Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G_2 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 $\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ 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). (5). Finally, we study compact 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 7-dimensional manifold. We develop foundational results for the flow, both in terms of analytic and geometric aspects.
1.1. Basic theory. Let M be a 7-manifold. A G 2 structure on M is defined by a 3-form ϕ on M satisfying a certain nondegeneracy condition. To any such ϕ, one associates a unique metric g and orientation on M , and thus a Hodge star operator * ϕ . If ∇ is the Levi-Civita connection of g, we interpret ∇ϕ as the torsion of the G 2 structure ϕ. Thus, if ∇ϕ = 0, which is equivalent to dϕ = d * ϕ ϕ = 0, we say ϕ is torsion-free and (M, ϕ) is a G 2 manifold.
The key property of torsion-free G 2 structures is that the holonomy group of the associated metric satisfies Hol(g) ⊂ G 2 , and hence (M, g) is Ricci-flat. If (M, ϕ) is a compact G 2 manifold, then Hol(g) = G 2 if and only if π 1 (M ) is finite, and thus finding torsion-free G 2 structures is essential for constructing compact manifolds with holonomy G 2 . Notice that the torsion-free condition is a nonlinear PDE on ϕ, since * ϕ depends on ϕ, and thus finding torsion-free G 2 structures is a challenging problem.
Bryant [4] used the theory of exterior differential systems to first prove the local existence of holonomy G 2 metrics. This was soon followed by the first explicit complete holonomy G 2 manifolds in work of Bryant-Salamon [7]. In ground-breaking work, Joyce [22] developed a fundamental existence theory for torsion-free G 2 structures by perturbing closed G 2 structures with "small" torsion which, together with a gluing method, led to the first examples of compact 7-manifolds with holonomy G 2 . This theory has formed the cornerstone of the programme for constructing compact holonomy G 2 manifolds, of which there are now many examples (see [13,29]).
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. Such data is always close to "degenerate", arising from a gluing procedure in known examples, 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 7manifold 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. [5]), 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 [5]) introduced the Laplacian flow for closed G 2 structures:    ∂ ∂t ϕ = ∆ ϕ ϕ, dϕ = 0, ϕ(0) = ϕ 0 , where ∆ 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 [5]), 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.
We note that there are other proposals for geometric flows of G 2 structures in various settings, which may also potentially find torsion-free G 2 structures (e.g. [16,24,36]). The study of these flows is still in development.
An essential ingredient in studying the Laplacian flow (1.1) is a short time existence result: this was claimed in [5] and the proof given in [8].
Theorem 1.1. For a compact 7-manifold M , the initial value problem (1.1) has a unique solution for a short time t ∈ [0, ǫ) with ǫ depending on ϕ 0 .
To prove Theorem 1.1, Bryant-Xu showed that the flow (1.1) is (weakly) parabolic in the direction of closed forms. This is not a typical form of parabolicity, and so standard theory does not obviously apply. It is also surprising since the flow is defined by the Hodge Laplacian (which is nonnegative) and thus appears at first sight to have the wrong sign for parabolicity. Nonetheless, the theorem follows by applying DeTurck's trick and the Nash-Moser inverse function theorem.
This short time existence result naturally motivates the study of the long time behavior of the flow. Here little is known, apart from a compact example computed by Bryant [5] where the flow exists for all time but does not converge, and recently, Fernández-Fino-Manero [15] constructed some noncompact examples where the flow converges to a flat G 2 structure.
1.2. Shi-type estimates. After some preliminary material on closed G 2 structures in §2 and deriving the essential evolution equations along the flow in §3, we prove our first main result in §4: Shi-type derivative estimates for the Riemann curvature and torsion tensors along the Laplacian flow.
