Local curvature estimates for the Laplacian flow

Abstract

In this paper we give local curvature estimates for the Laplacian flow on closed $G_{2}$-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed $G_{2}$-structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed $G_{2}$-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature $R_{g(t)}$ is equal to the Laplacian of $R_{g(t)}$, plus an extra term which can be written as the difference of two nonnegative quantities.

1 Introduction

Let $\mathcal {M}$ be a smooth 7-manifold. The Laplacian flow for closed $G_{2}$-structures on $\mathcal {M}$ introduced by Bryant [1] is to study the torsion-free $G_{2}$-structures

$$\begin{aligned} \partial _{t}\varphi (t)=\Delta _{\varphi (t)}\varphi (t) , \ \ \ \varphi (0)=\varphi , \end{aligned}$$

(1.1)

where $\Delta _{\varphi (t)}\varphi (t)=dd^{*}_{\varphi (t)} \varphi (t)+d^{*}_{\varphi (t)} d\varphi (t)$ is the Hodge Laplacian of $g_{\varphi (t)}$ and $\varphi$ is an initial closed $G_{2}$-structure. Since $d\partial _{t}\varphi (t)=\partial _{t}d\Delta _{\varphi (t)}\varphi (t) =0$, we see that the flow (1.1) preserves the closedness of $\varphi (t)$. For more background on $G_{2}$-structures, see Sect. 2. When $\mathcal {M}$ is compact, the flow (1.1) can be viewed as the gradient flow for the Hitchin functional introduced by Hitchin [18]

$$\begin{aligned} \mathscr {H}: [\overline{\varphi }]_{+}\longrightarrow \mathbb {R}^{+}, \ \ \ \varphi \longmapsto \frac{1}{7}\int _{\mathcal {M}}\varphi \wedge \psi =\int _{\mathcal {M}} *_{\varphi }1. \end{aligned}$$

(1.2)

Here $\overline{\varphi }$ is a closed $G_{2}$-structure on $\mathcal {M}$ and $[\overline{\varphi }]_{+}$ is the open subset of the cohomology class $[\overline{\varphi }]$ consisting of $G_{2}$-structures. Any critical point of $\mathscr {H}$ gives a torsion-free $G_{2}$-structure.

The study of Laplacian flows on some special 7-manifolds, Laplacian solitons, and other flows on $G_{2}$-structures can be found in [13,14,15,16, 19, 24, 29, 33, 34, 38, 39].

Recently, Donaldson [7,8,9,10] studied the co-associative Kovalev-Lefschetz fibrations $G_{2}$-manifolds and $G_{2}$-manifolds with boundary.

1.1 Notions and conventions

To state the main results, we fix our notions used throughout this paper. Let $\mathcal {M}$ be as before a smooth 7-manifold. The space of smooth functions and the space of smooth vector fields are denoted respectively by $C^{\infty }(\mathcal {M})$ and $\mathfrak {X}(\mathcal {M})$. The space of k-tenors (i.e., (0, k)-covariant tensor fields) and k-forms on $\mathcal {M}$ are denoted, respectively, by $\otimes ^{k}(\mathcal {M}) =C^{\infty }(\otimes ^{k}(T^{*}\mathcal {M}))$ and $\wedge ^{k}(\mathcal {M})=C^{\infty }(\wedge ^{k}(T^{*} \mathcal {M}))$. For any k-tensor field $\varvec{T}\in \otimes ^{k}(\mathcal {M})$, we locally have the expression $\varvec{T}=\varvec{T}_{i_{1}\cdots i_{k}}dx^{i_{1}}\otimes \cdots \otimes dx^{i_{k}} =:\varvec{T}_{i_{1}\cdots i_{k}}dx^{i_{1}\otimes \cdots \otimes i_{k}}$. A k-form $\alpha$ on $\mathcal {M}$ can be written in the standard form as $\alpha =\frac{1}{k!}\alpha _{i_{1}\cdots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=:\frac{1}{k!}\alpha _{i_{1}\cdots i_{k}}dx^{i_{1} \wedge \cdots \wedge i_{k}}$, where $\alpha _{i_{1}\cdots i_{k}}$ is fully skew-symmetric in its indices. Using the standard forms, if we take the interior product $X\lrcorner \alpha$ of a k-form $\alpha \in \wedge ^{k}(\mathcal {M})$ with a vector field $X\in \mathfrak {X}(\mathcal {M})$, we obtain the $(k-1)$-form $X\lrcorner \alpha =\frac{1}{(k-1)!}X^{m}\alpha _{mi_{1}\cdots i_{k-1}} dx^{i_{1}\wedge \cdots \wedge i_{k-1}}$ which is also in the standard form. In particular, consider the vector space $\otimes ^{2}(\mathcal {M})$ of 2-tensors. For any 2-tensor $\varvec{A}=\varvec{A}_{ij}dx^{i\otimes j}$, define $\varvec{A}^{\odot }:=\frac{1}{2}(\varvec{A}_{ij}+\varvec{A}_{ji})dx^{i\otimes j}\equiv \varvec{A}^{\odot }_{ij} dx^{i\otimes j}$ and $\varvec{A}^{\wedge }:=\frac{1}{2}(\varvec{A}_{ij}-\varvec{A}_{ji}) dx^{i\otimes j}\equiv \varvec{A}^{\wedge }_{ij}dx^{i\otimes j}$. Then $\varvec{A}^{\odot }$ is an element of $\odot ^{2}(\mathcal {M})$, the space of symmetric 2-tensors. Since¹$dx^{i\wedge j}=dx^{i\otimes j}-dx^{j\otimes i}$, it follows that $\varvec{A}^{\wedge }=\frac{1}{2}\varvec{A}_{ij}dx^{i\wedge j}$. Define $\alpha ^{\varvec{A}}:=\frac{1}{2}\alpha ^{\varvec{A}}_{ij}dx^{i\wedge j}$ with $\alpha ^{\varvec{A}}_{ij}:= \varvec{A}_{ij}$. Then we see that $\alpha ^{\varvec{A}}=\varvec{A}^{\wedge }\in \wedge ^{2}(\mathcal {M})$ and $\otimes ^{2}(\mathcal {M})=\odot ^{2}(\mathcal {M}) \oplus \wedge ^{2}(\mathcal {M})$.

A given Riemannian metric g on $\mathcal {M}$ determines two isomorphisms between vector fields and 1-forms: $\flat _{g}: \mathfrak {X}(\mathcal {M})\longrightarrow \wedge ^{1} (\mathcal {M})$ and $\sharp _{g}: \wedge ^{1}(\mathcal {M})\longrightarrow \mathfrak {X}(\mathcal {M})$, where, for every vector field $X=X^{i}\frac{\partial }{\partial x^{i}}$ and 1-form $\alpha =\alpha _{i}dx^{i}$, $\flat _{g}(X)=X^{i}g_{ij}dx^{j}\equiv X_{j}dx^{j}$ and $\sharp _{g}(\alpha ) =\alpha _{i}g^{ij}\frac{\partial }{\partial x^{j}}\equiv \alpha ^{j}\frac{\partial }{\partial x^{j}}$. Using these two natural maps, we can frequently raise or lower indices on tensors. The metric g also induces a metric on k-forms $g(dx^{i_{1}\wedge \cdots \wedge i_{k}},dx^{j_{1} \wedge \cdots \wedge j_{k}})= \det (g(dx^{i_{a}},dx^{j_{b}}))=\sum _{\sigma \in \mathfrak {S}_{7}}\mathrm{sgn}(\sigma )g^{i_{1}j_{\sigma (1)}} \cdots g^{i_{k}j_{\sigma (k)}}$ where $\mathfrak {S}_{7}$ is the group of permutations of seven letters and $\mathrm{sgn}(\sigma )$ denotes the sign $(\pm 1)$ of an element $\sigma$ of $\mathfrak {S}_{7}$. The inner product $\langle \cdot , \cdot \rangle _{g}$ of two k-forms $\alpha ,\beta \in \wedge ^{k} (\mathcal {M})$ now is given by $\langle \alpha ,\beta \rangle _{g}=\frac{1}{k!} \alpha _{i_{1}\cdots i_{k}}\beta ^{i_{1}\cdots i_{k}} =\frac{1}{k!}\alpha _{i_{1}\cdots i_{k}}\beta _{j_{1}\cdots j_{k}} g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}$.

