2 ≤ i ≤ k + 1 we have (4.24) (4.28) Using (2.10) and (2.12), we can estimate ∇ i ψ. We see from (2.12) that Then from (2.10) and (2.12) we schematically have Using the same equations we see that

schematically, and thus by hypothesis
.
A straightforward induction then shows that for i ≥ 2 we have (4.29) Combining (4.24)-(4.29), using (4.10) for 0 ≤ j < k and the assumption tK ≤ 1, the third line of (4.23) can be estimated by where the last term arises from the estimated terms in (4.26), (4.28) and (4.29). We can estimate the last line of (4.23) similarly. We conclude that where we again used Young's inequality to estimate Combining (4.22) and (4.30), we have (4.31)

GAFA LAPLACIAN FLOW FOR CLOSED G 2 STRUCTURES 195
Using Young's inequality once again, we know that for any > 0 we have We deduce from these estimates and (4.31) that, by choosing > 0 sufficiently small (depending on C), we have (4.32) Given these calculations, we now define for some constants β k to be determined later and α k i = (k−1)! (k−i)! . Assuming (4.10) holds for all 1 ≤ i < k, then by a similar calculation to those leading to (4.32), we have where here we do not require the corresponding last term in (4.32), since by assumption (4.10) holds, so we have From (4.32) and (4.34), we may calculate Collecting terms we see that where we used the facts α k i (k − i) − α k i+1 = 0, Kt ≤ 1 and chose β k sufficiently large. Since f k (0) = β k α k k (|Rm| 2 + |∇T | 2 ) ≤ β k α k k K 2 , applying the maximum principle to (4.35) gives Then from the definition of f k , we obtain that This completes the inductive step and finishes the proof of Theorem 4.2.
From Proposition 4.1, we know the assumption Λ(x, t) ≤ K in Theorem 4.2 is reasonable, since Λ(x, t) cannot blow up quickly along the flow. Note that the estimate (4.10) blows up as t approaches zero, but the short-time existence result (Theorem 1.1) already bounds all derivatives of Rm and T for a short time. In fact, when Λ(x, t) ≤ K, from (4.16) we have for t ∈ [0, ] if sufficiently small. Using (4.22)-(4.23) and the maximum principle, we may deduce that such estimates also hold for higher order derivatives, so max Mt (|∇ k Rm| 2 + |∇ k+1 T | 2 ) is also bounded in terms of its initial value and K for a short time.
Remark 4.3. One can ask whether the growth of the constants C k in Theorem 4.2 can be controlled in terms of k. The authors show this is indeed the case in [LoWe] and as a consequence deduce that the Laplacian flow is real analytic in space for each fixed positive time.
We can also prove a local version of Theorem 4.2, stated below. Since we already established evolution inequalities for the relevant geometric quantities in the proof of Theorem 4.2, the proof just follows by applying a similar argument to Shi [Shi89] (see also [Ham95]) in the Ricci flow case, so we omit it. (4.36) Remark 4.5. By Proposition 2.4 and Corollary 2.5, we can bound |∇T | using bounds on |Rm|, and hence we can, if we wish, replace the bound on Λ in (1.2) in Theorems 4.2 and 4.4 by a bound on |Rm|.

Long Time Existence I
Given an initial closed G 2 structure ϕ 0 , there exists a solution ϕ(t) of Laplacian flow on a maximal time interval [0, T 0 ), where maximal means that either T 0 = ∞, or that T 0 < ∞ but there do not exist > 0 and a smooth Laplacian flowφ(t) for t ∈ [0, T 0 + ) such thatφ(t) = ϕ(t) for t ∈ [0, T 0 ). We call T 0 the singular time.
In this section, we use the global derivative estimates (1.3) for Rm and ∇T to prove Theorem 1.3, i.e. Λ(x, t) given in (1.2) will blow up at a finite time singularity along the flow. We restate Theorem 1.3 below.

