Laplacian Flow for Closed $\mathrm{G}_2$ Structures

Abstract

This is an expository article based on the author’s talk in Workshop on $\mathrm{G}_2$ Manifolds and Related Topics held in August 2017 at The Fields Institute. The aim is to explain the results obtained recently by the author and Jason D. Lotay on the Laplacian flow for closed $\mathrm{G}_2$ structures and some related progress.

1 $G_2$ Structures on 7-Manifolds

The group $\mathrm{G}_2$ is one of the exceptional holonomy groups and is defined as the stabilizer of the following 3-form on the 7-dimensional Euclidean space $\mathbb {R}^7$:

$$\begin{aligned} \phi =e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}, \end{aligned}$$

where $e^{ijk}=e^i\wedge e^j\wedge e^k$ with respect to the basis $\{e^1,e^2,\ldots ,e^7\}$ of $\mathbb {R}^7$. The group $\mathrm{G}_2$ is a compact, connected, simply-connected, simple Lie subgroup of $\mathrm{SO}(7)$ of dimension 14. The group $\mathrm{G}_2$ acts irreducibly on $\mathbb {R}^7$ and preserves the Euclidean metric and orientation on $\mathbb {R}^7$. If $*_{\phi }$ denotes the Hodge star determined by the metric and orientation, then $\mathrm{G}_2$ also preserves the 4-form $*_{\phi }\phi$.

Let M be a 7-manifold. We say a 3-form $\varphi$ on M is definite if for $x\in M$ there exists an homomorphism $u\in \text {Hom}_{\mathbb {R}}(T_xM,\mathbb {R}^7)$ such that $u^*\phi =\varphi _x$. The space of definite 3-forms on M will be denoted by $\Omega _+^3(M)$. Since $\phi$ is invariant under the action of the group $G_2$, each definite 3-form will define a $\mathrm{G}_2$ structure on M. The existence of $\mathrm{G}_2$ structures is equivalent to the property that the manifold M is oriented and spin. Note that as $\mathrm{G}_2$ is a subgroup of $\mathrm{SO}(7)$, a $\mathrm{G}_2$ structure $\varphi$ defines a unique Riemannian metric $g=g_{\varphi }$ on M and an orientation such that

$$\begin{aligned} g_{\varphi }(u,v)\mathrm {vol}_{g_{\varphi }}=\frac{1}{6}(u\lrcorner \varphi )\wedge (v\lrcorner \varphi )\wedge \varphi ,\quad \forall ~u,v\in C^{\infty }(TM). \end{aligned}$$

The metric and orientation determine the Hodge star operator $*_{\varphi }$, and we define $\psi =*_{\varphi }\varphi$, which is sometimes called a positive 4-form. Notice that the relationship between $g_{\varphi }$ and $\varphi$, and hence between $\psi$ and $\varphi$, is nonlinear.

1.1 Type Decomposition of k-Forms

The group $\mathrm{G}_2$ acts irreducibly on $\mathbb {R}^7$ (and hence on $\Lambda ^1(\mathbb {R}^7)^*$ and $\Lambda ^6(\mathbb {R}^7)^*$), but it acts reducibly on $\Lambda ^k(\mathbb {R}^7)^*$ for $2\le k\le 5$. Hence a $\mathrm{G}_2$ structure $\varphi$ induces splittings of the bundles $\Lambda ^kT^*M$ ($2\le k\le 5$) into direct summands, which we denote by $\Lambda ^k_l(T^*M,\varphi )$ so that l indicates the rank of the bundle. We let the space of sections of $\Lambda ^k_l(T^*M,\varphi )$ be $\Omega ^k_l(M)$. We have that

$$\begin{aligned} \Omega ^2(M)=&\, \Omega ^2_7(M)\oplus \Omega ^2_{14}(M), \qquad \Omega ^3(M)= \Omega ^3_1(M)\oplus \Omega ^3_7(M)\oplus \Omega ^3_{27}(M), \end{aligned}$$

where

$$\begin{aligned} \Omega ^2_7(M)&= \{\beta \in \Omega ^2(M)|\beta \wedge \varphi =2*_{\varphi }\beta \} =\{X\lrcorner \varphi |X\in C^{\infty }(TM)\},\\ \Omega ^2_{14}(M)&= \{\beta \in \Omega ^2(M)|\beta \wedge \varphi =-*_{\varphi }\beta \} =\{\beta \in \Omega ^2(M)|\beta \wedge \psi =0\}, \end{aligned}$$

and

$$\begin{aligned} \Omega ^3_1(M)&=\{f\varphi |f\in C^{\infty }(M)\}, \quad \Omega ^3_7(M)=\{X\lrcorner \psi |X\in C^{\infty }(TM)\},\\ \Omega ^3_{27}(M)&=\{\gamma \in \Omega ^3(M)|\gamma \wedge \varphi =0=\gamma \wedge \psi \}. \end{aligned}$$

Hodge duality gives corresponding decompositions of $\Omega ^4(M)$ and $\Omega ^5(M)$.

The space $\Omega ^3_{27}(M)$ deserves more attention. As in [3] we define a map $\mathrm {i}_{\varphi }: \mathrm {Sym}^2(T^*M)\rightarrow \Omega ^3(M)$ from the space of symmetric 2-tensors to the space of 3-forms, given locally by

$$\begin{aligned} \mathrm {i}_{\varphi }(h)&=\frac{1}{2}h^l_i\varphi _{ljk}dx^i\wedge dx^j\wedge dx^k \end{aligned}$$

(1.1)

where $h=h_{ij}dx^idx^j\in \mathrm {Sym}^2(T^*M)$. Then $C^{\infty }(M)\otimes g_{\varphi }$ is mapped isomorphically onto $\Omega ^3_1(M)$ under the map $\mathrm {i}_{\varphi }$ with $\mathrm {i}_{\varphi }(g_{\varphi })=3\varphi$, and the space of trace-free symmetric 2-tensors $\mathrm {Sym}^2_0(T^*M)$ is mapped isomorphically onto the space $\Omega ^3_{27}(M)$.