Given two 2-tensors $\varvec{A}, \varvec{B}\in \otimes ^{2}(\mathcal {M})$, with the forms $\varvec{A}=\varvec{A}_{ij}dx^{i\otimes j}$ and $\varvec{B}=\varvec{B}_{ij}dx^{i\otimes j}$. Define $\langle \langle \varvec{A}, \varvec{B}\rangle \rangle _{g}:=\varvec{A}_{ij}\varvec{B}^{ij}$. There are two special cases which will be used later:

• (1) $\alpha =\frac{1}{2}\alpha _{ij}dx^{i\wedge j}\in \wedge ^{2}(\mathcal {M})$ and $\varvec{B}=\varvec{B}_{ij}dx^{i\otimes j}\in \otimes ^{2}(\mathcal {M})$. In this case, $\alpha$ can be written as a 2-tensor $\varvec{A}^{\alpha }=\varvec{A}^{\alpha }_{ij}dx^{i\otimes j}$ with $\varvec{A}^{\alpha }_{ij} =\alpha _{ij}$. Then $\langle \langle \alpha ,\varvec{B}\rangle \rangle _{g}:=\langle \langle \varvec{A}^{\alpha }, \varvec{B}\rangle \rangle _{g}=\alpha _{ij}\varvec{B}^{ij}$.

• (2) $\alpha =\frac{1}{2}\alpha _{ij}dx^{i\wedge j}$ and $\beta =\frac{1}{2}\beta _{ij} dx^{i\wedge j}\in \wedge ^{2}(\mathcal {M})$. In this case, $\alpha , \beta$ can be both written as 2-tensors $\varvec{A}^{\alpha }=\varvec{A}^{\alpha }_{ij} dx^{i\otimes j}$ and $\varvec{B}^{\beta }=\varvec{B}^{\beta }_{ij}dx^{i\otimes j}$ with $\varvec{A}^{\alpha }_{ij} =\alpha _{ij}$ and $\varvec{B}^{\beta }_{ij}=\beta _{ij}$. Then $\langle \langle \alpha ,\beta \rangle \rangle _{g}:=\langle \langle \varvec{A}^{\alpha }, \varvec{B}^{\beta }\rangle \rangle _{g} =\alpha _{ij}\beta ^{ij}=2\langle \alpha ,\beta \rangle _{g}$.

The norm of $\varvec{A}\in \otimes ^{2}(\mathcal {M})$ is defined by $||\varvec{A}||^{2}_{g}:=\langle \langle \varvec{A},\varvec{A}\rangle \rangle _{g} =\varvec{A}_{ij}\varvec{A}^{ij}$, while the norm of $\alpha \in \wedge ^{k} (\mathcal {M})$ is $|\alpha |^{2}_{g}:=\langle \alpha ,\alpha \rangle _{g} =\frac{1}{k!}\alpha _{i_{1}\cdots i_{k}}\alpha ^{i_{1}\cdots i_{k}}$. In particular, $||X||^{2}_{g}=X_{i}X^{i}=|\flat _{g}(X)|^{2}_{g}$ and $||\alpha ||^{2}_{g}=2|\alpha |^{2}_{g}$, for any vector field $X\in \mathfrak {X}(\mathcal {M})$ and 2-form $\alpha$.

The Levi–Civita connection associated to a given Riemannian metric g is denoted by $\nabla _{g}$ or simply $\nabla$. Our convention on Riemann curvature tensor is $R^{m}_{ijk}\frac{\partial }{\partial x^{m}}$ $:=\mathrm{Rm}(\frac{\partial }{\partial x^{i}},\frac{\partial }{\partial x^{j}} )\frac{\partial }{\partial x^{k}}=(\nabla _{i}\nabla _{j} -\nabla _{j}\nabla _{i})\frac{\partial }{\partial x^{k}}$ and $R_{ijk\ell }:=R^{m}_{ijk}g_{m\ell }$. The Ricci curvature of g is given by $R_{jk}:=R_{ijk\ell }g^{i\ell }$. We use $dV_{g}$ and $*_{g}$ to denote the volume form and Hodge star operator, respectively, on $\mathcal {M}$ associated to a metric g and an orientation.

We use the standard notion $A*B$ to denote some linear combination of contractions of the tensor product $A\otimes B$ relative to the metric g(t) associated the $\varphi (t)$. In Theorem 1.4 and its proof, all universal constants c, C below depend only on the given real number p.

1.2 Main results

Applying De Turck’s trick and Hamilton’s Nash-Moser inverse function theorem, Bryant and Xu [2] proved the following local time existence for (1.1).

Theorem 1.1

(Bryant-Xu [2]) For a compact 7-manifold $\mathcal {M}$, the initial value problem (1.1) has a unique solution for a short time interval $[0,T_{\max })$ with the maximal time $T_{\max }\in (0,\infty ]$ depending on $\varphi$.

As in the Ricci flow, we can prove following results on the long time existence for the Laplacian flow (1.1).

Theorem 1.2

(Lotay-Wei [32]) Let $\mathcal {M}$ be a compact 7-manifold and $\varphi (t)$, $t\in [0,T)$, where $T<\infty$, be a solution to the flow (1.1) for closed $G_{2}$-structures with associated metric $g(t)=g_{\varphi (t)}$ for each t.

• (a) If the velocity of the flow satisfies

  $$\begin{aligned} \sup _{\mathcal {M}\times [0,T)}||\Delta _{g(t)}\varphi (t) ||_{g(t)}<\infty , \end{aligned}$$

  then the solution $\varphi _{t}$ can be extended past time T.

• (b) If $T=T_{\max }$, then

  $$\begin{aligned} \limsup _{t\rightarrow T_{\max }}\max _{\mathcal {M}}\left( ||\mathrm{Rm}_{g(t)}||^{2}_{g(t)} +||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}\right) =\infty . \end{aligned}$$

  Here $\varvec{T}(t)$ is the torsion of $\varphi (t)$ [see (2.14)].

In this paper, we give a new elementary proof of Theorem 1.2, based on the idea of [25] and the structure of the Eq. (1.1).

Theorem 1.3

Let $\mathcal {M}$ be a compact 7-manifold and $\varphi (t)$, $t\in [0,T)$, where $T<\infty$, be a solution to the flow (1.1) for closed $G_{2}$-structures with associated metric $g(t)=g_{\varphi (t)}$ for each t. Suppose that

$$\begin{aligned} K:=\sup _{\mathcal {M}\times [0,T)} ||\mathrm{Ric}_{g(t)}||_{g(t)}<\infty , \ \ \ \Lambda :=\max _{\mathcal {M}} ||\mathrm{Rm}_{g(0)}||_{g(0)}. \end{aligned}$$

Then

$$\begin{aligned} \sup _{\mathcal {M}\times [0,T)}||\mathrm{Rm}_{g(t)}||_{g(t)}<\infty , \end{aligned}$$

where the bound depends only on n, K, T and $\Lambda$.

When $\mathcal {M}$ is compact, the theorem immediately implies the part (a) in Theorem 1.2. Indeed, we shall show that [see (3.10) and (3.29)]

$$\begin{aligned} \sup _{\mathcal {M} \times [0,T)}||\Delta _{g(t)} \varphi (t)||_{g(t)}<\infty \Longleftrightarrow \sup _{\mathcal {M} \times [0,T)}||\mathrm{Ric}_{g(t)}||_{g(t)}<\infty . \end{aligned}$$

In the compact case, Theorem 1.3 shows that, if the conclusion in part (a) does not hold, then $T=T_{\max }$ and $\sup _{\mathcal {M} \times [0,T_{\max })}||\mathrm{Rm}_{g(t)}||_{g(t)}<\infty$ which implies the quantity $\sup _{ \mathcal {M}\times [0,T_{\max })}$ $(||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}+||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)})$ is finite, since the norm $||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}$ can be controlled by $||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}$ [see (3.58)]. However, by part (b) in Theorem 1.2, it is impossible. Therefore, the conclusion in part (a) is true.

As remarked in [25], to prove Theorem 1.3, it suffices to establish the following integral estimate.

Theorem 1.4

Let $\mathcal {M}$ be a smooth 7-manifold and $\varphi (t)$, $t\in [0,T)$, where $T<\infty$, be a solution to the flow (1.1) for closed $G_{2}$-structures with associated metric $g(t) =g_{\varphi (t)}$ for each t. Assume that there exist constants $A, K>0$ and a point $x_{0}\in \mathcal {M}$ such that the geodesic ball $B_{g(0)}(x_{0}, A/\sqrt{K})$ is compactly contained in $\mathcal {M}$ and that

$$\begin{aligned} |\mathrm{Ric}_{g(t)}|_{g(t)}\le K \ \ \ \text {on} \ B_{g(0)}\left( x_{0},\frac{A}{\sqrt{K}}\right) \times [0,T]. \end{aligned}$$

Then, for any $p\ge 5$, there exists $c=c(p)>0$ so that

$$\begin{aligned}&\int _{B_{g(0)}(x_{0},A/2\sqrt{K})} ||\mathrm{Rm}_{g(t)}||^{p}_{g(t)}dV_{t}\nonumber \\&\quad \le c(1+K) e^{cKT}\int _{B_{g(0)}(x_{0},A/\sqrt{K})} ||\mathrm{Rm}_{g(0)}||^{p}_{g(0)}dV_{g(0)}\nonumber \\&\qquad + \ c K^{p}\left( 1+A^{-2p}\right) e^{cKT} \mathrm{Vol}_{g(t)}\left( B_{g(0)} \left( x_{0},\frac{A}{\sqrt{K}}\right) \right) \end{aligned}$$

(1.3)

for all $t\in [0,T]$.