Moreover, we have a lower bound on the blow-up rate,
for some constant C > 0.
where g(t) is the metric determined by ϕ(t). Then, in particular, we have the uniform curvature bound sup ≤ CK.
We fix a background metricḡ = g(0), the metric determined by ϕ(0). From (5.5) and the uniform equivalence of the metricsḡ and g(t), we have For any 0 < t 1 < t 2 < T 0 , which implies that ϕ(t) converges to a 3-form ϕ(T 0 ) continuously as t → T 0 . We may similarly argue using (3.6) and (5.4) that the uniformly equivalent Riemannian GAFA LAPLACIAN FLOW FOR CLOSED G 2 STRUCTURES 199 metrics g(t) converge continuously to a Riemannian metric g(T 0 ) as t → T 0 , since all the g(t) are uniformly equivalent toḡ. By (2.1), for each t ∈ [0, T 0 ) we have Let t → T 0 in (5.8). Recall that we have argued above that g(t) → g(T 0 ) which is a Riemannian metric and thus vol g(t) → vol g(T0) which is a volume form. Therefore the left hand side of (5.8) tends to a positive definite 7-form valued bilinear form. Thus, the right-hand side of (5.8) has a positive definite limit, and thus the limit 3-form ϕ(T 0 ) is positive, i.e. ϕ(T 0 ) is a G 2 structure on M . Moreover, note that dϕ(t) = 0 for all t means that the limit G 2 structure ϕ(T 0 ) is also closed. In summary, the solution ϕ(t) of the Laplacian flow for closed G 2 structures can be extended continuously to the time interval [0, T 0 ]. We now show that the extension is actually smooth, thus obtaining our required contradiction. We beginning by showing that we can uniformly bound the derivatives of the metric and 3-form with respect to the background Levi-Civita connection along the flow.

Claim 5.2. There exist constants
where ∇ is the Levi-Civita connection with respect toḡ.
Proof of Claim 5.3. We begin with m = 1. At any ( where we denote A = ∇ − ∇ as the difference of two connections, which is a tensor. Then in a fixed chart around x we have Integrating in time t, we get since t < T 0 is finite and |∇Ric| + |∇T ||T | is bounded by (4.10) and (5.4). Furthermore, we can derive from Claim 5.2 that For m ≥ 2, we can prove by induction that It then follows from (4.10), (5.11) and (5.13) that (5.14) Then Claim 5.3 follows from (5.14) by integration.
Now we continue the proof of Theorem 5.1. We have that a continuous limit of closed G 2 structures ϕ(T 0 ) exists, and in a fixed local coordinate chart U it satisfies Let α = (a 1 , . . . , a r ) be any multi-index with |α| = m ∈ N. By Claim 5.3 and (5.14), we have that ∂ m ∂x α ϕ ijk and are uniformly bounded on U × [0, T 0 ). Then from (5.15) we have that ∂ m ∂x α ϕ ijk (T 0 ) is bounded on U and hence ϕ(T 0 ) is a smooth closed G 2 structure. Moreover, and thus ϕ(t) → ϕ(T 0 ) uniformly in any C m norm as t → T 0 , m ≥ 2. Now, Theorem 1.1 gives a solutionφ(t) of the Laplacian flow (1.1) withφ(0) = ϕ(T 0 ) for a short time 0 ≤ t < . Since ϕ(t) → ϕ(T 0 ) smoothly as t → T 0 , this gives thatφ is a solution of (1.1) with initial valueφ(0) = ϕ(0) for t ∈ [0, T 0 + ), which is a contradiction to the maximality of T 0 . So we have lim sup t T0 (5.18) We now prove (5.1) by replacing the lim sup in (5.18) by lim. Suppose, for a contradiction, that (5.1) does not hold. Then there exists a sequence t i T 0 such that Λ(t i ) ≤ K 0 for some constant K 0 . By the doubling time estimate in Proposition 4.1, T0) Λ(x, t) ≤ 2K 0 , (5.20) but we already showed above that this leads to a contradiction to the maximality of T 0 . This completes the proof of (5.1). We conclude by proving the lower bound of the blow-up rate (5.2). Applying the maximum principle to (4.8) we have We already proved that lim Integrating (5.21) from t to t ∈ (t, T 0 ) and passing to the limit t → T 0 , we obtain This completes the proof of Theorem 5.1.
Combining Theorem 5.1 and Proposition 4.1 gives us the following corollary on the estimate of the minimal existence time.
Corollary 5.4. Let ϕ 0 be a closed G 2 structure on a compact manifold M 7 with