For a solution ϕ(t) of the Laplacian flow (1.1), we define the quantity Λ(x, t) = |∇T (x, t)| 2 g(t) + |Rm(x, t)| 2 where T is the torsion tensor of ϕ(t) (see §2 for a definition) and Rm denotes the Riemann curvature tensor of the metric g(t) determined by ϕ(t). We show that a bound on Λ(x, t) will induce a priori bounds on all derivatives of Rm and ∇T for positive time. More precisely, we have the following. We call the estimates (1.3) Shi-type (perhaps, more accurately, Bernstein-Bando-Shi) estimates for the Laplacian flow, because they are analogues of the well-known Shi derivative estimates in the Ricci flow. In Ricci flow, a Riemann curvature bound will imply bounds on all the derivatives of the Riemann curvature: this was proved by Bando [3] and comprehensively by Shi [34] independently. The techniques used in [3,34] 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 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 §4 by giving a local version of Theorem 1.2.
In §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 blowup 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.

1.3.
Uniqueness. In §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 [18] and another relies on DeTurck's trick and the harmonic map flow (see [19]). Recently, Kotschwar [27] provided a new approach to prove forward uniqueness. The idea in [27] 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 [26], 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 [26] is using Carleman-type estimates inspired by [1,38]. Recently, Kotschwar [28] gave a simpler proof of the general backward uniqueness theorem in [26].
Here we will use the ideas in [26,27] to give a new proof of forward uniqueness (given in [8]) 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 torsion-free G 2 structure ϕ on M , this gives a test for when the Laplacian flow with a given initial condition could converge. 1.4. Compactness. In the study of Ricci flow, Hamilton's compactness theorem [20] is an essential tool to study the behavior of the flow near a singularity. In §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 p i ∈ M i . For each (M i , ϕ i (t)), 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 ϕ i 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)).
Theorem 1.5. Let M i be a sequence of compact 7-manifolds and let p i ∈ M i for each i. Suppose that, for each i, ϕ i (t) is a solution to the Laplacian flow 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 §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 [20].
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. [32]) to obtain the injectivity radius estimate (1.6). Even without such an estimate, we can use Theorem 1.5 to 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 [33], 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 [14] and Kähler-Ricci flow [39]. In §2.2, we see that for a closed G 2 structure ϕ, we have ∆ ϕ ϕ = i ϕ (h), where i ϕ : S 2 T * M → Λ 3 T * M is an injective map defined in (2.6) 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.
1.5. Solitons. In §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. 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. [25,30,37]. In particular, Lin [30] 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. We give the following result for compact Laplacian solitons. Proposition 1.7. There is no compact Laplacian soliton of the type ∆ ϕ ϕ = λϕ unless ϕ is torsion free.
Combining Lin's [30] result and the above proposition, any nontrivial Laplacian soliton on a compact manifold M (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 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 9.3). In particular, from this result, we can give a new short proof of the main result in [30].
To conclude the paper in §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 2 structures
We collect some facts on closed G 2 structures, mainly based on [5,24].
2.1. Definitions. Let {e 1 , e 2 , · · · , e 7 } denote the standard basis of R 7 and let {e 1 , e 2 , · · · , e 7 } be its dual basis. Write e ijk = e i ∧ e j ∧ e k for simplicity and define the 3-form 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 . Using the ε-notation in [5] we write φ and * φ φ as where here and throughout the paper we use summation convention. The symbol ε satisfies the following identities: is thus an open subbundle of Λ 3 T * M . 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 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.
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 a positive definite bilinear form and a nowhere vanishing 7-form which defines a unique metric g with volume form vol g as follows: 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 of Λ k l (T * M, ϕ) be Ω k l (M ). We have that To study the Laplacian flow, it is convenient to write key quantities in local coordinates. We write a k-form α as where α i 1 i 2 ···i k is totally skewsymmetric in its indices. In particular, we write ϕ, ψ in local coordinates as Note that the metric g on M induces an inner product of two k-forms α, β, given locally by Here we use the orientation in [5] rather than [24].

2.2.