Now by the standard De Giorgi–Nash–Moser iteration (our manifold is compact and the Ricci curvature is uniformly bounded), under the condition in Theorem 1.4, we can prove

$$\begin{aligned} ||\mathrm{Rm}_{g(T)}||_{g(T)}(x_{0}) \le d_{1}(d_{2}+\Lambda _{0}), \end{aligned}$$

(1.4)

where $d_{1}, d_{2}$ are constants depending on K, T, A, and

$$\begin{aligned} \Lambda _{0}:=\sup _{B_{g(0)}(x_{0},A/\sqrt{K})} ||\mathrm{Rm}_{g(0)}||_{g(0)}. \end{aligned}$$

Actually, this follows from the same argument in [25] by noting that

$$\begin{aligned} (\Delta _{g(t)} -\partial _{t})||\mathrm{Rm}_{g(t)}||_{g(t)}\ge -c||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}. \end{aligned}$$

(1.5)

To verify (1.5), we use (2.26), (3.56) and (3.60) to deduce that

$$\begin{aligned} ||\nabla _{g(t)}\varvec{T}(t)||_{g(t)}\le c||\mathrm{Rm}_{g(t)}||_{g(t)} \end{aligned}$$

and

$$\begin{aligned} ||\nabla ^{2}_{g(t)}\varvec{T}(t)||_{g(t)} \le c||\nabla _{g(t)}\mathrm{Rm}_{g(t)}||_{g(t)}+c||\mathrm{Rm}_{g(t)}||^{3/2}_{g(t)}. \end{aligned}$$

Then, by (3.23) and the Cauchy inequality

$$\begin{aligned} ||\nabla _{g(t)}\mathrm{Rm}_{g(t)}||^{2}_{g(t)}\le & {} -\frac{1}{2}(\partial _{t}-\Delta _{g(t)})||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}+c||\mathrm{Rm}_{g(t)}||^{3}_{g(t)}\\&+ \ c||\mathrm{Rm}_{g(t)}||^{3/2}_{g(t)} ||\nabla _{g(t)}\mathrm{Rm}_{g(t)}||_{g(t)}\\\le & {} -\frac{1}{2}(\partial _{t}-\Delta _{g(t)})||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}\\&+ \ c||\mathrm{Rm}_{g(t)}||^{3}_{g(t)}+||\nabla _{g(t)}\mathrm{Rm}_{g(t)}||^{2}_{g(t)} \end{aligned}$$

which implies (1.5). Now the estimate (1.4) yields Theorem 1.3.

The analogue of Theorem 1.2 in the Ricci flow was proved by Hamilton [17] (for part (b)) and Sesum [37] (for part (a)). It is an open question (due to Hamilton, see [3]) that the Ricci flow will exist as long as the scalar curvature remains bounded. For the Kähler–Ricci flow [40] or type-I Ricci flow [11], this question was settled. For the general case, some partial result on Hamilton’s conjecture was carried out in [3].

For the Ricci-harmonic flow introduce by List [30, 31] (see also, [35, 36]), the analogue of Theorem 1.2 was proved in [30, 31] (see also, [35, 36]) and [4] (see [28] for another proof). The author [26, 27] extended Cao’s result [3] to the Ricci-harmonic flow. The same Hamilton’s conjecture was asked by the author in [26, 27].

We can ask the same question for the Laplacian flow on closed $G_{2}$-structures. In [32] (see p. 171, line -6 to -3, or Open Problem (3) in p. 230), Lotay and Wei asked that whether the Laplacian flow on closed $G_{2}$-structures will exist as long as the torsion tensor or scalar curvature remains bounded. Let g(t) be the associated metric of $\varphi (t)$. Then the evolution equation for $g_{t}$ is given by

$$\begin{aligned} \partial _{t}g_{ij}=-2R_{ij}-\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)} g_{ij} -4\varvec{T}_{i}{}^{k}\varvec{T}_{kj}. \end{aligned}$$

(1.6)

For the Laplacian flow on closed $G_{2}$-structures, the torsion $\varvec{T}(t)$ is actually a 2-form for each t, hence we use the norm $|\cdot |_{g(t)}$ in (1.6). The standard formula for the scalar curvature $R_{g(t)}$ gives [see (3.15)]

$$\begin{aligned} \partial _{t}R_{g(t)}=\Delta _{g(t)}R_{g(t)} +2||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}-\frac{2}{3}R^{2}_{g(t)} +4R_{ijk\ell }\varvec{T}^{ik}\varvec{T}^{j\ell }+4(\nabla ^{j}\varvec{T}^{ik}) (\nabla _{i}\varvec{T}_{jk}). \end{aligned}$$

(1.7)

Now the above mentioned open problem states that

$$\begin{aligned} \text {Is it ture that} \ \limsup _{t\rightarrow T_{\max }}R_{g(t)}=-\infty ? \end{aligned}$$

The “minus infinity” comes from the fact that along the Laplacian flow on closed $G_{2}$-structures the scalar curvature is always nonpositive [see (2.26)]. The following Proposition 1.5 is motivate to solve this problem, and starts from the basic evolution Eq. (1.7) where the last two terms on the right-hand side do not have good signature. However, using the closedness of $\varphi (t)$ [in particular, the identity (3.15)], we can prove the following interesting evolution equation for $R_{g(t)}$.

Proposition 1.5

Let $\mathcal {M}$ be a smooth 7-manifold and $\varphi (t)$, $t\in [0,T)$, where $T\in (0,\infty ]$, be a solution to the flow (1.1) for closed $G_{2}$-structures with associated metric $g(t)= g_{\varphi (t)}$ for each t. Then the scalar curvature $R_{g(t)}$ satisfies

$$\begin{aligned} \partial _{t}R_{g(t)}= & {} \Delta _{g(t)}R_{g(t)}+\bigg \{2\left| \left| R_{ij}+\frac{2}{3} |\varvec{T}(t)|^{2}_{g(t)}g_{ij}\right| \right| ^{2}_{g(t)} +\frac{1}{2}\left| \left| R_{ijab}R^{ij}{}_{mn} -\psi _{abmn}\right| \right| ^{2}_{g(t)}\nonumber \\&+ \ \frac{1}{2} \left| \left| 2\varvec{T}_{ia}\varvec{T}_{jb}R^{ij}{}_{mn} -\psi _{abmn}\right| \right| ^{2}_{g(t)}\nonumber \\&+\frac{1}{2}\left| \left| 2\widehat{\varvec{T}}_{am}\widehat{\varvec{T}}_{bn} -\psi _{abmn}\right| \right| ^{2}_{g(t)}+2||\widehat{\varvec{T}}(t)||^{2}_{g(t)} \nonumber \\&+ \ 4||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}\bigg \}-\bigg \{||\mathrm{Rm}_{g(t)}||^{2}_{g(t)} +\frac{26}{9}R^{2}_{g(t)}+\frac{1}{2} \left| \left| R_{ijab}R^{ij}{}_{mn}\right| \right| ^{2}_{g(t)}\nonumber \\&+ \ 2\left| \left| \varvec{T}_{ia}\varvec{T}_{jb}R^{ij}{}_{mn} \right| \right| ^{2}_{g(t)} +2||\widehat{\varvec{T}}_{g(t)}||^{4}_{g(t)}+210\bigg \}. \end{aligned}$$

(1.8)

Here $\widehat{\varvec{T}}_{ij}=\varvec{T}_{i}{}^{k}\varvec{T}_{kj}$.

The evolution Eq. (1.8) can be written simply as

$$\begin{aligned} \partial _{t}R_{g(t)}=\Delta _{g(t)}R_{g(t)}+A(t)-B(t) \end{aligned}$$

(1.9)

for some suitable time-dependent nonnegative functions A(t) and B(t). By the maximum principle we obtain

$$\begin{aligned} R_{\max }(0)+\int ^{t}_{0}\max _{\mathcal {M}}[A(\tau )-B(\tau )]d\tau \ge R_{g(t)}\ge R_{\min }(0)+\int ^{t}_{0}\min _{\mathcal {M}} [A(\tau )-B(\tau )]d\tau . \end{aligned}$$

Here $R_{\max }(0):=\max _{\mathcal {M}}R_{g(0)}$ and $R_{\min }(0):= \min _{\mathcal {M}}R_{g(0)}$. Observe that the above well-arranged evolution equation can give us a weakly lower bound for $R_{g(t)}$, which can not prove or disprove the conjecture of Lotay and Wei.