Uniqueness
In this section, we will use the ideas in [Kot10,Kot14]  Λ(x, t) + Λ(x, t) ≤ K 0 , (6.1) adopting the obvious notation for quantities determined by ϕ(t) andφ(t). By the Shi-type estimate (1.3), there is a constant K 1 depending on K 0 such that The uniform curvature bounds from (6.2) imply that g(t) andg(t) are uniformly equivalent on M × [0, ], so the norms | · | g(t) and | · |g (t) only differ by a uniform constant on M × [0, ]. We deduce the following from (6.2).
We will use this fact frequently in the following calculation. We continue to let A * B denote some contraction of two tensors A, B using g(t). We also recall that if ϕ(s) =φ(s) for some s ∈ [0, ], then the induced metrics also satisfy g(s) =g(s).  [Kot14], is to define an energy quantity E(t) by and show that E(t) satisfies a differential inequality which implies that E(t) vanishes identically if E(0) = 0 initially. Here in the definition (6.3) of E(t), In local coordinates, we have We begin by deriving inequalities for the derivatives of the quantities in the integrand defining E(t).
In the above inequalities, ∇, Δ and div are the Levi-Civita connection, Laplacian and divergence on M with respect to g(t) and C denotes uniform constants depending on K 1 given in (6.2).
Proof. We have the following basic facts: The above equations can be expressed schematically as This satisfies the estimate ∂ ∂t φ where we used the fact that | T | g(t) is bounded due to Lemma 6.1. We thus obtain the inequality (6.4). From the evolution equation (3.6) for the metric, we have in coordinates we obtain from (6.11) that whereŠ ik = S j ijk . Then (6.5) follows from (6.12) and Lemma 6.1. Recall that under the evolution (3.6) of g(t), the connection evolves by where schematically which gives (6.6). From the evolution equation (3.13) of T , we have Noting that |ψ − ψ| ≤ C|φ − ϕ| = C|φ|, we see that (6.7) follows from the evolution equation for U . We next compute the evolution of V using (4.2). We start by seeing that The second terms from (4.2) give schematically that Similarly, the third and fourth terms from (4.2) yield and We now observe that and, by virtue of (6.10), the last term is given schematically as (6.14) Hence, the fifth terms in (4.2) give The sixth terms in (4.2) yield For the remaining terms in (4.2) we observe that Altogether, we find the evolution equation for V : We thus obtain (6.8) as claimed. Finally, we compute the evolution of S using the evolution (3.18) for Rm: where we used We thus obtain (6.9) as required.
We now use Lemma 6.2 to obtain a differential inequality for E(t).
Lemma 6.3. The quantity E(t) defined by (6.3) satisfies where C is a uniform constant depending only on K 0 given in (6.1).
Proof. Under the curvature and torsion bounds (6.2), the evolution equations of the metric (3.6) and volume form (3.8) imply ≤ C. (6.16) For any tensor P (t) we therefore have Hence, We also observe that, by integration by parts, we have and, if P(t) is another tensor, Using Lemma 6.2, including the estimates for V and S, we may calculate: The second term is clearly bounded by CE(t). Now we use the negative third integral in the inequality to crucially cancel the terms involving ∇V and ∇S arising from GAFA LAPLACIAN FLOW FOR CLOSED G 2 STRUCTURES 209 the fourth and fifth integrals via Young's inequality. Concretely, for any > 0, we have so by choosing sufficiently we obtain as claimed.
The forward uniqueness property in Theorem 1.4 now follows immediately from Lemma 6.3. If ϕ(s) =φ(s) for some s ∈ [0, ], then E(s) = 0. Thus for t ∈ [s, ], we can integrate the differential inequality in Lemma 6.3 to obtain and that the Ricci curvature of the metric g(t) is bounded below by a uniform constant, i.e. Ric(g(t)) ≥ −Kg(t) for some K ≥ 0. Let X and Y be finite direct sums of the bundles T k l (M ), and