1.2 Torsion of $G_2$ Structures

Given a $\mathrm{G}_2$ structure $\varphi \in \Omega ^3_+(M)$, if $\nabla$ denotes the Levi-Civita connection with respect to $g_{\varphi }$, we can interpret $\nabla \varphi$ as the torsion of the $\mathrm{G}_2$ structure $\varphi$. Following [25], we see that $\nabla \varphi$ lies in the space $\Omega ^1_7(M)\otimes \Omega ^3_7(M)$. Thus we can define a 2-tensor T which we shall call the full torsion tensor such that

$$\begin{aligned} \nabla _i\varphi _{jkl} =T_{im}g^{mn}\psi _{njkl}. \end{aligned}$$

(1.2)

Using the decomposition of the spaces of forms on M determined $\varphi$, we can also decompose $d\varphi$ and $d\psi$ into types. Bryant [3] showed that there exist unique differential forms $\tau _0\in \Omega ^0(M), \tau _1\in \Omega ^1(M), \tau _{2}\in \Omega ^2_{14}(M)$ and $\tau _{3}\in \Omega ^3_{27}(M)$ such that

$$\begin{aligned} d\varphi&= \tau _0\psi +3\tau _1\wedge \varphi +*_{\varphi }\tau _{3}, \end{aligned}$$

(1.3)

$$\begin{aligned} d\psi&= 4\tau _1\wedge \psi +\tau _{2}\wedge \varphi . \end{aligned}$$

(1.4)

We call $\{\tau _0,\tau _1,\tau _2,\tau _3\}$ the intrinsic torsion forms of the $\mathrm{G}_2$ structure $\varphi$. The full torsion tensor $T_{ij}$ is related to the intrinsic torsion forms by the following (see [25]):

$$\begin{aligned} T_{ij}=\frac{\tau _0}{4}g_{ij}-(\tau _1^{\#}\lrcorner \varphi )_{ij}-(\bar{\tau }_3)_{ij}-\frac{1}{2}(\tau _2)_{ij}, \end{aligned}$$

(1.5)

where $\bar{\tau }_3$ is the trace-free symmetric 2-tensor such that $\tau _3=\mathrm {i}_{\varphi }(\bar{\tau }_3)$.

If $\nabla \varphi =0$, we say the $\mathrm{G}_2$ structure $\varphi$ is torsion-free on M. The torsion-free condition clearly implies that $d\varphi =0=d^{*}_{\varphi }\varphi$ on M. Fernández and Gray [12] showed that $d\varphi =0=d^{*}_{\varphi }\varphi$ also implies $\nabla \varphi =0$ on M, which also follows from the Eq. (1.5). The key property of a torsion-free $\mathrm{G}_2$ structure $\varphi$ is that the holonomy group $\text {Hol}(g_{\varphi })\subseteq \mathrm{G}_2$, and thus the manifold $(M,g_{\varphi })$ is Ricci-flat. Moreover, one can characterise the compact $\mathrm{G}_2$ manifolds (i.e., compact manifolds with torsion-free $\mathrm{G}_2$ structures) with $\text {Hol}(g_{\varphi })= \mathrm{G}_2$ as those with finite fundamental group. Thus understanding torsion-free $\mathrm{G}_2$ structures is crucial for constructing Riemannian manifolds with holonomy $\mathrm{G}_2$.

While there are some explicit examples of manifolds which admit torsion-free $\mathrm{G}_2$ structures for which the holonomy of the induced metric is properly contained in $\mathrm{G}_2$, for example the product of circle $\mathcal {S}^1$ with a Calabi-Yau 3-fold and the product of 3-torus $\mathbb {T}^3$ with a Calabi-Yau 2-fold, the construction of manifolds which admit torsion-free $\mathrm{G}_2$ structures with holonomy equal to $\mathrm{G}_2$ is a hard and important problem. The first local existence result of metrics with holonomy $\mathrm{G}_2$ was obtained by Bryant [2] using the theory of exterior differential systems. Then Bryant–Salamon [4] constructed the first complete non-compact manifolds with holonomy $\mathrm{G}_2$, which are the spinor bundle of $\mathcal {S}^3$ and the bundles of anti-self-dual 2-forms on $\mathcal {S}^4$ and $\mathbb {C}\mathbb {P}^2$. In [22], Joyce constructed the first examples of compact 7-manifolds with holonomy $\mathrm{G}_2$ and many further compact examples have now been constructed [7, 24, 29].

1.3 Closed $G_2$ Structures

If $\varphi$ is closed, i.e. $d\varphi =0$, then (1.3) implies that $\tau _0,\tau _{1}$ and $\tau _3$ are all zero, so the only non-zero torsion form is $\tau _2\in \Omega ^2_{14}(M)$. In this case, we write $\tau =\tau _2$ for simplicity. Then from (1.5) we have that the full torsion tensor satisfies $T_{ij}=-\frac{1}{2}\tau _{ij}$ and is a skew-symmetric 2-tensor. By (1.4) and $\tau \in \Omega ^2_{14}(M)$, we have $d\psi =\tau \wedge \varphi =-\!*_{\varphi }\!\tau$, which implies that

$$\begin{aligned} d^*\tau =*_{\varphi }d*_{\varphi }\tau =-*_{\varphi }d^2\psi =0 \end{aligned}$$

(1.6)

and the Hodge Laplacian of $\varphi$ is equal to $\Delta _{\varphi }\varphi = -d*_{\varphi }d\psi =d\tau$. We computed in [34] (see also [3]) that

$$\begin{aligned} \Delta _{\varphi }\varphi ~=~\mathrm {i}_{\varphi }(h)~\in ~\Omega _1^3(M)\oplus \Omega ^3_{27}(M) \end{aligned}$$

(1.7)

where h is the symmetric 2-tensor given as follows:

$$\begin{aligned} h_{ij}=-\nabla _mT_{ni}\varphi _{j}^{\,\,mn}-\frac{1}{3}|T|^2g_{ij}-T_{ik}g^{kl}T_{lj}. \end{aligned}$$

(1.8)

Since $\varphi$ determines a unique metric $g=g_{\varphi }$ on M, we then have the Riemann curvature tensor $\mathrm {Rm}=\{R_{ijkl}\}$, the Ricci tensor $R_{ij}=g^{kl}R_{ijkl}$ and the scalar curvature $\mathrm {R}=g^{ij}R_{ij}$ of $(M,g_{\varphi })$. For closed $\mathrm{G}_2$ structure $\varphi$, we computed in [34] that the Ricci curvature is equal to

$$\begin{aligned} R_{ij}=\nabla _mT_{ni}\varphi _{j}^{\,\,mn}-T_{ik}g^{kl}T_{lj}, \end{aligned}$$

(1.9)

and then the scalar curvature $R=-|T|^2$. With (1.9) we can write the symmetric tensor h in (1.8) as

$$\begin{aligned} h_{ij}=-R_{ij}-\frac{1}{3}|T|^2g_{ij}-2T_{ik}g^{kl}T_{lj}. \end{aligned}$$