We give an outline of the current paper. We review the basic theory in Sect. 2 about $G_{2}$-structures, $G_{2}$-decompositions of 2-forms and 3-forms, and general flows on $G_{2}$-structures. In Sect. 3, we rewrite results in Sect. 2 for closed $G_{2}$-structures, and the local curvature estimates will be given in the last subsection.

2 Basic theory of $G_{2}$-structures

In this section, we view some basic theory of $G_{2}$-structures, following [1, 20,21,22,23, 32]. Let $\{e_{1},\ldots , e_{7}\}$ denote the standard basis of $\mathbb {R}^{7}$ and let $\{e^{1},\ldots , e^{7}\}$ be its dual basis. Define the 3-form

$$\begin{aligned} \phi :=e^{1\wedge 2\wedge 3}+e^{1\wedge 4\wedge 5}+e^{1\wedge 6\wedge 7} +e^{2\wedge 4\wedge 6}-e^{2\wedge 5\wedge 7}-e^{3\wedge 4\wedge 7} -e^{3\wedge 5\wedge 6}, \end{aligned}$$

where $e^{ i\wedge j\wedge k}:= e^{i}\wedge e^{j}\wedge e^{k}$. The subgroup $G_{2}$, which fixes $\phi$, of $\mathbf{GL}(7,\mathbb {R})$ is the 14-dimensional Lie subgroup of $\mathbf{SO}(7)$, acts irreducibly on $\mathbb {R}^{7}$, and preserves the metric and orientation for which $\{e_{1},\cdots , e_{7}\}$ is an oriented orthonormal basis. Note that $G_{2}$ also preserves the 4-form

$$\begin{aligned} *_{\phi }\phi =e^{4\wedge 5\wedge 6\wedge 7}+e^{2\wedge 3\wedge 6 \wedge 7}+e^{2\wedge 3\wedge 4\wedge 5}+e^{1\wedge 3\wedge 5\wedge 7}-e^{1\wedge 3\wedge 4\wedge 6}-e^{1\wedge 2\wedge 5\wedge 6}-e^{1\wedge 2\wedge 4\wedge 7}. \end{aligned}$$

where the Hodge star operator $*_{\phi }$ is determined by the metric and orientation.

For a smooth 7-manifold $\mathcal {M}$ and a point $x\in \mathcal {M}$, define as in [32]

$$\begin{aligned} \wedge ^{3}_{+}(T^{*}_{x}\mathcal {M}):= \left\{ \varphi _{x}\in \wedge ^{3}(T^{*}_{x}\mathcal {M}): \begin{array}{cc} \textsf {u}^{*} \phi =\varphi _{x} \ \text {for some invertible}\\ \text {map} \ \textsf {u}\in \mathrm{Hom}_{\mathbb {R}}(T_{x}\mathcal {M}, \mathbb {R}^{7}) \end{array} \right\} \end{aligned}$$

and the bundle

$$\begin{aligned} \wedge ^{3}_{+}(T^{*}\mathcal {M}):=\bigsqcup _{x\in \mathcal {M}} \wedge ^{3}_{+}(T^{*}_{x}\mathcal {M}). \end{aligned}$$

We call a section $\varphi$ of $\wedge ^{3}_{+}(T^{*}\mathcal {M})$ a positive 3-form on $\mathcal {M}$ or a $G_{2}$-structure on $\mathcal {M}$, and denote the space of positive 3-forms by $\wedge ^{3}_{+}(\mathcal {M})$. The existence of $G_{2}$-structures is equivalent to the property that $\mathcal {M}$ is oriented and spin, which is equivalent to the vanishing of the first and second Stiefel–Whitney classes. From the definition of $G_{2}$-structures, we see that any $\varphi \in \wedge ^{3}_{+}(\mathcal {M})$ uniquely determines a Riemannian metric $g_{\varphi }$ and an orientation $dV_{\varphi }$, hence the Hodge star operator $*_{\varphi }$ and the associated 4-form

$$\begin{aligned} \psi :=*_{\varphi }\varphi . \end{aligned}$$

(2.1)

We also have the isomorphisms $\flat _{\varphi }:=\flat _{g_{\varphi }}$ and $\sharp _{\varphi }:= \sharp _{g_{\varphi }}$. For a given $G_{2}$-structure $\varphi \in \wedge ^{3}_{+}(\mathcal {M})$, we denote by $\langle \cdot ,\cdot \rangle _{\varphi }$, $\langle \langle \cdot ,\cdot \rangle \rangle$, $|\cdot |_{\varphi }$, $||\cdot ||_{\varphi }$, the corresponding inner products $\langle \cdot ,\cdot \rangle _{g_{\varphi }}$, $\langle \langle \cdot ,\cdot \rangle \rangle _{g_{\varphi }}$ and norms $|\cdot |_{g_{\varphi }}$, $||\cdot ||_{g_{\varphi }}$.

Given a $G_{2}$-structure $\varphi \in \wedge ^{3}_{+}(\mathcal {M})$. We say that $\varphi$ is torsion-free if $\varphi$ is parallel with respect to the metric $g_{\varphi }$. Equivalently, $\varphi$ is torsion-free if and only if ${}^{\varphi }\nabla \varphi =0$, where ${}^{\varphi } \nabla$ is the Levi–Civita connection of $g_{\varphi }$.

Theorem 2.1

(Fernández-Gray [12]) The $G_{2}$-structure $\varphi$ is torsion-free if and only if $\varphi$ is both closed (i.e., $d\varphi =0$) and co-closed (i.e., $d*_{\varphi } \varphi =d\psi =0$).

When $\mathcal {M}$ is compact, the above theorem says that a $G_{2}$-structure $\varphi$ is torsion-free if and only if $\varphi$ is harmonic with respect to the induces metric $g_{\varphi }$.

We say that a $G_{2}$-structure $\varphi$ is closed (resp., co-closed) if $d\varphi =0$ (resp., $d\psi =0$). Theorem 2.1 can be restated as that a $G_{2}$-structure is torsion-free if and only if it is both closed and co-closed.

2.1 $G_{2}$-decompositions of $\wedge ^{2}(\mathcal {M})$ and $\wedge ^{3}(\mathcal {M})$

A $G_{2}$-structure $\varphi$ induces splittings of the bundles $\wedge ^{k}(T^{*}\mathcal {M})$, $2\le k\le 5$, into direct summands, which we denote by $\wedge ^{k}_{\ell } (T^{*}\mathcal {M},\varphi )$ with $\ell$ being the rank of the bundle. We let the space of sections of $\wedge ^{k}_{\ell }(T^{*}\mathcal {M},\varphi )$ by $\wedge ^{k}_{\ell }(\mathcal {M},\varphi )$. Define the natural projections

$$\begin{aligned} \pi ^{k}_{\ell }: \wedge ^{k}(\mathcal {M})\longrightarrow \wedge ^{k}_{\ell }(\mathcal {M},\varphi ), \ \ \ \alpha \longmapsto \pi ^{k}_{\ell }(\alpha ). \end{aligned}$$

(2.2)

We mainly focus on the $G_{2}$–decompositions of $\wedge ^{2} (\mathcal {M})$ and $\wedge ^{3} (\mathcal {M})$. Recall that

$$\begin{aligned} \wedge ^{2}(\mathcal {M})= & {} \wedge ^{2}_{7}(\mathcal {M},\varphi ) \oplus \wedge ^{2}_{14}(\mathcal {M},\varphi ), \end{aligned}$$

(2.3)

$$\begin{aligned} \wedge ^{3}(\mathcal {M})= & {} \wedge ^{3}_{1}(\mathcal {M},\varphi ) \oplus \wedge ^{3}_{7}(\mathcal {M},\varphi ) \oplus \wedge ^{3}_{27}(\mathcal {M},\varphi ). \end{aligned}$$

(2.4)