Hodge Laplacian of ϕ. Since dϕ = 0, from (2.11) and (2.12) 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 (2.20) as (2.21) We can decompose ∆ 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 ∆ ϕ ϕ, and using the identity (2.19) (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 (2.8), (2.10) 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 (2.21), we have where in the above calculation we used (2.15), (2.18), (2.19) and the totally skew-symmetry in ϕ ijk and ψ ijkl . So X = 0 and thus the Ω 3 7 (M ) part of ∆ ϕ ϕ is zero. To find h, using the decomposition (2.22), X = 0 and the contraction identities (2.8) and (2.9), we have (as in [17]) The left-hand side of the above equation can be calculated using (2.21): (2.19) and that for closed G 2 structures, ∇ m τ ni ϕ mn j is symmetric in i, j (see Remark 2.3). Then We conclude that 2.3. 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 [24] derived the following second Bianchi-type identity for the full torsion tensor.
Proof. The proof of (2.27) in [24] is indirect, but as remarked there, (2.27) can also be established directly using (2.13)-(2.15) and the Ricci identity. We provide the detail here for completeness.
where in the third equality we used (2.13), (2.15) and (2.26), and in the forth equality we used the contraction identity (2.8).
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 [5,12,24]. We give the general result from [24] here.
Proposition 2.2. The Ricci tensor of the associated metric g of the G 2 structure ϕ is given locally as In particular, for a closed G 2 structure ϕ, we have Proof. We multiply (2.27) by −ϕ jp k : where the last equality is due to (2.25). The formula (2.28) follows. For a closed G 2 structure, we have Then we obtain which is (2.29).
Remark 2.3. By (2.29), for a closed G 2 structure, ∇ j T sp ϕ jp k is symmetric in s, k, since R sk and T j s T jk are symmetric in s, k.
We can then deduce a useful known formula for the scalar curvature of the metric given by a closed G 2 structure.
Corollary 2.4. The scalar curvature of a metric associated to a closed G 2 structure satisfies (2.30) Proof. By taking trace in (2.29), using T ij = − 1 2 τ ij and (2.19), we obtain the scalar curvature

Evolution equations
In this section we derive evolution equations for several geometric quantities under the Laplacian flow, including the torsion tensor T , Riemann curvature tensor Rm, Ricci tensor Ric and scalar curvature R. These are fundamental equations for understanding the flow.
Recall that the Laplacian flow for a closed G 2 structure is where h is the symmetric 2-tensor given in (2.24). We may write h in terms the full torsion tensor T ij as follows: For closed ϕ, the Ricci curvature is equal to where h(t), X(t) are a time-dependent symmetric 2-tensor and vector field on M respectively, it is well known that (see [5,23] and explicitly [24]) 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 [5]. 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 [5]).

Evolution of torsion.
By [24,Theorem 3.7], the evolution of the full torsion tensor T under the flow (3.5) is given by Using the contraction identity (2.9) and Ricci identity (2.26), the first term on the right hand side of (3.10) is equal to where we used ∇ m T mi = 0 in the last equality. Using the contraction identity (2.10) and (2.13), 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 mk = 0 and T k m ϕ mq The above evolution 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 ∂ ∂t g(t) = η(t), (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. [10, Lemma 6.5]) where ∆ 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 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 ≤ |Rm|, we have Similarly, substituting (3.6) into (3.16) and (3.17), we obtain the evolution equation of the Ricci tensor (3.20) and the evolution equation of the scalar curvature

Derivative estimates of curvature and torsion
In this section, we use the evolution equations derived in §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.
Proof. We will calculate an 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 (2.13) and (2.15) in the form 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 ≤ |Rm| in the last inequality. Now, using (3.19), (4.3) and the Cauchy-Schwarz inequality, we obtain The idea behind the calculation of (4.4) is that there are enough negative gradient terms appearing in the evolution equations of |∇T | 2 and |Rm| 2 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 (4.4), we deduce that in the sense of lim sup of forward difference quotients. We conclude that 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).
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), = ∇∆Rm + Rm * ∇Rm + ∇Rm * T * T + Rm * T * ∇T where in the last equality we used the commuting formula ∇∆Rm = ∆∇Rm + Rm * ∇Rm.