(1.10)

2 Laplacian Flow for Closed $G_2$ Structures

Since Hamilton [16] introduced the Ricci flow in 1982, geometric flows have been an important tool in studying geometric structures on manifolds. For example, Ricci flow was instrumental in proving the Poincaré conjecture and the $\frac{1}{4}$-pinched differentiable sphere theorem, and Kähler–Ricci flow has proved to be a useful tool in Kähler geometry, particularly in low dimensions. In 1992, Bryant (see [3]) proposed the Laplacian flow for closed $\mathrm{G}_2$ structures

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial }{\partial t}\varphi (t) ~~=&~~\Delta _{\varphi (t)}\varphi (t),\\ d\varphi (t) ~~=&~~0, \\ \varphi (0) ~~=&~~\varphi _0, \end{aligned}\right. \end{aligned}$$

(2.1)

where $\Delta _{\varphi }=dd^*_{\varphi }+d^*_{\varphi }d$ is the Hodge Laplacian with respect to $g_{\varphi }$ and $\varphi _0$ is an initial closed $\mathrm{G}_2$ structure. The stationary points of the flow are harmonic $\varphi$, which on a compact manifold are precisely the torsion-free $\mathrm{G}_2$ structures, so the Laplacian flow provides a tool for studying the existence of torsion-free $\mathrm{G}_2$ structures on a manifold admitting closed $\mathrm{G}_2$ structures. The goal is to understand the long-time behavior of the Laplacian flow on compact manifolds M; specifically, to understand conditions under which the flow will converge to a torsion-free $\mathrm{G}_2$ structure. We remark that there are other proposed flows which also have torsion-free $\mathrm{G}_2$ structures as stationary points (e.g. [15, 26, 42]).

2.1 Gradient Flow of Volume Functional

Another motivation for studying the Laplacian flow comes from work of Hitchin [19] (see also [5]), which demonstrates its relationship to a natural volume functional. Let $\bar{\varphi }$ be a closed $\mathrm{G}_2$ structure on a compact 7-manifold M and let $[\bar{\varphi }]_+$ be the open subset of the cohomology class $[\bar{\varphi }]$ consisting of $\mathrm{G}_2$ structures. Define the volume functional on M by

$$\begin{aligned} \mathrm {Vol}(M,\varphi )~=~\frac{3}{7}\int _M\varphi \wedge *_{\varphi }\varphi ,\quad \varphi \in [\bar{\varphi }]_+. \end{aligned}$$

(2.2)

In the arXiv version of [19], Hitchin showed that $\varphi \in [\bar{\varphi }]_+$ is a critical point of $\mathrm {Vol}(M,\varphi )$ if and only if $d *_{\varphi }\!\varphi =0$, i.e. $\varphi$ is torsion-free.

Moreover, the Laplacian flow (2.1) can be viewed as the gradient flow of the volume functional (2.2). Since $\varphi (t)$ evolves in the same cohomology class with the initial data $\varphi _0$, we can write $\varphi (t)=\varphi _0+d\eta (t)$ for some time dependent 2-form $\eta (t)$. To calculate the variation of the volume functional, we need to compute the variation of $*_{\varphi (t)}\varphi (t)$. This has already been computed in [3, 23]:

$$\begin{aligned} \frac{\partial }{\partial t}(*_{\varphi (t)}\varphi (t))~=~\frac{4}{3} *_{\varphi (t)}\pi _1\left( \frac{\partial \varphi (t)}{\partial t}\right) +*_{\varphi (t)}\pi _7\left( \frac{\partial \varphi (t)}{\partial t}\right) -*_{\varphi (t)}\pi _{27}\left( \frac{\partial \varphi (t)}{\partial t}\right) , \end{aligned}$$

(2.3)

where $\pi _k$’s are the respective projections to the invariant subspaces of $\Omega ^3(M)$ and are determined by $\varphi (t)$. Then

$$\begin{aligned} \frac{d}{dt}\mathrm {Vol}(M,\varphi (t)) =&\frac{3}{7}\int _M\left( \frac{\partial \varphi (t)}{\partial t}\wedge *_{\varphi (t)}\varphi (t)+\varphi (t)\wedge \frac{\partial }{\partial t}(*_{\varphi (t)}\varphi (t))\right) \\ =&\int _M \frac{\partial \varphi (t)}{\partial t}\wedge *_{\varphi (t)}\varphi (t)\\ =&\int _M\langle \frac{\partial \eta (t)}{\partial t}, d^*_{\varphi (t)}\varphi (t)\rangle *_{\varphi (t)}1. \end{aligned}$$

Thus gradient flow of the volume functional within the same cohomology class is given by

$$\begin{aligned} \frac{\partial \varphi (t)}{\partial t}=d\frac{\partial \eta (t)}{\partial t}=dd^*_{\varphi (t)}\varphi (t)=\Delta _{\varphi (t)}\varphi (t), \end{aligned}$$

which is exactly the Laplacian flow. Then along the Laplacian flow, the volume will increase unless $\varphi (t)$ is torsion-free. By examining the second variation of the volume functional, Bryant [3] showed that if $\bar{\varphi }$ is torsion-free, then $\mathrm {Diff}^0(M)\cdot \bar{\varphi }$ is a local maximum of the volume functional on the moduli space $\mathrm {Diff}^0(M)\setminus [\bar{\varphi }]_+$. This gives rise to the following natural question:

Question 2.1

([3]) Starting from a initial data $\varphi _0\in [\bar{\varphi }]_+$ which is sufficiently close to $\bar{\varphi }$ in an appropriate norm, does the Laplacian flow converge to a point on $\mathrm {Diff}^0(M)\cdot \bar{\varphi }$?

In the statement of Question 2.1, we assumed the existence of a torsion-free $\mathrm{G}_2$ structure $\bar{\varphi }$ on M. In 1996, Joyce [22] proved a criterion for the existence of torsion-free $\mathrm{G}_2$ structures, which says that if one can find a $\mathrm{G}_2$ structure $\varphi$ with $d\varphi =0$ on a compact 7-manifold M, whose torsion is sufficiently small in a certain sense, then there exists a torsion-free $\mathrm{G}_2$-structure $\bar{\varphi }\in [\varphi ]$ on M which is close to $\varphi$. This result has been used to construct compact examples of manifolds with $\mathrm{G}_2$ holonomy. It would be interesting to give a new proof of Joyce’s result [22] using the Laplacian flow.