Here each component is determined by

$$\begin{aligned} \wedge ^{2}_{7}(\mathcal {M},\varphi )= & {} \{X\lrcorner \varphi : X\in \mathfrak {X}(\mathcal {M})\} \ \ = \ \ \{\beta \in \wedge ^{2}(\mathcal {M}): *_{\varphi }(\varphi \wedge \beta )=2\beta \},\\ \wedge ^{2}_{14}(\mathcal {M},\varphi )= & {} \{\beta \in \wedge ^{2}(\mathcal {M}): \psi \wedge \beta =0\} \ \ = \ \ \{\beta \in \wedge ^{2}(\mathcal {M}): *_{\varphi }(\varphi \wedge \beta )=-\beta \},\\ \wedge ^{3}_{1}(\mathcal {M},\varphi )= & {} \{f\varphi : f\in C^{\infty }(\mathcal {M})\},\\ \wedge ^{3}_{7}(\mathcal {M},\varphi )= & {} \left\{ *_{\varphi }(\varphi \wedge \alpha ): \alpha \in \wedge ^{1}(\mathcal {M})\right\} \ \ = \ \ \left\{ X\lrcorner \psi : X\in \mathfrak {X}(\mathcal {M}) \right\} ,\\ \wedge ^{3}_{27}(\mathcal {M},\varphi )= & {} \{\eta \in \wedge ^{3}(\mathcal {M}): \eta \wedge \varphi =\eta \wedge \psi =0\}. \end{aligned}$$

For any 2-form $\beta =\frac{1}{2}\beta _{ij}dx^{i\wedge j} \in \wedge ^{2}(\mathcal {M})$, its two components $\pi ^{2}_{7}(\beta )$ and $\pi ^{2}_{14}(\beta )$ are determined by

$$\begin{aligned} \pi ^{2}_{7}(\beta )= & {} \frac{\beta +*_{\varphi }(\varphi \wedge \beta )}{3} \ \ = \ \ \frac{1}{2}\left( \frac{1}{3}\beta _{ab}+\frac{1}{6}\beta ^{\ell m} \psi _{\ell m ab}\right) dx^{ab}, \end{aligned}$$

(2.5)

$$\begin{aligned} \pi ^{2}_{14}(\beta )= & {} \frac{2\beta -*_{\varphi }(\varphi \wedge \beta )}{3} \ \ = \ \ \frac{1}{2}\left( \frac{2}{3}\beta _{ab}-\frac{1}{6}\beta ^{\ell m} \psi _{\ell mab}\right) dx^{ab}. \end{aligned}$$

(2.6)

To decompose 3-forms, recall two maps introduce by Bryant [1]

$$\begin{aligned} \textsf {i}_{\varphi }: \odot ^{2}(\mathcal {M})\longrightarrow \wedge ^{3}(\mathcal {M}), \ \ \ \textsf {j}_{\varphi }: \wedge ^{3}(\mathcal {M})\longrightarrow \odot ^{2}(\mathcal {M}), \end{aligned}$$

(2.7)

where

$$\begin{aligned} \textsf {i}_{\varphi }(h):= & {} h_{ij}g^{j\ell }dx^{i} \wedge \left( \frac{\partial }{\partial x^{\ell }}\lrcorner \varphi \right) \ \ = \ \ \frac{1}{2}h_{i\ell }\varphi ^{\ell }{}_{jk}dx^{ijk}\nonumber \\= & {} \frac{1}{6}\left( h_{i\ell }\varphi ^{\ell }{}_{jk} +h_{j\ell }\varphi _{i}{}^{\ell }{}_{k}+h_{k\ell }\varphi _{ij}{}^{\ell } \right) dx^{ijk}, \ \ \ h=h_{ij}dx^{ij}\in \odot ^{2}(\mathcal {M}), \end{aligned}$$

(2.8)

and

$$\begin{aligned} \left( \textsf {j}_{\varphi }(\eta )\right) (X,Y):= *_{\varphi }\left( (X\lrcorner \varphi )\wedge (Y\lrcorner \varphi ) \wedge \eta \right) . \end{aligned}$$

(2.9)

Then $\textsf {i}_{\varphi }$ is injective and is isomorphic onto $\wedge ^{3}_{1} (\mathcal {M},\varphi )\oplus \wedge ^{3}_{27}(\mathcal {M},\varphi )$, and $\textsf {j}_{\varphi }$ is an isomorphism between $\wedge ^{3}_{1}(\mathcal {M},\varphi )\oplus \wedge ^{3}_{27}(\mathcal {M}, \varphi )$ and $\odot ^{2}(\mathcal {M})$. Moreover, for any 3-form $\eta \in \wedge ^{3}(\mathcal {M})$, we have

$$\begin{aligned} \eta =\textsf {i}_{\varphi }(h)+X\lrcorner \psi \end{aligned}$$

(2.10)

for some symmetric 2-tensor $h\in \odot ^{2}(\mathcal {M})$ and vector field $X\in \mathfrak {X}(\mathcal {M})$. Then

$$\begin{aligned} \eta= & {} h_{i}{}^{\ell }dx^{i}\wedge \left( \frac{\partial }{\partial x^{\ell }}\lrcorner \varphi \right) +X^{\ell }\left( \frac{\partial }{\partial x^{\ell }}\lrcorner \psi \right) \ \ = \ \ \frac{1}{2}h_{i}{}^{\ell }\varphi _{\ell jk}dx^{ijk}+\frac{1}{6}X^{\ell } \psi _{\ell ijk}dx^{ijk}\\= & {} \frac{1}{6}\left( 3 h_{i}{}^{\ell }\varphi _{\ell jk} +X^{\ell }\psi _{\ell ijk}\right) dx^{ijk} \ \ = \ \ \frac{1}{6}\eta _{ijk} dx^{ijk}. \end{aligned}$$

Write h as $h_{ij}=\mathring{h}_{ij}+\frac{1}{7}\mathrm{tr}_{\varphi }(h)\!\ g_{\varphi }$, where $\mathring{h}\in \odot ^{2}_{0}(\mathcal {M})$ is the trace-free part of h, one has

$$\begin{aligned} \eta =\underbrace{\frac{3}{7}\left( \mathrm{tr}_{\varphi }(h)\right) \varphi }_{ \pi ^{3}_{1}(\eta )} +\underbrace{\frac{1}{2}\mathring{h}_{i}{}^{\ell }\varphi _{\ell jk} dx^{ijk}}_{\pi ^{3}_{27}(\eta )}+ \underbrace{\frac{1}{6}X^{\ell }\psi _{\ell ijk}dx^{ijk}}_{\pi ^{3}_{7}(\eta )}. \end{aligned}$$

(2.11)

2.2 The torsion tensors of a $G_{2}$-structure

By Hodge duality we obtain the $G_{2}$-decompositions of 4-forms $\wedge ^{4}(\mathcal {M})=\wedge ^{4}_{1}(\mathcal {M},\varphi ) \oplus \wedge ^{4}_{7}(\mathcal {M},\varphi )\oplus \wedge ^{4}_{27}(\mathcal {M}, \varphi )$ and 5-forms $\wedge ^{5}(\mathcal {M})=\wedge ^{5}_{7}(\mathcal {M},\varphi ) \oplus \wedge ^{5}_{14}(\mathcal {M},\varphi )$, respectively. By definition, we can find forms $\tau _{0}\in C^{\infty }(\mathcal {M})$, $\tau _{1}, \widetilde{\tau }_{1}\in \wedge ^{1}(\mathcal {M})$, $\tau _{2}\in \wedge ^{2}_{14}(\mathcal {M},\varphi )$, and $\tau _{3}\in \wedge ^{3}_{27}(\mathcal {M},\varphi )$ such that

$$\begin{aligned} d\varphi =\tau _{0}\psi +3\tau _{1}\wedge \varphi +*_{\varphi }\tau _{3}, \ \ \ d\psi =4\widetilde{\tau }_{1}\wedge \psi -*_{\varphi }\tau _{2}. \end{aligned}$$

(2.12)

Since $\tau _{2}\in \wedge ^{2}_{14}(\mathcal {M},\varphi )$, it follows that $\tau _{2}\wedge \varphi =-*_{\varphi }\tau _{2}$. Then (2.12) can be written as in the sense of Bryant [1]

$$\begin{aligned} d\varphi =\tau _{0}\psi +3\tau _{1}\wedge \varphi +*_{\varphi }\tau _{3}, \ \ \ d\psi =4\widetilde{\tau }_{1}\wedge \psi +\tau _{2}\wedge \varphi . \end{aligned}$$

(2.13)

It can be proved that $\tau _{1}=\widetilde{\tau }_{1}$ (see [23]). We call $\tau _{0}$ the scalar torsion, $\tau _{1}$ the vector torsion, $\tau _{2}$ the Lie algebra torsion, and $\tau _{3}$ the symmetric traceless torsion. We also call $\varvec{\tau }_{\varphi }:=\{\tau _{0},\tau _{1},\tau _{2}, \tau _{3}\}$ the intrinsic torsion forms of the $G_{2}$-structure $\varphi$.