(t) is the Laplacian with respect to g(t) acting on tensors. Then
Suppose ϕ(s) =φ(s) for some s ∈ [0, ]. For our purpose, we let where φ, h, A, U, V, S are defined as in Sect. 6.1 and Then To be able to apply Theorem 6.4, we need to show that X(t), Y(t) defined in (6.19)-(6.20) satisfy the system of differential inequalities (6.17)-(6.18). We begin with the following. Proof. At the beginning of this section, we argued that the metrics g(t) andg(t) are uniformly equivalent on M × [0, ]. We immediately deduce that |h(t)| g(t) = |g(t) −g(t)| g(t) is bounded. From (6.2) and the uniform equivalence of g(t) andg(t), we further have are bounded on M × [0, ]. Recall |T | 2 g = −R, where R is the scalar curvature of g. Thus we also have that |U | g(t) = |T −T | g(t) is bounded on M × [0, ].
Since ϕ(s) =φ(s) for some s ∈ [0, ], we have Since g(t) andg(t) are uniformly equivalent on M × [0, ], we know that Hence, by virtue of (2.14) and (2.18), we have Therefore, by (6.2) and the fact that s, t ∈ [0, ], there is a uniform constant C depending on K 1 such that Finally, we show A, B are bounded on M × [0, ]. Since A(s) = 0, we have du .
In (6.13) we showed that Thus, by the uniform equivalence of g(t) andg(t) and (6.2), we have a uniform constant C depending on K 1 such that Similarly, we can bound We derived the evolution equations of φ, h, A, U, V, S in Sect. 6.1, so now we compute the evolutions of B, W, Q.
Lemma 6.6. We have the following estimates on the evolution of B, W, Q: (6.23) Proof. Since A, as a difference of connections, is a tensor, (4.1) gives Since g is uniformly bounded and ∇Rm, T and ∇T are uniformly bounded in light of (6.2), we immediately deduce from the evolution equation (3.6) for g that For the first term we observe from (6.10) that ∇h and ∇g −1 are bounded by C|A| as well sinceg andg −1 are uniformly bounded by the uniform equivalence ofg and g and Lemma 6.1. Using these observations together with the uniform boundedness of derivatives of Rm, Rm, T , T by (6.2), Lemma 6.1 and the boundedness of A by Lemma 6.5, we may apply ∇ to the evolution equation (6.13) for A to deduce that ).
(Note that there is no ∇S term since ∇g = 0.) We then observe that This gives the inequality (6.21). The inequalities (6.22) and (6.23) follow from similar calculations using (4.12) and (4.14).

Recall the elementary inequality
By taking the divergence of (6.14) and using the uniform boundedness ofg −1 , derivatives of T and A by Lemmas 6.1 and 6.5, we have ).
We deduce from these observations and the evolution equation (6.8) for V that We now observe that by taking the divergence of (6.15) we have an estimate for div S: ).
Hence, using the evolution equation (6.9) for S together with the above estimate, we have: Recall the definition of X(t) and Y(t) in (6.19). We see from Lemma 6.6, (6.24) and (6.25) we have estimates of the form for P = V, W, S, Q. Moreover, we have from Lemma 6.2 that ≤ C(|X(t)| 2 g(t) + |∇X(t)| 2 g(t) + |Y(t)| 2 g(t) ), GAFA and we also observe that Hence, X(t) satisfies (6.17) in Theorem 6.4. Similarly, from Lemma 6.2 and Lemma 6.6, we have estimates of the form for P = φ, h, A, B. Thus, Y(t) satisfies (6.18) in Theorem 6.4. Overall, since M is compact and we have the estimates (6.2), we have demonstrated that all of the conditions in Theorem 6.4 are satisfied.

Applications.
We finish this section with two applications of Theorem 1.4; specifically, to the isotropy subgroup of the G 2 structure under the flow, and to solitons.
Let M be a 7-manifold and let D be the group of diffeomorphisms of M isotopic to the identity. For a G 2 structure ϕ on M , we let I ϕ denote the subgroup of D fixing ϕ. We now study the behaviour of I ϕ under the Laplacian flow. Proof. Let Ψ ∈ I ϕ(0) andφ(t) = Ψ * ϕ(t). Thenφ(t) is closed for all t and ∂ ∂tφ (t) = Ψ * ∂ ∂t ϕ(t) = Ψ * Δ ϕ(t) ϕ(t) = Δ Ψ * ϕ(t) Ψ * ϕ(t) = Δφ (t)φ (t), soφ(t) is also a solution to the flow (1.1). Sinceφ(0) = Ψ * ϕ(0) = ϕ(0) as Ψ ∈ I ϕ(0) , the forward uniqueness in Theorem 1.4 implies thatφ(t) = ϕ(t) for all t ∈ [0, ]. Thus, Ψ ∈ I ϕ(t) for all t ∈ [0, ]. Similarly, using the backward uniqueness in Theorem 1.4, we can show if s ∈ [0, ] and Ψ ∈ I ϕ(s) , then Ψ ∈ I ϕ(t) for all t ∈ [0, s]. Therefore, for all t ∈ [0, ], I ϕ(0) ⊂ I ϕ(t) ⊂ I ϕ(0) , which means I ϕ(t) = I ϕ(0) . Irreducible compact G 2 manifolds (M, ϕ) cannot have continuous symmetries and so I ϕ is trivial. Since the symmetry group I ϕ is not expected to become smaller at an infinite time limit, Corollary 6.7 suggests an immediate test on a closed G 2 GAFA LAPLACIAN FLOW FOR CLOSED G 2 STRUCTURES 215 structure ϕ 0 to determine when the Laplacian flow starting at ϕ 0 can converge to an irreducible torsion-free G 2 structure. We can also use Theorem 1.4 in a straightforward way to deduce the following result, which says that any Laplacian flow satisfying the Laplacian soliton equation at some time must in fact be a Laplacian soliton.