Generally, one cannot expect that the Laplacian flow will converge to a torsion-free $\mathrm{G}_2$ structure, even if it has long-time existence. There are compact 7-manifolds with closed $\mathrm{G}_2$ structures that cannot admit holonomy $\mathrm{G}_2$ metrics for topological reasons (c.f. [9, 10]), and Bryant [3] showed that the Laplacian flow starting with a particular one of these examples will exist for all time but it does not converge; for instance, the volume of the associated metrics will increase without bound. Some explicit examples of the solution to the Laplacian flow which exist for all time and converge can be found in [11, 13, 21].

2.2 Short Time Existence

Recall that the Hodge Laplacian $\Delta _{\varphi }$ is related to the analyst’s Laplacian $\Delta =g^{ij}\nabla _i\nabla _j$ by the Weitzenbock formula:

$$\begin{aligned} \Delta _{\varphi }\omega ~=~-\Delta \omega +\mathcal {R}(\omega ) \end{aligned}$$

(2.4)

for any (0, k)-tensor $\omega$, where $\mathcal {R}$ is the Weitzenbock curvature operator. Since the Laplacian flow (2.1) is defined by the Hodge Laplacian, it appears at first sight to have the wrong sign for the parabolicity. However, if $d\varphi =0$, using definition (1.2) of the torsion tensor and the divergence-free property (1.6) of $\tau$, we see that $\Delta \varphi$ involves only up to first order derivatives of $\varphi$ and thus the second order part of the Hodge Laplacian $\Delta _{\varphi }\varphi$ lies in the part $\mathcal {R}(\varphi )$ of (2.4). Using DeTurck’s trick in the Ricci flow, Bryant–Xu [5] modified the Laplacian flow by an operator of the form $\mathcal {L}_{V(\varphi )}\varphi =d(V\lrcorner \varphi )+V\lrcorner d\varphi = d(V\lrcorner \varphi )$ for some vector field $V(\varphi )$ and showed that the Laplacian–DeTurck flow

$$\begin{aligned} \frac{\partial \varphi (t)}{\partial t}=\Delta _{\varphi (t)}\varphi (t)+\mathcal {L}_{V(\varphi )}\varphi (t) \end{aligned}$$

(2.5)

is strictly parabolic in the direction of closed forms by choosing a special vector field $V(\varphi )$. In fact, if $d\theta =0$, they calculated that the linearization of RHS of (2.5) is

$$\begin{aligned} \frac{d}{d\epsilon }\bigg |_{\epsilon =0}\left( \Delta _{\varphi +\epsilon \theta }(\varphi +\epsilon \theta )+\mathcal {L}_{V(\varphi +\epsilon \theta )}(\varphi +\epsilon \theta )\right) ~=~-\Delta _{\varphi }\theta +d\Phi (\theta ) \end{aligned}$$

(2.6)

where $d\Phi (\theta )$ is algebraic linear in $\theta$ and $d\Phi (\theta )=0$ if $\varphi$ is torsion-free. However, no existing theory of parabolic equations can be used directly since the parabolicity of (2.5) is only true in the direction of closed forms. Fortunately, by using the Nash Moser inverse function theorem [17] for tame Féchet spaces, Bryant and Xu proved the following short time existence theorem.

Theorem 2.2

(Bryant–Xu [5]) Assume that M is compact and $\varphi _0$ is a closed $\mathrm{G}_2$ structure on M. Then the Laplacian flow has a unique solution for a short time $t\in [0,\varepsilon )$ with $\varepsilon$ depending on $\varphi _0$.

As in the Ricci flow, we can also write the Laplacian–DeTurck flow (2.5) explicitly in local coordinates. Let $\tilde{g}$ be a fixed Riemannian metric on M and $\tilde{\nabla }, \tilde{\Gamma }_{ij}^k$ be the corresponding Levi-Civita connection and Christoffel symbols. We know that the difference $\Gamma _{kl}^j-\tilde{\Gamma }_{kl}^j$ of the Levi-Civita connections of the metrics g and $\tilde{g}$ is a well-defined tensor on M. This gives us a vector field V on M with

$$\begin{aligned} V_i=g_{ij}g^{kl}(\Gamma _{kl}^j-\tilde{\Gamma }_{kl}^j), \end{aligned}$$

(2.7)

which is just the vector field chosen in Ricci-DeTurck flow [41]. By a direct but lengthy computation, we can show that if $d\varphi =0$, the Laplacian–DeTurck flow Eq. (2.5) with V given by (2.7) has the following expression in local coordinates:

$$\begin{aligned} \frac{\partial }{\partial t}\varphi _{ijk} =&g^{pq}\tilde{\nabla }_p\tilde{\nabla }_q\varphi _{ijk}+l.o.t \end{aligned}$$

(2.8)

and the associated metric $g_{ij}$ evolves by

$$\begin{aligned} \frac{\partial }{\partial t}g_{ij}=&g^{pq}\tilde{\nabla }_p\tilde{\nabla }_qg_{ij}+l.o.t \end{aligned}$$

(2.9)

where the lower order terms only involve the $\varphi , g, \tilde{\nabla }g$ and $\tilde{\nabla }\varphi$ and can be written down explicitly. The readers may find that the vector field V is different at first sight with the one chosen by Bryant–Xu [5]. However, we can see that they are essentially the same by considering the linearization of V in the direction of closed forms (see also [15, pp. 400–401]).

2.3 Evolution Equations

Since each $\mathrm{G}_2$ structure induces a unique Riemannian metric on the manifold, the Laplacian flow (2.1) induces a flow for the associated Riemannian metric $g(t)=g_{\varphi (t)}$. Recall that under a general flow for $\mathrm{G}_2$ structures

$$\begin{aligned} \frac{\partial }{\partial t}\varphi (t)=\mathrm {i}_{\varphi (t)}(h(t))+X\lrcorner \psi (t), \end{aligned}$$

(2.10)

where $h(t)\in \mathrm {Sym}^2(T^*M)$ and $X(t)\in C^{\infty }(TM)$, it is well known that (see [3, 23] and explicitly [25]) the associated metric tensor g(t) evolves by

$$\begin{aligned} \frac{\partial }{\partial t}g(t)=2h(t). \end{aligned}$$

(2.11)

By (1.7) and (1.10), we deduce that the associated metric g(t) of the solution $\varphi (t)$ of the Laplacian flow evolves by

$$\begin{aligned} \frac{\partial }{\partial t}g_{ij}=-2R_{ij}-\frac{2}{3}|T|^2g_{ij}-4T_{ik}g^{kl}T_{lj}, \end{aligned}$$