Recall that a $G_{2}$-structure $\varphi$ is torsion-free if and only if $d \varphi =d\psi =0$ by Theorem 2.1. From (2.12) we see that $\varphi$ is torsion-free if and only if the intrinsic torsion forms $\varvec{\tau }_{\varphi }\equiv =0$; that is, $\tau _{0}=\tau _{1}=\tau _{2} =\tau _{3}=0$.

Lemma 2.2

(Fernández-Gray, [12]) For any $X\in \mathfrak {X}(\mathcal {M})$, the 3-form $\nabla _{X} \varphi$ lines in the space $\wedge ^{3}_{7}(\mathcal {M},\varphi )$. Therefore the covariant derivative $\nabla \varphi \in \wedge ^{1}(\mathcal {M}) \otimes \wedge ^{3}_{7}(\mathcal {M})$.

Consequently, there exists a 2-tensor $\varvec{T}=\varvec{T}_{ij}dx^{i\otimes j}$, called the full torsion tensor, such that

$$\begin{aligned} \nabla _{\ell }\varphi =\varvec{T}_{\ell }{}^{n}\psi _{nabc}. \end{aligned}$$

(2.14)

Equivalently,

$$\begin{aligned} \varvec{T}_{\ell m}=\frac{1}{24}(\nabla _{\ell }\varphi _{abc}) \psi _{m}{}^{abc}. \end{aligned}$$

(2.15)

Write

$$\begin{aligned} \tau _{1}= & {} (\tau _{1})_{i}dx^{i}\in \wedge ^{1}(\mathcal {M}), \end{aligned}$$

(2.16)

$$\begin{aligned} \tau _{2}= & {} \frac{1}{2}(\tau _{2})_{ab}dx^{ab}\in \wedge ^{2}_{14}(\mathcal {M}), \end{aligned}$$

(2.17)

$$\begin{aligned} \tau _{3}= & {} \frac{1}{2}(\tau _{3})_{i}{}^{\ell }\varphi _{\ell ij} dx^{ijk}\in \wedge ^{3}_{27}(\mathcal {M},\varphi ). \end{aligned}$$

(2.18)

The associated 2-tensor $\varvec{\tau }_{3}:=(\tau _{3})_{ij}dx^{i\otimes j}$ of $\tau _{3}$ lies in the space $\odot ^{2}_{0}(\mathcal {M})$. With this convenience, the full torsion tensor $\varvec{T}_{\ell m}$ is determined by

$$\begin{aligned} \varvec{T}_{\ell m}=\frac{\tau _{0}}{4}g_{\ell m}-(\varvec{\tau }_{3})_{\ell m} -\left( \sharp _{\varphi }(\tau _{1})\lrcorner \varphi \right) _{\ell m} -\frac{1}{2}(\tau _{2})_{\ell m} \end{aligned}$$

(2.19)

or as 2-tensors,

$$\begin{aligned} \varvec{T}=\frac{\tau _{0}}{4}g_{\varphi } -\varvec{\tau }_{3}-\sharp _{\varphi }(\tau _{1})\lrcorner \varphi -\frac{1}{2}\tau _{2}. \end{aligned}$$

(2.20)

Here the 2-form $\sharp _{\varphi }(\tau _{1})\lrcorner \varphi$ is defined by

$$\begin{aligned} \sharp _{\varphi }(\tau _{1})\lrcorner \varphi =\frac{1}{2}\left( \sharp _{\varphi }(\tau _{1})\lrcorner \varphi \right) dx^{a\wedge b}=\frac{1}{2}\left( (\tau _{1})_{k}\varphi ^{k}{}_{ab} \right) dx^{a\wedge b}. \end{aligned}$$

As an application, this gives another proof of Theorem 2.1.

For fixed indices i and j, set

$$\begin{aligned} R_{ij|k\ell }:=R_{ijk\ell } \ \text {is skew-symmetric in} \ k \ \text {and} \ \ell , \end{aligned}$$

(2.21)

where

$$\begin{aligned} R_{ij|\bullet \bullet }:=\frac{1}{2}R_{ij|k\ell }dx^{k\ell } =\frac{1}{2}R_{ijk\ell }dx^{k\ell }\in \wedge ^{2} (\mathcal {M}). \end{aligned}$$

(2.22)

Then, according to (2.5) and (2.6)

$$\begin{aligned} R_{ijk\ell }=R_{ij|k\ell }=\left( \pi ^{2}_{7}(R_{ij|\bullet \bullet }) \right) _{k\ell }+\left( \pi ^{2}_{14}(R_{ij|\bullet \bullet })\right) _{k\ell }, \end{aligned}$$

where

$$\begin{aligned} \left( \pi ^{2}_{7}(R_{ij|\bullet \bullet })\right) _{k\ell }= & {} \frac{1}{3}R_{ij|k\ell }+\frac{1}{6}R_{ij|ab}\psi ^{ab}{}_{k\ell } \ \ = \ \ \frac{1}{3}R_{ijk\ell }+\frac{1}{6}R_{ijab}\psi ^{ab}{}_{k\ell },\\ \left( \pi ^{2}_{14}(R_{ij|\bullet \bullet })\right) _{k\ell }= & {} \frac{2}{3}R_{ij|k\ell }-\frac{1}{6}R_{ij|ab}\psi ^{ab}{}_{k\ell } \ \ = \ \ \frac{1}{3}R_{ijk\ell }-\frac{1}{6}R_{ijab}\psi ^{ab}{}_{k\ell }. \end{aligned}$$

Karigiannis [23] (see also the equivalent formula obtained by Bryant in [1]) proved that the Ricci curvature is given by

$$\begin{aligned} R_{jk}= & {} R_{ijk\ell }g^{i\ell } \ \ = \ \ 3\left( \pi ^{2}_{7}(R_{ij|\bullet \bullet })\right) _{k\ell }g^{i\ell } \ \ = \ \ \frac{3}{2}\left( \pi ^{2}_{14}(R_{ij|\bullet \bullet })\right) _{k\ell } g^{i\ell }\nonumber \\= & {} -\left( \nabla _{i}\varvec{T}_{jm}-\nabla _{j}\varvec{T}_{im}\right) \varphi ^{m}{}_{ k}{}^{i}-\varvec{T}_{j}{}^{i}\varvec{T}_{ik}+\left( \mathrm{tr}_{\varphi }\varvec{T}\right) \varvec{T}_{jk} +\varvec{T}_{jb}\varvec{T}_{ia}\psi ^{iab}{}_{k},\nonumber \\= & {} -\nabla _{i}\left( \varvec{T}_{j}{}^{n}\varphi _{nk}{}^{i}\right) +\nabla _{j}\left( \varvec{T}_{i}{}^{n}\varphi _{nk}{}^{i}\right) -\varvec{T}_{j}{}^{i}\varvec{T}_{ik}+\left( \mathrm{tr}_{\varphi }\varvec{T}\right) \varvec{T}_{jk}-\varvec{T}_{jb}\varvec{T}_{ia}\psi ^{iab}{}_{k}.\nonumber \\ \end{aligned}$$

(2.23)

Cleyton and Ivanov [6] also derived a formula for the Ricci tensor for closed $G_{2}$-structures in terms of $d^{*}_{\varphi }\varphi$. Taking the trace of (2.23), we obtain Btyant’s formula [1] for the scalar curvature

$$\begin{aligned} R= & {} -12\nabla ^{\ell }(\tau _{1})_{\ell } +\frac{21}{8}\tau ^{2}_{0}-||\varvec{\tau }_{3}||^{2}_{\varphi } +5||\sharp _{\varphi }(\tau _{1})\lrcorner \varphi ||^{2}_{\varphi }-\frac{1}{4}||\tau _{2}||^{2}_{\varphi }, \nonumber \\= & {} -12\nabla ^{\ell }(\tau _{1})_{\ell } +\frac{21}{8}\tau ^{2}_{0}-||\varvec{\tau }_{3}||^{2}_{\varphi } +30|\tau _{1}|^{2}_{\varphi }-\frac{1}{2}|\tau _{2}|^{2}_{\varphi }, \end{aligned}$$

(2.24)

For a closed $G_{2}$-structure, we have $\tau _{0}=\tau _{1}=\tau _{3}=0$ and then $R=-\frac{1}{4}||\tau _{2}||^{2}_{\varphi }\le 0$. On the other hand, we have $(\tau _{2})_{ij}=-2\varvec{T}_{ij}$ by (2.20). Thus the full torsion tensor $\varvec{T}$ is actually a 2-form

$$\begin{aligned} \varvec{T}=\frac{1}{2}\varvec{T}_{ij}dx^{ij}\in \wedge ^{2}(\mathcal {M}) \end{aligned}$$

(2.25)

and the scalar curvature can be written in terms of T

$$\begin{aligned} R=-||\varvec{T}||^{2}_{\varphi }=-2|\varvec{T}|^{2}_{\varphi }\le 0. \end{aligned}$$

(2.26)

Hence, for closed $G_{2}$-structures, scalar curvatures are always non-positive.