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 (4.7) of f , we obtain (4.6) 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 (4.6) holds for all 1 ≤ j < k. From (4.1), for any time-dependent tensor A(t) we have (4.14) By (3.6), (3.18) and (4.14), we have where in the last equality we used the Ricci identity Using (4.15), the evolution of the squared norm of ∇ k Rm is: Applying (4.6) for 1 ≤ j < k to (4.17) and using Cauchy-Schwarz, we get where the constant C depends on the constants C j , 1 ≤ j < k in (4.6). Similarly, we have The second line of (4.19) can be estimated using the second line of (4.17).
To estimate the third line of (4.19), for 2 ≤ i ≤ k + 1 we have (4.20) For i = 1, where Similarly for i = 0, we have Using (2.13) and (2.15), we can estimate ∇ i ψ: Combining (4.20)-(4.25), using (4.6) for 0 ≤ j < k and the assumption tK ≤ 1, the third line of (4.19) can be estimated by We can estimate the last line of (4.19) similarly. We conclude that (4.26) Combining (4.18) and (4.26), we have Given these calculations, we now define for some constants β k to be determined later and α k i = (k−1)! (k−i)! . Assuming (4.6) holds for all 1 ≤ i < k, then by a similar calculation to those leading to (4.27), we have From (4.27) and (4.29), we may calculate where we used the facts applying the maximum principle to (4.30) 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) can not blow up quickly along the flow. Note that the estimate (4.6) 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 (4.12) we have if ǫ sufficiently small.Using (4.18)-(4.19) 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.
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 [34] (see also [19]) in the Ricci flow case, so we omit it. For any k ∈ N, there exists a constant C = C(K, r, k) such that if Λ(x, t) ≤ K for all x ∈ U and t ∈ [0, 1 K ], then for all y ∈ B g(0) (p, r/2) and t ∈ [0, 1 K ], we have (4.31)

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 ǫ > 0 and a smooth Laplacian flowφ(t) for t ∈ [0, T 0 + ǫ) 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.
where g(t) is the metric determined by ϕ(t). Then, in particular, we have the uniform curvature bound ≤ CK.
(Keep in mind that |T | 2 = −R). Then all the metrics g(t) (0 ≤ t < T 0 ) are uniformly equivalent (see e.g. [ for some uniform positive constant C. 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 metrics g(t) converge continuously to a Riemannian metric g(T 0 ) as t → T 0 . By (2.5), for each t ∈ [0, T 0 ) we have Let t → T 0 in (5.8). The left hand side tends to a positive definite 7-form valued bilinear form as the limit metric g(T 0 ) is still a Riemannian metric. Thus, the right-hand side 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.
where ∇ is the Levi-Civita connection with respect toḡ.
Proof of Claim 5.2. Since g(t) evolves by (3.6), the proof of the claim is similar to the Ricci flow case, see e.g. [10, §6.7], so we omit the detail here.
Proof of Claim 5.3. We begin with m = 1. At any (x, t) ∈ M × [0, T 0 ), 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 For m ≥ 2, we can prove by induction that It then follows from (4.6), (5.11) and (5.13) that 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 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) with ϕ(0) = ϕ(T 0 ) for a short time 0 ≤ t < ǫ. Since ϕ(t) → ϕ(T 0 ) smoothly as t → T 0 , this gives that We now prove (5.1) by replacing the lim sup in (5.18) by lim. Suppose, for a contradication, that (5.1) does not hold. Then there exists a sequence t i ր T 0 such that Λ(t i ) ≤ K 0 for some constant K 0 . By the doubling time estimate in Proposition 4.1, but we already showed above that this leads to a contradication 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 (4.4) we have We already proved that lim t→T 0 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.

Uniqueness
In this section, we will use the ideas in [26,27]  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) and g(t) are uniformly equivalent on M × [0, ǫ], so the norms | · | g(t) and | · |g (t) only differ by a uniform constant on M × [0, ǫ]. 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, ǫ], then the induced metrics also satisfy g(s) =g(s). 6.1. Forward uniqueness. We begin by showing forward uniqueness of the flow as claimed in Theorem 1.4; namely, that if ϕ(s) =φ(s) for some s ∈ [0, ǫ] then ϕ(t) =φ(t) for all t ∈ [s, ǫ]. The strategy to show this, inspired by [27], 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, ∇, ∆ 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 Then (6.5) follows from (6.12) and Lemma 6.1.