(2.12)

which corresponds to the Ricci flow plus some lower order terms involving the torsion tensor, as already observed in [3]. Then it’s easy to see that the volume form $\mathrm {vol}_{g(t)}$ evolves by

$$\begin{aligned} \frac{\partial }{\partial t}\mathrm {vol}_{g(t)}=&\frac{1}{2}\mathrm {tr}_g(\frac{\partial }{\partial t}g(t))\mathrm {vol}_{g(t)}=\frac{2}{3}|T|^2\mathrm {vol}_{g(t)}, \end{aligned}$$

(2.13)

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 (as already noted in Sect. 2.1). Since the torsion tensor T is defined by the first covariant derivative of $\varphi$ and the Riemannian curvature tensor $\mathrm {Rm}$ involves up to second order derivatives of the metric, we calculated in [34] that the evolution equations of the torsion tensor and Riemannian curvature tensor along the Laplacian flow are of the form

$$\begin{aligned} \frac{\partial }{\partial t}T=\Delta T+Rm*T+Rm*T*\psi +\nabla T*T*\varphi +T*T*T, \end{aligned}$$

(2.14)

$$\begin{aligned} \frac{\partial }{\partial t} Rm=\Delta Rm+Rm*Rm+Rm*T*T+\nabla ^2T*T+\nabla T*\nabla T, \end{aligned}$$

(2.15)

where we use $*$ to mean some contraction using the metric g(t) associated with $\varphi (t)$.

3 Foundational Results of Laplacian Flow

In this section, we discuss several foundational results on the Laplacian flow, which are important for further studies.

3.1 Shi-type Estimates

The first result is the derivative estimates of the solution to the Laplacian flow. For a solution $\varphi (t)$ of the Laplacian flow (2.1), we define the quantity

$$\begin{aligned} \Lambda (x,t)=\left( |\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right) ^{\frac{1}{2}}. \end{aligned}$$

(3.1)

Notice that the torsion tensor T is determined by the first order derivative of $\varphi$ and the curvature tensor Rm is second order in the metric $g_{\varphi }$, so both Rm and $\nabla T$ are second order in $\varphi$. We show that a bound on $\Lambda (x,t)$ will induce a priori bounds on all derivatives of Rm and $\nabla T$ for positive time. More precisely, we have the following.

Theorem 3.1

([34]) Suppose that $K>0$ and $\varphi (t)$ is a solution of the Laplacian flow (2.1) for closed $\mathrm{G}_2$ structures on a compact manifold $M^7$ for $t\in [0,\frac{1}{K}]$. For all $k\in \mathbb {N}$, there exists a constant $C_k$ such that if $\Lambda (x,t)\le K$ on $M^7\times [0,\frac{1}{K}]$, then

$$\begin{aligned} |\nabla ^kRm(x,t)|_{g(t)}+|\nabla ^{k+1}T(x,t)|_{g(t)}\le {C_k}{t^{-\frac{k}{2}}}K,\quad t\in (0,\frac{1}{K}]. \end{aligned}$$

(3.2)

We call the estimates (3.2) Shi-type 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 [1] and comprehensively by Shi [41] independently. The techniques used in [1, 41] were introduced by Bernstein (in the early twentieth century) for proving gradient estimates via the maximum principle, and was also the key in [34] to prove Theorem 3.1. A key motivation for defining $\Lambda (x,t)$ as in (3.1) is that the evolution equations of $|\nabla T(x,t)|^2$ and $|Rm(x,t)|^2$ both have some bad terms, but the chosen combination kills these terms and yields an effective evolution equation for $\Lambda (x,t)$ which looks like

$$\begin{aligned} \frac{\partial }{\partial t}\Lambda (x,t)^2&\le ~\Delta \Lambda (x,t)^2+C\Lambda (x,t)^{3} \end{aligned}$$

for some positive constant C. This shows that the quantity $\Lambda$ has similar properties to Riemann curvature under Ricci flow. Moreover, it implies that the assumption $\Lambda (x,t)\le K$ in Theorem 3.1 is reasonable as $\Lambda (x,t)$ cannot blow up quickly. We remark that the constant $C_k$ depends on the order of differentiation. In a joint work with Lotay [36], we showed that $C_k$ are of sufficiently slow growth in the order k and then we deduced that the $\mathrm{G}_2$ structure $\varphi (t)$ and associated metric $g_{\varphi (t)}$ are real analytic at each fixed time $t>0$.

The Shi-type estimates could be used to study finite-time singularities of the Laplacian flow. Given an initial closed $\mathrm{G}_2$ structure $\varphi _0$ on a compact 7-manifold, Theorem 2.2 tells us there exists a solution $\varphi (t)$ 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 (3.2), we have the following long time existence result on the Laplacian flow.

Theorem 3.2

([34]) If $\varphi (t)$ is a solution of the Laplacian flow (2.1) on a compact manifold $M^7$ in a maximal time interval $[0,T_0)$ with $T_0<\infty$, then

$$\begin{aligned} \lim _{t\nearrow T_0}\sup _{x\in M}\Lambda (x,t)= \infty . \end{aligned}$$

Moreover, there exists a positive constant C such that the blow-up rate satisfies

$$\begin{aligned} \sup _{x\in M} \Lambda (x,t)\ge \frac{C}{T_0-t}. \end{aligned}$$

In other words, Theorem 3.2 shows that the solution $\varphi (t)$ of the Laplacian flow for closed $\mathrm{G}_2$ structures will exist as long as the quantity $\Lambda (x,t)$ in (3.1) remains bounded.

3.2 Uniqueness

Given a closed $\mathrm{G}_2$ structure $\varphi _0$ on a compact 7-manifold, Theorem 2.2 says that there exists a unique solution to the Laplacian flow for a short time interval $t\in [0,\varepsilon )$. The proof in [5] relies on the Nash–Moser inverse function theorem [16] and the DeTurck’s trick. In [34], we gave a new proof the forward uniqueness by adapting an energy approach used previously by Kotschwar [28] for Ricci flow. The idea is to define an energy quantity $\mathcal {E}(t)$ in terms of the differences of the $\mathrm{G}_2$ structures, metrics, connections, torsion tensors and Riemann curvatures of two Laplacian flows, which vanishes if and only if the flows coincide. By deriving a differential inequality for $\mathcal {E}(t)$, it can be shown that $\mathcal {E}(t)=0$ if $\mathcal {E}(0)=0$, which gives the forward uniqueness. We also proved in [34] a backward uniqueness result for the solution of Laplacian flow by applying a general backward uniqueness theorem in [27] for time-dependent sections of vector bundles satisfying certain differential inequalities.