Compactness
In this section, we prove a Cheeger-Gromov-type compactness theorem for solutions to the Laplacian flow for closed G 2 structures.

Compactness for G 2 structures.
We begin by proving a compactness theorem for the space of G 2 structures. Let M i be a sequence of 7-manifolds and let p i ∈ M i for each i. Suppose that ϕ i is a G 2 structure on M i for each i such that the associated metrics g i on M i are complete. Let M be a 7-manifold with p ∈ M and let ϕ be a G 2 structure on M . We say that We may thus give our compactness theorem for G 2 structures. Proof. In the proof we always use the convention that, after taking a subsequence, we will continue to use the index i.
By the Cheeger-Gromov compactness theorem [Ham95, Theorem 2.3] for complete pointed Riemannian manifolds, there exists a complete Riemannian 7-manifold (M, g) and p ∈ M such that, after passing to a subsequence, (7. 2) The convergence in (7.2) means that, as above, there exist nested compact sets Ω i ⊂ M exhausting M with p ∈ int(Ω i ) for all i and diffeomorphisms F i : smoothly as i → ∞ on any compact subset of M .
Fix i sufficiently large. For j ≥ 0 we have Ω i ⊂ Ω i+j and a diffeomorphism F i+j : Ω i+j → F i+j (Ω i+j ) ⊂ M i+j . We can then define a restricted diffeomorphism The convergence (7.2) implies that the sequence {g i,j = F * i,j g i+j } ∞ j=0 of Riemannian metrics on Ω i converges to g i,∞ = g on Ω i as j → ∞.
Let ∇, ∇ gi,j be the Levi-Civita connections of g, g i,j on Ω i respectively. As before, let h = g − g i,j and A = ∇ − ∇ gi,j be the difference of the metrics and their connections, respectively. It is straightforward to see locally that Since g i,j → g smoothly on Ω i as j → ∞, g i,j and g are equivalent for sufficiently large j, and |∇ k h| g tends to zero as j → ∞ for all k ≥ 0. Hence, A is uniformly bounded with respect to g for all large j. Moreover, Thus there exist constants c k for k ≥ 0 such that |∇ k A| g ≤ c k for all j ≥ 0. Using each diffeomorphism F i,j , we can define a G 2 structure ϕ i,j = F * i,j ϕ i+j on Ω i by pulling back the G 2 structure ϕ i+j on M i+j . We next estimate |∇ k ϕ i,j | g . First, since g and g (i,j) are all equivalent for large j, |ϕ i,j | g ≤ c 0 |ϕ i,j | gi,j ≤ 7c 0 =c 0 for some constants c 0 ,c 0 . We next observe trivially that ∇ϕ i,j = ∇ gi,j ϕ i,j + (∇ − ∇ gi,j )ϕ i,j , GAFA LAPLACIAN FLOW FOR CLOSED G 2 STRUCTURES 217 so, since A is uniformly bounded, there is a constantc 1 such that |∇ϕ i,j | g ≤ c 0 |∇ gi,j ϕ i,j | gi,j + C|A| g |ϕ i,j | g ≤c 1 .
For k ≥ 2, we have the estimate By an inductive argument, using the estimate |∇ k A| g ≤ c k and the assumption (7.1), we can show the existence of constantsc k for k ≥ 0 such |∇ k ϕ i,j | g ≤c k on Ω i for all j, k ≥ 0. The Arzelà-Ascoli theorem (see, e.g. [AH11, Corollary 9.14]) now implies that there exists a 3-form ϕ i,∞ and a subsequence of ϕ i,j in j, which we still denote by ϕ i,j , that converges to ϕ i,∞ smoothly on Ω i , i.e. |∇ k (ϕ i,j − ϕ i,∞ )| g → 0 as j → ∞ (7.3) uniformly on Ω i for all k ≥ 0. Since each ϕ i,j is a G 2 structure on Ω i with associated metric g i,j , the 7-form valued bilinear form Since the Cheeger-Gromov compactness theorem guarantees the limit metric g i,∞ = g is a Riemannian metric on Ω i , (7.5) implies that ϕ i,∞ is a positive 3-form and hence defines a G 2 structure on Ω i with associated metric g i,∞ = g.
We now denote the inclusion map of Ω i into Ω k for k ≥ i by