We next compute the evolution of V using (4.2): (6.14) We thus obtain (6.8) as claimed. Finally, we compute the evolution of S using the evolution (3.18) for Rm: We thus obtain (6.9) as required.
We now use Lemma 6.2 to obtain a differential inequality for E(t).
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 Using Lemma 6.2, we may calculate the required estimate directly: where in the last inequality we used Cauchy-Schwarz, particularly to use the negative third integral in the inequality to crucially cancel the terms involving ∇V and ∇S arising from the fourth and fifth integrals.

Backward uniqueness.
To complete the proof of Theorem 1.4, we need to show backward uniqueness of the flow; i.e. if ϕ(s) =φ(s) for some s ∈ [0, ǫ], then ϕ(t) =φ(t) for all t ∈ [0, s]. To this end, we apply a general backward uniqueness theorem [26, Theorem 3.1] for time-dependent sections of vector bundles satisfying certain differential inequalities. Since we only consider compact manifolds, we state [26, Theorem 3.1] here for this setting.
Theorem 6.4. Let M be a compact manifold and g(t), t ∈ [0, ǫ] be a family of smooth Riemannian metrics on M with Levi-Civita connection ∇ = ∇ g(t) . Assume that there exists a positive constant C such that ≤ C, 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 X(t) ∈ C ∞ (X ), Y(t) ∈ C ∞ (Y), for t ∈ [0, ǫ], be smooth families of sections satisfying for some constant C ≥ 0, where ∆ g(t) X(t) = g ij (t)∇ i ∇ j X(t) is the Laplacian with respect to g(t) acting on tensors. Then X(ǫ) ≡ 0, Y(ǫ) ≡ 0 implies Suppose ϕ(s) =φ(s) for some s ∈ [0, ǫ]. For our purpose, we let where φ, h, A, U, V, S are defined as in §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, ǫ]. 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 Since ϕ(s) =φ(s) for some s ∈ [0, ǫ], we have where C is a uniform constant depending on K 1 , and we used the uniform equivalence of where we used (6.2), (6.13) and the uniform equivalence of g(t) andg(t). Similarly, we can bound B = ∇A on M × [0, ǫ].
We derived the evolution equations of φ, h, A, U, V, S in §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: Proof. Since A, as a difference of connections, is a tensor, (4.1) gives From (6.10), (6.13), Lemma 6.1 and Lemma 6.5, we then have , where in the last inequality we used This gives the inequality (6.21). The inequalities (6.22) and (6.23) follow from similar calculations using (4.8) and (4.10).
Irreducible compact G 2 manifolds (M, ϕ) have I ϕ finite, and it is often trivial in examples. Corollary 6.7 thus gives an immediate test on a closed G 2 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. 7.1. 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 in the sense that F * i ϕ i − ϕ and its covariant derivatives of all orders (with respect to any fixed metric) converge uniformly to zero on every compact subset of M .
We may thus give our compactness theorem for G 2 structures.
Theorem 7.1. Let M i be a sequence of smooth 7-manifolds and for each i we let p i ∈ M i and ϕ i be a G 2 structure on M i such that the metric g i on M i induced by ϕ i is complete on M i . Suppose that where T i , Rm g i are the torsion and curvature tensor of ϕ i and g i respectively, and inj(M i , g i , p i ) denotes the injectivity radius of (M i , g i ) at p i . Then there exists a 7-manifold M , a G 2 structure ϕ on M and a point p ∈ M such that, after passing to a subsequence, we have 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 [20,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, 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 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 ∇, ∇ g i,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 = ∇ − ∇ g i,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 | g i,j ≤ 7c 0 =c 0 for some constants c 0 ,c 0 . We next observe trivially that so, since A is uniformly bounded, there is a constantc 1 such that Similarly, we have and so, since A, ∇A are uniformly bounded, there is a constantc 2 such that 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. [2, 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.