Theorem 3.3

([34]) Suppose $\varphi (t)$, $\tilde{\varphi }(t)$ are two solutions to the Laplacian flow (2.1) on a compact manifold $M^7$ for $t\in [0,\epsilon ]$, $\epsilon >0$. If $\varphi (s)=\tilde{\varphi }(s)$ for some $s\in [0,\epsilon ]$, then $\varphi (t)=\tilde{\varphi }(t)$ for all $t\in [0,\epsilon ]$.

An application of Theorem 3.3 is that on a compact manifold $M^7$, the subgroup $I_{\varphi (t)}$ of diffeomorphisms of M isotopic to the identity and fixing $\varphi (t)$ is unchanged along the Laplacian flow. Since $I_{\varphi }$ is strongly constrained for a torsion-free $\mathrm{G}_2$ structure $\varphi$ on M, this gives a test for when the Laplacian flow with a given initial condition could converge.

3.3 Compactness and $\kappa$-Non-collapsing

In the study of Ricci flow, Hamilton’s compactness theorem [18] and Perelman’s $\kappa$-non-collapsing estimate [38] are two essential tools to study the behavior of the flow near a singularity. We also have the analogous results for the Laplacian flow, which were proved by the author and Lotay [34] and Chen [6] respectively.

Theorem 3.4

([34]) Let $M_i$ be a sequence of compact 7-manifolds and let $p_i\in M_i$ for each i. Suppose that, for each i, $\varphi _i(t)$ is a solution to the Laplacian flow (2.1) on $M_i$ for $t\in (a,b)$, where $-\infty \le a<0<b\le \infty$. Suppose that

$$\begin{aligned} \sup _i\sup _{x\in M_i,t\in (a,b)}\Lambda _{\varphi _i}(x,t)<\infty \end{aligned}$$

(3.3)

and

$$\begin{aligned} \inf _i \text {inj}(M_i,g_i(0),p_i)>0. \end{aligned}$$

(3.4)

Then there exists a 7-manifold M, a point $p\in M$ and a solution $\varphi (t)$ of the Laplacian flow on M for $t\in (a,b)$ such that, after passing to a subsequence, $(M_i,\varphi _i(t),p_i)$ converge to $(M,\varphi (t),p)$ as $i\rightarrow \infty$.

To prove Theorem 3.4, we first proved in [34] a Cheeger–Gromov-type compactness theorem for the space of $\mathrm{G}_2$ structures, which states that the space of $\mathrm{G}_2$ structures with bounded $|\nabla ^{k+1}T|+|\nabla ^kRm|, k\ge 0$, and bounded injectivity radius is compact. Given this, Theorem 3.4 follows from a similar argument for the analogous compactness theorem in Ricci flow as in [18], with the help of the Shi-type estimate in Theorem 3.1.

The $\kappa$-non-collapsing estimate is an estimate on the volume ratio which only involves the Riemannian metric. A Riemannian metric g on a manifold M is $\kappa$-non-collapsed relative to an upper bound on the scalar curvature of the metric on the scale $\rho$ if for any geodesic ball $B_g(p,r)$ with $r<\rho$ such that $\sup _{B_g(p,r)}R_g\le r^{-2}$, there holds $\mathrm {Vol}(B_g(p,r))\ge \kappa r^n$. By using the same $\mathcal {W}$ functional, Chen [6] generalized Perelman’s $\kappa$-non-collapsing theorem [38] for Ricci flow to any flow

$$\begin{aligned} \frac{\partial }{\partial t}g(t)~=~-2\mathrm {Ric}(g(t))+E(t) \end{aligned}$$

(3.5)

for the Riemannian metric g(t), where E(t) is a symmetric 2-tensor.

Theorem 3.5

([6]) If $|E(t)|_{g(t)}$ is bounded along the flow (3.5) for $t\in [0,s)$ with $s<\infty$, then there exists $\kappa >0$ such that for all $t\in [0,s)$, g(t) is $\kappa$-non-collapsed relative to the upper bound on the scalar curvature on the scale $\rho =\sqrt{s}$.

Theorem 3.5 applies effectively to our Laplacian flow since the induced metric flow is just a perturbation of the Ricci flow, see (2.12). The $\kappa$-non-collapsing estimate is useful to estimate the lower bound on the injectivity radius, which together with the Shi-type estimate in Theorem 3.1 guarantees the condition of the compactness theorem for the purpose of the blow up analysis.

3.4 Solitons

Given a 7-manifold M, a Laplacian soliton on M is a triple $(\varphi ,X,\lambda )$ satisfying

$$\begin{aligned} \Delta _{\varphi }\varphi =\lambda \varphi +\mathcal {L}_X\varphi , \end{aligned}$$

(3.6)

where $d\varphi =0$, $\lambda \in \mathbb {R}$, X is a vector field on M and $\mathcal {L}_X\varphi$ is the Lie derivative of $\varphi$ in the direction of X. Laplacian solitons give self-similar solutions to the Laplacian flow. Specifically, suppose $(\varphi _0,X,\lambda )$ satisfies (3.6). Define $\rho (t)=(1+\frac{2}{3}\lambda t)^{\frac{3}{2}}, X(t)=\rho (t)^{-\frac{2}{3}}X$, and let $\phi _t$ be the family of diffeomorphisms generated by the vector fields X(t) such that $\phi _0$ is the identity. Then $\varphi (t)$ defined by $\varphi (t)=\rho (t)\phi _t^*\varphi _0$ is a solution of the Laplacian flow (2.1), which only differs by a scaling factor $\rho (t)$ and pull-back by a diffeomorphism $\phi _t$ for different times t. We say a Laplacian soliton $(\varphi ,X, \lambda )$ is expanding if $\lambda >0$; steady if $\lambda =0$; and shrinking if $\lambda <0$.