Finally, we mention a Bianchi type identity

$$\begin{aligned} \nabla _{i}\varvec{T}_{j\ell }-\nabla _{j}\varvec{T}_{i\ell } =-\frac{1}{2}R_{ijab}\varphi ^{ab}{}_{\ell } -\varvec{T}_{ia}\varvec{T}_{jb}\varphi ^{ab}{}_{\ell } =-\left( \frac{1}{2}R_{ijab}+\varvec{T}_{ia}\varvec{T}_{jb}\right) \varphi ^{ab}{}_{\ell }. \end{aligned}$$

(2.27)

The proof can be found in [23].

2.3 Basic theory of closed $G_{2}$-structures

Let $\wedge ^{3}_{+,\bullet }(\mathcal {M})\subset \wedge ^{3}_{+} (\mathcal {M},\varphi )$ be the set of all closed $G_{2}$-structures on $\mathcal {M}$. If $\varphi \in \wedge ^{3}_{+,\bullet }(\mathcal {M})$ is closed, i.e., $d\varphi =0$, then $\tau _{0}, \tau _{1}, \tau _{3}$ are all zero, so the only nonzero torsion form is

$$\begin{aligned} \varvec{\tau }\equiv \tau _{2}=\frac{1}{2}(\tau _{2})_{ij}dx^{ij} =\frac{1}{2}\varvec{\tau }_{ij} dx^{ij}. \end{aligned}$$

(2.28)

According to (2.20) and (2.25), we have $\varvec{T}_{ij}=-\frac{1}{2}\varvec{\tau }_{ij}$ so that

$$\begin{aligned} \varvec{T}\equiv \frac{1}{2}\varvec{T}_{ij}dx^{ij} \ \ \ \text {or equivalently} \ \ \ \varvec{T}=-\frac{1}{2}\varvec{\tau }, \end{aligned}$$

(2.29)

is a 2-form. Since $d\psi =\varvec{\tau }\wedge \varphi =-*_{\varphi } \varvec{\tau }$, we get $d^{*}_{\varphi }\varvec{\tau }=*_{\varphi }d*_{\varphi } \varvec{\tau }=-*_{\varphi }d^{2}\psi =0$ which is given in local coordinates by

$$\begin{aligned} \nabla ^{i}\varvec{\tau }_{ij}=0 \end{aligned}$$

(2.30)

For a closed $G_{2}$-structure $\varphi$, according to (2.23), the Ricci curvature is given by (in this case $\varvec{T}_{ij}$ is a 2-form)

$$\begin{aligned} R_{jk}=\left( \nabla _{j}\varvec{T}_{im}-\nabla _{i}\varvec{T}_{jm}\right) \varphi ^{m}{}_{k}{}^{i} -\varvec{T}_{j}{}^{i}\varvec{T}_{ik}+\varvec{T}_{jb}\varvec{T}_{ia}\psi ^{iab}{}_{k}. \end{aligned}$$

Since $\varvec{\tau }\in \wedge ^{2}_{14}(\mathcal {M},\varphi )$ and $\varvec{T}_{ij}=-\frac{1}{2}\varvec{\tau }_{ij}$, it follows from [32] (see pp. 179–180) that

$$\begin{aligned} (\nabla _{j}\varvec{T}_{im})\varphi ^{m}{}_{k}{}^{i} =2\varvec{T}_{j}{}^{\ell }\varvec{T}_{\ell k}. \end{aligned}$$

(2.31)

and therefore, for a closed $G_{2}$-structure $\varphi$, the Ricci curvature is given by

$$\begin{aligned} R_{jk}=-(\nabla _{i}\varvec{T}_{jm})\varphi _{k}{}^{im} -\varvec{T}_{j}{}^{i}\varvec{T}_{ik}. \end{aligned}$$

(2.32)

Taking the trace of (2.32) yields (2.26). Moreover, the factor $\nabla _{i}\varvec{T}_{jm}$ in (3.6) can be expressed as (see Proposition 2.4 in [32])

$$\begin{aligned} \nabla _{i}\varvec{T}_{jk}= & {} -\frac{1}{4}R_{ijmn}\varphi _{k}{}^{mn} -\frac{1}{4}R_{kjmn}\varphi _{i}{}^{mn}+\frac{1}{4}R_{ikmn}\varphi _{j}{}^{mn}\nonumber \\&- \ \frac{1}{2}\varvec{T}_{im}\varvec{T}_{jn}\varphi _{k}{}^{mn} -\frac{1}{2}\varvec{T}_{km}\varvec{T}_{jn}\varphi _{i}{}^{mn} +\frac{1}{2}\varvec{T}_{im}\varvec{T}_{kn}\varphi _{j}{}^{mn}. \end{aligned}$$

(2.33)

If $\varphi$ is a closed $G_{2}$-structure, Section 2.2 in [32] shows that $\pi ^{3}_{7}(\Delta _{\varphi }\varphi )=0$ and hence, according to (2.10),

$$\begin{aligned} \Delta _{\varphi }\varphi =\textsf {i}_{\varphi }(h)\in \wedge ^{3}_{1}(\mathcal {M},\varphi ) \oplus \wedge ^{3}_{27}(\mathcal {M},\varphi ), \end{aligned}$$

(2.34)

where

$$\begin{aligned} h_{ij}=\frac{1}{2}\nabla _{m}\varvec{\tau }_{ni}\varphi _{j}{}^{mn} -\frac{1}{6}|\varvec{\tau }|^{2}_{\varphi }g_{ij}-\frac{1}{4}\varvec{\tau }_{i}{}^{\ell } \varvec{\tau }_{\ell j}=-R_{ij}-\frac{2}{3}|\varvec{T}|^{2}_{\varphi }g_{ij} -2\varvec{T}_{i}{}^{k}\varvec{T}_{kj}. \end{aligned}$$

(2.35)

Here $|\varvec{T}|^{2}_{\varphi }=\frac{1}{2}\varvec{T}_{k\ell }\varvec{T}^{k\ell } =\frac{1}{2}||\varvec{T}||^{2}_{\varphi }$.

2.4 General flows on $G_{2}$-structures

For any family $\varphi (t)$ of $G_{2}$-structures, according to the decomposition (2.10), we can consider the general flow

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

(2.36)

where $h(t)\in \odot ^{2}(\mathcal {M})$ and $X(t)\in \mathfrak {X} (\mathcal {M})$. The general flow (2.36) locally can be written as

$$\begin{aligned} \partial _{t}\varphi _{ijk}=h_{i}{}^{\ell }\varphi _{\ell jk} +h_{j}{}^{\ell }\varphi _{i\ell k}+h_{k}{}^{\ell }\varphi _{ij\ell } +X^{\ell }\psi _{\ell ijk}. \end{aligned}$$

(2.37)

We write for g(t) and $dV_{g(t)}$ the metric and volume form associated to $\varphi (t)$, respectively.

Theorem 2.3

Under the general flow (2.36), we have

$$\begin{aligned} \partial _{t}g_{ij}= & {} 2h_{ij}, \end{aligned}$$

(2.38)

$$\begin{aligned} \partial _{t}g^{ij}= & {} -2h^{ij}, \end{aligned}$$

(2.39)

$$\begin{aligned} \partial _{t}dV_{g(t)}= & {} \left( \mathrm{tr}_{g(t)}h(t)\right) dV_{g(t)}, \end{aligned}$$

(2.40)

$$\begin{aligned} \partial _{t}\varvec{T}_{pq}= & {} \varvec{T}_{p}{}^{m}h_{mq} -\varvec{T}_{p}{}^{m}X^{k}\varphi _{kmq}-(\nabla _{k}h_{ip})\varphi ^{ki}{}_{q} +\nabla _{p}X_{q}. \end{aligned}$$

(2.41)

These evolution equations can be found in [23].

3 Laplacian flows on closed $G_{2}$-structures

We now consider the Laplacian flow for closed $G_{2}$-structures

$$\begin{aligned} \partial _{t}\varphi (t)=\Delta _{\varphi (t)}\varphi (t)=\Delta _{g(t)}\varphi (t) , \ \ \ \varphi (0)=\varphi , \end{aligned}$$

(3.1)

where $\Delta _{\varphi (t)}\varphi (t)=dd^{*}_{\varphi (t)} \varphi (t)+d^{*}_{\varphi (t)} d\varphi (t)$ is the Hodge Laplacian of $g_{\varphi (t)}$ and $\varphi$ is an initial closed $G_{2}$-structure. The short time existence for (3.1) on compact manifolds was proved by Bryant and Xu [2], see also Theorem 1.1.