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 is positive definite for each j and satisfies where u, v are any vector fields on Ω i ⊂ M . Letting j → ∞ in (7.4) gives 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 3form 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 For each Ω k , we can argue as before to define g k,j , ϕ k,j which converge to g k,∞ , ϕ k,∞ respectively as j → ∞, after taking a subsequence. By definition, I * ik g k,j = g i,j and I * ik ϕ k,j = ϕ i,j . Since I * i,k is independent of j, by taking j → ∞ here we find that I * ik g k,∞ = g i,∞ and I * ik ϕ k,∞ = ϕ i,∞ . (7.6) From (7.6), we see that there exists a 3-form ϕ on M , which is a G 2 structure with associated metric g, such that where I i : Ω i → M is the inclusion map. Finally, we show that (M i , ϕ i , p i ) converges to (M, ϕ, p). For any compact subset Ω ⊂ M , there exists i 0 such that Ω is contained in Ω i for all i ≥ i 0 . Fixing i such that Ω ⊂ Ω i , on Ω we have by (7.3) that for all k ≥ 0, as required.

7.2.
Compactness for the Laplacian flow. Now we can prove Theorem 1.5, the compactness theorem for the Laplacian flow for closed G 2 structures, which we restate here for convenience.
where T i and Rm g i (t) denote the torsion and curvature tensors determined by ϕ i (t) respectively, and the injectivity radius of (M i , g i (0)) at p i satisfies There exists a 7-manifold M , p ∈ M and a solution ϕ(t) of the flow (1.1) on M for t ∈ (a, b) such that, after passing to a subsequence, we have The proof is an adaptation of Hamilton's argument in the Ricci flow case [20].
Proof. By a usual diagonalization argument, without loss of generality, we can assume a, b are finite. From the Shi-type estimates in §4 and (7.8), we have (7.10) Assumption (7.9) allows us to apply Theorem 7.1 to extract a subsequence of (M i , ϕ i (0), p i ) which converges to a complete limit (M,φ ∞ (0), p) in the sense described above. Using the notation of Theorem 7.1, we have smoothly on any compact subset Ω ⊂ M as i → ∞. Since each ϕ i (0) is closed, we see that dφ ∞ (0) = 0. × (a, b), and let i be sufficiently large that Ω ⊂ Ω i , in the notation of Theorem 7.1. Theñ ϕ i (t) is a sequence of solutions of the Laplacian flow on Ω ⊂ M defined for t ∈ [c, d], with associated metricsg i (t) = F * i g i (t). By Claims 5.2 and 5.3, we may deduce from (7.10) that there exist constants C k , independent of i, such that since the time derivatives can be written in terms of spatial derivatives via the Laplacian flow evolution equations. It follows from the Arzelá-Ascoli theorem that there exists a subsequence ofφ i (t) which converges smoothly on Ω × [c, d]. A diagonalization argument then produces a subsequence that converges smoothly on any compact subset of M × (a, b) to a solutionφ ∞ (t) of the Laplacian flow.
As in Ricci flow, we would want to use our compactness theorem for the Laplacian flow to analyse singularities of the flow as follows.
Let M be a compact 7-manifold and let ϕ(t) be a solution of the Laplacian flow (1.1) on a maximal time interval [0, T 0 ) with T 0 < ∞. Theorem 1.3 implies that Λ(t) given in (1.4) satisfies lim Λ(t) = ∞ as t ր T 0 . Choose a sequence of points (x i , t i ) such that t i ր T 0 and where T and Rm are the torsion and curvature tensor as usual.