The soliton solutions of the Laplacian flow are expected to play a role in understanding the behavior of the flow near singularities. Thus the classification is an important problem. In this direction, Lin [30] proved that there are no compact shrinking solitons, and the only compact steady solitons are given by torsion-free $\mathrm{G}_2$ structures. In [34], we show that any Laplacian soliton that is an eigenform (i.e., $X=0$ in (3.6)) must be an expander or torsion-free. Hence, stationary points of the Laplacian flow on 7-manifold (not necessarily compact) are given by torsion-free $\mathrm{G}_2$ structures. Moreover, we show that there are no compact Laplacian solitons that are eigenforms unless $\varphi$ is torsion-free. Combining this with Lin’s result, any nontrivial Laplacian soliton on a compact manifold M (if it exists) must satisfy (3.6) for $\lambda >0$ and $X\ne 0$. This phenomenon is somewhat surprising, since it is very different from Ricci solitons $\mathrm {Ric}+\mathcal {L}_Xg=\lambda g$: when $X=0$, the Ricci soliton equation is just the Einstein equation and there are many examples of compact Einstein metrics.

Since a $\mathrm{G}_2$ structure $\varphi$ determines a unique metric g, it is natural to ask what condition the Laplacian soliton Eq. (3.6) on $\varphi$ will impose on g. By writing $\mathcal {L}_X\varphi$ with respect to the type decomposition of 3-forms, we derived from the Laplacian soliton Eq. (3.6) that the induced metric $g_{\varphi }$ satisfies, in local coordinates,

$$\begin{aligned} -R_{ij}-\frac{1}{3}|T|^2g_{ij}-2T_{ik}g^{kl}T_{lj}=\frac{1}{3}\lambda g_{ij}+\frac{1}{2}(\mathcal {L}_Xg)_{ij} \end{aligned}$$

(3.7)

and the vector field X satisfies $d^*(X\lrcorner \varphi )=0$. In particular, we deduce that any Laplacian soliton $(\varphi ,X,\lambda )$ must satisfy $7\lambda +3\mathrm{div}(X)=2|T|^2\ge 0$, which leads to a new short proof of Lin’s result [30] for the closed case.

Remark 3.6

We remark that there are many new results concerning the soliton solutions of the Laplacian flow. We refer the readers to [11, 31,32,33, 37] for details.

4 Extension Theorem

As we said in Sect. 3, the compactness theorem and the non-collapsing estimate could be used to study the singularities of the Laplacian flow. Theorem 3.2 already characterized the finite time singularities as the points where the quantity $\Lambda (x,t)$ (defined in (3.1)) blow up. This means that the solution of the Laplacian flow exists as long as $\Lambda (x,t)$ remains bounded. The quantity $\Lambda (x,t)$ consists of the full information of the $\mathrm{G}_2$ structure $\varphi (t)$ up to second derivatives. It’s interesting to see whether some weaker quantity can control the behavior of the flow. Using the compactness theorem, we improved Theorem 3.2 to the following desirable result, which states that the Laplacian flow will exist as long as the velocity of the flow remains bounded.

Theorem 4.1

([34]) Let M be a compact 7-manifold and $\varphi (t)$, $t\in [0,T_0)$, where $T_0<\infty$, be a solution to the Laplacian flow (2.1) with associated metric g(t) for each t. If the velocity of the flow satisfies $\sup _{M\times [0,T_0)}|\Delta _{\varphi }\varphi (x,t)|_{g(t)}<\infty$, then the solution $\varphi (t)$ can be extended past time $T_0$.

Note that for closed $\mathrm{G}_2$ structures, the velocity $\Delta _{\varphi }\varphi =d\tau$ is just some components of the first derivative of the torsion tensor. Theorem 4.1 is the $\mathrm{G}_2$ analogue of Sesum’s [39] theorem that the Ricci flow exists as long as the Ricci tensor remains bounded. It is an open question whether the scalar curvature (the trace of the Ricci tensor) is enough to control the behavior of the Ricci flow, though it is known for Type-I Ricci flow [8] and Kähler–Ricci flow [44]. For a closed $\mathrm{G}_2$ structure $\varphi$, the velocity $\Delta _{\varphi }\varphi =\mathrm {i}_{\varphi }(h)$ is equivalent to a symmetric 2-tensor h with trace equal to $\frac{2}{3}|T|^2$. Since the scalar curvature of the metric induced by $\varphi$ is $-|T|^2$, comparing with Ricci flow one may ask whether the Laplacian flow for closed $\mathrm{G}_2$ 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 $\mathrm{G}_2$ geometry. However, even though $-|T|^2$ is the scalar curvature, it is only first order in $\varphi$, rather than second order like $\Delta _{\varphi }\varphi$, so it would be a major step forward to control the Laplacian flow using just a bound on the torsion tensor.

The Proof of Theorem 4.1 involves a standard blow up analysis using the compactness theorem in Sect. 3. However, the non-collapsing estimate is not required for the proof. In fact, for a closed $\mathrm{G}_2$ structure $\varphi$, $\Delta _{\varphi }\varphi =\mathrm {i}_{\varphi }(h)$ and $|\Delta _{\varphi }\varphi |^2_g=(\mathrm {tr}_g(h))^2+2|h|^2$ with h given by (1.10). Then the condition $|\Delta _{\varphi (t)}\varphi (t)|_{g(t)}<\infty$ is equivalent to $\sup _{M\times [0,T_0)}|h(t)|<\infty$, which implies the uniform continuity of the metric g(t). A desired injectivity radius estimate then follows and the blow up analysis works.

Remark 4.2

By applying the compactness theorem and the non-collapsing estimate and using the method in [43], Chen [6] improved the result in Theorem 4.1. See [6] for the details. Moreover, Fine and Yao studied in [14] the hypersymplectic flow on a compact 4-manifold X related to the Laplacian flow on the 7-manifold $X\times \mathbb {T}^3$ and proved that the flow extends as long as the scalar curvature of the corresponding $\mathrm{G}_2$ structure remains bounded.

5 Stability of Torsion-Free $G_2$ Structures

As we stated in Question 2.1, Bryant asked the question whether the Laplacian flow with initial $\mathrm{G}_2$ structure $\varphi _0$ which is sufficiently close to a torsion-free $\mathrm{G}_2$ structure $\bar{\varphi }$ will converge to a point in the diffeomorphism orbit of $\bar{\varphi }$. Jointly with Lotay, we gave a positive answer in [35].

Theorem 5.1

([35]) Let $\bar{\varphi }$ be a torsion-free $\mathrm{G}_2$ structure on a compact 7-manifold M. Then there is a neighborhood $\mathcal {U}$ of $\bar{\varphi }$ such that for any $\varphi _0\in [\bar{\varphi }]_+\cap \mathcal {U}$, the Laplacian flow (2.1) with initial value $\varphi _0$ exists for all $t\in [0,\infty )$ and converges to $\varphi _{\infty }\in \mathrm{Diff}^{0}(M)\cdot \bar{\varphi }$ as $t\rightarrow \infty$. In other words, torsion-free $\mathrm{G}_2$ structures are (weakly) dynamically stable along the Laplacian flow for closed $\mathrm{G}_2$ structures.

The Proof of Theorem 5.1 is inspired by the proof of an analogous result in Ricci flow: Ricci-flat metrics are dynamically stable along the Ricci flow. The idea is to combine arguments for the Ricci flow case [20, 40] with the particulars of the geometry of closed $\mathrm{G}_2$ structures and new higher order estimates for the Laplacian flow derived by the author with Lotay in [34]. We first look at the Laplacian–DeTurck flow (2.5). By linearizing (2.5) at the torsion-free $\mathrm{G}_2$ structure $\bar{\varphi }$, we have (see (2.6)):

$$\begin{aligned} \frac{d}{d\epsilon }\bigg |_{\epsilon =0}\left( \Delta _{\bar{\varphi }+\epsilon \theta }(\bar{\varphi }+\epsilon \theta )+\mathcal {L}_{V(\bar{\varphi }+\epsilon \theta )}(\bar{\varphi }+\epsilon \theta )\right) ~=~-\Delta _{\bar{\varphi }}\theta , \end{aligned}$$

(5.1)

where $\theta$ is an exact 3-form. Note that the operator $-\Delta _{\bar{\varphi }}$ is strictly negative on the space of exact 3-forms by Hodge decomposition theorem. Let $\tilde{\varphi }(t)$ be the solution of Laplacian–DeTurck flow and denote $\theta (t)=\tilde{\varphi }(t)-\bar{\varphi }$. By the linearization (5.1), there exists $\epsilon >0$ such that for all t for which $\Vert \theta (t)\Vert _{C^k_{\bar{g}}}<\epsilon$, we have

$$\begin{aligned} \frac{\partial }{\partial t}\theta (t) =&-\Delta _{\bar{\varphi }}\theta +dF(\bar{\varphi },\tilde{\varphi }(t),\theta (t),\bar{\nabla }\theta (t)), \end{aligned}$$

where F is a 2-form which is smooth in the first two arguments and linear in the last two arguments. The idea is that if $\theta (t)$ is sufficiently small, the behavior of the Laplacian–DeTurck flow is dominated by the linear term $-\Delta _{\bar{\varphi }}\theta$. If the initial $\varphi _0$ is sufficiently close to $\bar{\varphi }$, i.e., $\theta (0)$ is sufficiently small, by estimating the velocity of the Laplacian–DeTurck flow we can show that the solution exists and remains small at least for time $t\in [0,1]$. By using the strict negativity of the operator $-\Delta _{\bar{\varphi }}$, we show that $\theta (t)$ has an exponential decay in $L^2$ norm as long as the solution exists and remains small. By deriving higher order integral estimates, we can in fact show that the solution of the Laplacian–DeTurck flow exists for all time and also converges to $\bar{\varphi }$ exponentially and smoothly as time goes to infinity. The final step is to transform back to Laplacian flow via time-dependent diffeomorphisms $\phi (t)$ determined by the vector field $V(\tilde{\varphi }(t))$. The Shi-type esimate and compactness result apply here to show the smooth convergence of Laplacian flow and completes the proof.

As we mentioned in Sect. 2, Joyce [22] proved an existence result for torsion-free $\mathrm{G}_2$ structures, which states that if we control the $C^0$ and $L^2$-norms of $\gamma$ and the $L^{14}$-norm of $d^*_{\varphi _0}\gamma =d^*_{\varphi _0}\varphi _0$, we can deform $\varphi _0$ in its cohomology class to a unique $C^0$-close torsion-free $\mathrm{G}_2$ structure $\bar{\varphi }$. By choosing a neighbourhood $\mathcal {U}$ appropriately, controlling derivatives up to at least order 8, we can ensure that we can apply both the theory in [22] and Theorem 5.1, and thus deduce the following corollary.

Corollary 5.2

([35]) Let $\varphi _0$ be a closed $\mathrm{G}_2$ structure on a compact 7-manifold M. There exists an open neighbourhood $\mathcal {U}$ of 0 in $\Omega ^3(M)$ such that if $d^*_{\varphi _0}\varphi _0=d^*_{\varphi _0}\gamma$ for some $\gamma \in \mathcal {U}$, then the Laplacian flow (2.1) with initial value $\varphi _0$ exists for all time and converges to a torsion-free $\mathrm{G}_2$ structure.

The neighbourhood  $\mathcal {U}$ given by Corollary 5.2 is not optimal, and one would like to be able to prove this result directly using the Laplacian flow with optimal conditions and without recourse to [22], but nevertheless, Corollary 5.2 gives significant evidence that the Laplacian flow will play an important role in understanding the problem of existence of torsion-free $\mathrm{G}_2$ structures on 7-manifolds admitting closed $\mathrm{G}_2$ structures.

Our results also motivate us to study an approach to the following problem, as pointed out by Thomas Walpuski. The work of Joyce [22] shows that the natural map from the moduli space $\mathcal {M}$ of torsion-free $\mathrm{G}_2$ structures to $H^3(M)$ given by $\mathrm{Diff}^0(M)\cdot \bar{\varphi }\mapsto [\bar{\varphi }]$ is locally injective, but the question of whether this map is globally injective, raised by Joyce (c.f. [23]), is still open. Suppose we have two torsion-free $\mathrm{G}_2$ structures $\bar{\varphi }_0$ and $\bar{\varphi }_1$ which lie in the same cohomology class, so we can write $\bar{\varphi }_1=\bar{\varphi }_0+ d\eta$ for some 2-form $\eta$. We would like to see whether $\bar{\varphi }_1\in \mathrm{Diff}^0(M)\cdot \bar{\varphi }_0$. By our main theorem (Theorem 5.1) we know that the Laplacian flow starting at $\varphi _0(s)=\bar{\varphi }_0+sd\eta$ (which is closed) will exist for all time and converge to $\phi _s^*\bar{\varphi }_0$ for some $\phi _s\in \mathrm{Diff}^0(M)$ when s is sufficiently small. Similarly, the Laplacian flow starting at $\varphi _0(s)$ for s near 1 will also exist for all time and now converge to $\phi _s^*\bar{\varphi }_1$ for some $\phi _s\in \mathrm{Diff}^0(M)$. The aim would be to study long-time existence and convergence of the flow starting at any $\varphi _0(s)$.