We consider a sequence of parabolic dilations of the Laplacian flow and define From the doubling-time estimate (Proposition 4.1), there exists a uniform b > 0 such that sup Therefore, we obtain a sequence of solutions (M, ϕ i (t)) to the Laplacian flow defined on (a, b) for some a < 0, with If we can establish the injectivity radius estimate we can apply our compactness theorem (Theorem 1.5) and extract a subsequence of (M, ϕ i (t), x i ) which converges to a limit flow (M ∞ , ϕ ∞ (t), x ∞ ). Such a blow-up of the flow at the singularity will provide an invaluable tool for further study of the Laplacian flow.
8. Long time existence II Theorem 1.3 states that the Riemann curvature or the derivative of the torsion tensor must blow-up at a finite singular time of the Laplacian flow. However, based on Joyce's existence result for torsion-free G 2 structures [22], we would hope to be able to characterise the finite-time singularities of the flow via the blow-up of the torsion tensor itself.
In this section we will show that, under an additional continuity assumption on the metrics along the flow, that the Laplacian flow will exist as long as the torsion tensor remains bounded. From this result, stated below, our improvement Theorem 1.6 of Theorem 1.3 follows as a corollary.
Theorem 8.1. Let M 7 be a compact manifold and ϕ(t) for 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 g(t) is uniformly continuous and the torsion tensor T (x, t) of ϕ(t) satisfies then the solution ϕ(t) can be extended past time T 0 .
Here we say g(t) is uniformly continuous if for any ǫ > 0 there exists δ > 0 such that for any 0 ≤ t 0 < t < T 0 with t − t 0 ≤ δ we have which implies that, as symmetric 2-tensors, we have Before we prove Theorem 8.1, we deduce Theorem 1.6 from Theorem 8.1.
Therefore, if B g(t) (x, r) denotes the geodesic ball of radius r centred at x with respect to the metric g(t), we have Along the Laplacian flow, the volume form increases, so for any x ∈ M , r > 0 and t ∈ [t 0 , T 0 ). Then, for x ∈ M and r ≤ Λ(x i , t i ) for some uniform positive constant c. Hence we have for all x ∈ M and r ∈ [0, Λ(x i , t i ) 1 2 ]. Note that by definition of Λ ϕ i in (7.15) that on M . By the volume ratio bound (8.4) and [11,Theorem 5.42], we have a uniform injectivity radius estimate inj(M, g i (0), x i ) ≥ c for some constant c > 0. We can thus apply our compactness theorem (Theorem 1.5) to obtain a subsequence of (M, ϕ i (t), x i ) converging to a limit (M ∞ , ϕ ∞ (t), x ∞ ), t ∈ (−∞, 0] with |Λ ϕ∞ (x ∞ , 0)| = 1. By the assumption (8.1) that T remains bounded and Λ(x i , t i ) → ∞ as i → ∞, we have as i → ∞. Therefore, (M ∞ , ϕ ∞ (t)) has zero torsion for all t ∈ (−∞, 0]. Thus Ric g∞(t) ≡ 0 for all t ∈ (−∞, 0], where g ∞ (t) denotes the metric defined by ϕ ∞ (t), since torsion-free G 2 structures define Ricci-flat metrics. We can then argue as in [33] (see also [11, §6.4]) that g ∞ (0) has precisely Euclidean volume growth; i.e. for all r > 0, Since a Ricci-flat complete manifold with this property must be Euclidean space, Rm(g ∞ (0)) ≡ 0 on M ∞ . This contradicts the fact that where in the first equality we used the fact that the torsion of (M ∞ , ϕ ∞ (0)) vanishes. We have our required contradiction, so the result follows.
We may call a vector field X such that L X ϕ = 0 a symmetry of the G 2 structure ϕ. The following lemma shows that the symmetries of a closed G 2 structure correspond to certain Killing vector fields of the associated metric.
Lemma 9.2. Let ϕ be a closed G 2 structure on a compact manifold M with associated metric g and let X be a vector field on M . Then where i ϕ : S 2 T * M → Λ 3 T * M is the injective map given in (2.6). In particular, any symmetry X of the closed G 2 structure ϕ must be a Killing vector field of the associated metric g and satisfy d * (X ϕ) = 0 on M .
Proof. Since ϕ is closed, we have Denote β = X ϕ. Then β ij = X l ϕ lij and i.e., in index notation, we have We decompose L X 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, W is a vector field and η is a trace-free symmetric 2-tensor on M . We now calculate a, W and η, using a similar method to §2.2.
We can now derive the condition satisfied by the metric g induced by ϕ when (ϕ, X, λ) is a Laplacian soliton, which we expect to have further use. Proposition 9.3. Let (ϕ, X, λ) be a Laplacian soliton as defined by (9.1). Then the associated metric g of ϕ satisfies, in local coordinates, (9.12) and the vector field X satisfies d * (X ϕ) = 0.
Recall that Ricci solitons (g, X, λ) are given by Ric = λg + L X g, so we see that (9.12) can be viewed as a perturbation of the Ricci soliton equation using the torsion tensor T . We also re-iterate that the non-existence of compact Laplacian solitons of the form (ϕ, 0, λ) is somewhat surprising given that we have many compact Ricci solitons of the form (g, 0, λ) since these correspond to Einstein metrics.
As an application of Proposition 9.3, we can give a short proof of the main result in [30]. Proof. Taking the trace of (9.12), we have 2 3 |T | 2 = 7 3 λ + div(X).
When the soliton is defined on a compact manifold M , integrating the above equation gives λV ol g (M ) = 2 7 M |T | 2 vol g ≥ 0.
In Ricci flow, every compact Ricci soliton is a gradient Ricci soliton, meaning that the vector field X in that case satisfies X = ∇f for some function f . This was proved by Perelman using the W-functional and a logarithmic Sobolev inequality. In the Laplacian flow the situation is quite different and there is currently no reason to suspect that an analogous result to the Ricci flow will hold. In fact, we see from (9.9)-(9.10) and Proposition 9.3 that if (ϕ, ∇f, λ) is a Laplacian soliton then ∇f T = 0. It is thus currently an interesting open question whether any non-trivial compact Laplacian soliton is a gradient Laplacian soliton.

Concluding remarks
The research in this paper motivates several natural questions that form objectives for future study. We list some of these problems here.
(1) Show that torsion-free G 2 structures are dynamically stable under the Laplacian flow. This will be reported on in a forthcoming article by the authors [31]. (2) Prove a noncollapsing result along the Laplacian flow for closed G 2 structures as in Perelman's work [32] on Ricci flow. This would mean, in particular, that our compactness theory would give rise to well-defined blow-ups at finite-time singularities, which would further allow us to relate singularities of the flow to Laplacian solitons. (3) Study the behavior of the torsion tensor near the finite singular time T 0 of the Laplacian flow. Since for closed G 2 structures ϕ, we have ∆ ϕ ϕ = dτ , Theorem 1.6 says that dτ will blow up when t ր T 0 along the Laplacian flow. The question is whether the torsion tensor T , or equivalently τ , will blow up when t ր T 0 . Since |T | 2 = −R, this is entirely analogous to the question in Ricci flow as to whether the scalar curvature will blow up at a finite-time singularity. This is true for Type-I Ricci flow on compact manifolds by Enders-Müller-Topping [14] and Kähler-Ricci flow by Zhang [39], but it is still open in general and currently forms an active topic of research. (4) Find some conditions on the torsion tensor under which the Laplacian flow for closed G 2 structures will exist for all time and converge to a torsion-free G 2 structure. Based on the work of Joyce [23], it is expected that a reasonable condition to impose is that the initial G 2 structure ϕ 0 is closed and has sufficiently small torsion, in a suitable sense. The Laplacian flow would then provide a parabolic method for proving the fundamental existence theory for torsion-free G 2 structures (c.f. [23]).
(5) Study the space of gradient Laplacian solitons on a compact manifold. As mentioned earlier, this would show the similarities or differences with the analogous theory for Ricci solitons, which it would be instructive to study (see [9] for a recent survey on Ricci solitons). (6) Construct nontrivial examples of Laplacian solitons. Recent progress on this problem has been made by Bryant [6], and also forms a topic of current investigation by the authors.