A criterion for the long time existence for the Lapalcian flow on compact manifolds was given in Theorem 1.2. In this section, we give a new elementary proof of Lotay-Wei’s result in compact case.

3.1 Evolution equations along the Laplacian flow

Since the Laplacian flow (3.1) preserves the closedness of $\varphi (t)$, it follows from (3.10) that we have

$$\begin{aligned} \Delta _{\varphi (t)}\varphi (t)=\textsf {i}_{\varphi (t)}(h(t)) \in \wedge ^{3}_{1}(\mathcal {M}, \varphi (t))\oplus \wedge ^{3}_{27}(\mathcal {M},\varphi (t)), \end{aligned}$$

(3.2)

where

$$\begin{aligned} h_{ij}=-R_{ij}-\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij}-2\varvec{T}_{i}{}^{k}\varvec{T}_{kj}. \end{aligned}$$

(3.3)

From Theorem 2.3, we see that the associated metric tensor g(t) evolves by

$$\begin{aligned} \partial _{t}g_{ij}=2h_{ij}=-2R_{ij} -\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij}-4\varvec{T}_{i}{}^{k}\varvec{T}_{kj}. \end{aligned}$$

(3.4)

and the volume form $dV_{g(t)}$ evolves by

$$\begin{aligned} \partial _{t}dV_{g(t)}= & {} (\mathrm{tr}_{g(t)}h(t))dV_{g(t)} \ \ = \ \ \left( -R_{g(t)}-\frac{14}{3}|\varvec{T}(t)|^{2}_{g(t)}+4|\varvec{T}(t)|^{2}_{g(t)}\right) dV_{g(t)}\nonumber \\= & {} \left( 2-\frac{14}{3}+4\right) |\mathbf{T}(t)|^{2}_{g(t)}dV_{g(t)} \ \ = \ \ \frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}dV_{g(t)}. \end{aligned}$$

(3.5)

Hence, along the flow (3.1), the volume of g(t) is nondecreasing.

Introduce the following notions

$$\begin{aligned} \blacksquare _{g(t)}:=\partial _{t}-\blacktriangle _{g(t)}, \ \ \ |\cdot |_{g(t)}:=|\cdot |_{\varphi (t)}, \ \ \ \Delta _{g(t)}:=\Delta _{\varphi (t)}, \end{aligned}$$

(3.6)

where $\blacktriangle _{g(t)}:=g^{ij}\nabla _{i}\nabla _{j}$ is the usual Laplacian of g(t) and $\Delta _{g(t)}$ is the Hodge Laplacian of g(t), and also the 2-tenor $\mathrm{Sic}_{g(t)}$ with components

$$\begin{aligned} S_{ij}:=R_{ij}+\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij} +2\varvec{T}_{i}{}^{k}\varvec{T}_{kj}=-h_{ij}. \end{aligned}$$

(3.7)

Then the evolution Eq. (3.4) can be written as

$$\begin{aligned} \partial _{t}g_{ij}=-2S_{ij}. \end{aligned}$$

(3.8)

The trace of $\mathrm{Sic}_{g(t)}$ is exactly the scalar curvature, up to a multiplying constant,

$$\begin{aligned} S_{g(t)}:=\mathrm{tr}_{g(t)}\mathrm{Sic}_{g(t)}=R_{g(t)}+\frac{14}{3}|\varvec{T}(t)|^{2}_{g(t)} -4|\varvec{T}(t)|^{2}_{g(t)}=-\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}=\frac{2}{3}R_{g(t)}. \end{aligned}$$

(3.9)

It was proved in [32] that

$$\begin{aligned} |\Delta _{g(t)}\varphi (t)|^{2}_{g(t)} =(\mathrm{tr}_{g(t)}h(t))^{2}+2||h(t)||^{2}_{g(t)} =\frac{16}{9}|\varvec{T}(t)|^{4}_{g(t)}+2||\mathrm{Sic}_{g(t)}||^{2}_{g(t)}. \end{aligned}$$

(3.10)

This identity together with (2.26) shows that the boundedness of $\Delta _{g(t)}\varphi (t)$ is equivalent to the boundedness of $\mathrm{Ric}_{g(t)}$.

The evolution Eq. (2.41) implies that for the Laplacian flow on closed $G_{2}$-structures, the torsion $T_{ij}$ evolves by evolves

$$\begin{aligned} \partial _{t}\varvec{T}_{ij}=\varvec{T}_{i}{}^{k}h_{kj} -(\nabla _{m}h_{ni})\varphi _{j}{}^{mn}. \end{aligned}$$

(3.11)

Furthermore, we can prove

Proposition 3.1

Under the flow (3.1), we have

$$\begin{aligned} \blacksquare _{g(t)}\varvec{T}_{ij}= & {} 3R_{j}{}^{k}\varvec{T}_{ki}-R_{i}{}^{k}\varvec{T}_{kj} -\frac{1}{2}R_{ijmk}\varvec{T}^{mk}-\frac{1}{2}R_{mpi}{}^{k}R_{qk}\psi _{j}{}^{pqm}-\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)} \varvec{T}_{ij}\nonumber \\&+ \ \nabla _{p}\varvec{T}_{qi}\left( \varvec{T}^{pk}\varphi _{kj}{}^{q} -2\varvec{T}^{qk}\varphi _{kj}{}^{p}\right) - \frac{2}{3}\varphi _{ji}{}^{m} \nabla _{m}|\varvec{T}(t)|^{2}_{g(t)}-4T_{i}{}^{k}\varvec{T}_{k}{}^{m}\varvec{T}_{mj}.\nonumber \\ \end{aligned}$$

(3.12)

Proof

See [32].

For a geometric flow $\partial _{t}g_{ij}=\eta _{ij}$, where $\eta _{ij}$ is a family of symmetric 2-tensors, we have (e.g. see formula (2.66), (2.29), and (2.30) in [5])

$$\begin{aligned} \partial _{t} R^{\ell }_{ijk}= & {} \frac{1}{2}g^{\ell p}\bigg (\nabla _{i}\nabla _{j}\eta _{kp} +\nabla _{i}\nabla _{k}\eta _{jp}-\nabla _{i}\nabla _{p}\eta _{jk}\\&- \ \nabla _{j}\nabla _{i}\eta _{kp}-\nabla _{j}\nabla _{k} \eta _{ip}+\nabla _{j}\nabla _{p}\eta _{ik}\bigg ),\\ \partial _{t}R_{jk}= & {} \frac{1}{2}g^{pq} \left( \nabla _{q}\nabla _{j}\eta _{kp}+\nabla _{q}\nabla _{k}\eta _{jp} -\nabla _{q}\nabla _{p}\eta _{jk}-\nabla _{j}\nabla _{k}\eta _{qp}\right) ,\\ \partial _{t}R_{g(t)}= & {} -\blacktriangle _{g(t)}\mathrm{tr}_{g(t)}\eta (t) +\mathrm{div}_{g(t)}(\mathrm{div}_{g(t)}\eta (t)) -R_{ij}h^{ij}, \end{aligned}$$

where $(\mathrm{div}_{g(t)}\eta (t))_{j}=\nabla ^{i}\eta _{ij}$. Applying those evolution equations to $\eta _{ij}=-2R_{ij}-\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)} g_{ij}-4\varvec{T}_{i}{}^{k}\varvec{T}_{kj}=-2S_{ij}$ we have

$$\begin{aligned} \mathrm{tr}_{g(t)}\eta (t)= & {} -2R_{g(t)}-\frac{28}{3}|\varvec{T}(t)|^{2}_{g(t)} +8|\varvec{T}(t)|^{2}_{g(t)} \ \ = \ \ \frac{8}{3}|\varvec{T}(t)|^{2}_{g(t)},\\ (\mathrm{div}_{g(t)}\eta (t))_{j}= & {} -2\nabla ^{i}R_{ij}-\frac{4}{3}\nabla _{j}|\varvec{T}(t)|^{2}_{g(t)} -4\nabla ^{i}\widehat{\varvec{T}}_{ij}\\= & {} -\nabla _{j}R_{g(t)}-\frac{4}{3}\nabla _{j}|\varvec{T}(t)|^{2}_{g(t)}-4\nabla ^{i} \widehat{\varvec{T}}_{ij},\\ \mathrm{div}_{g(t)}(\mathrm{div}_{g(t)}\eta (t))= & {} \nabla ^{j}(\mathrm{div}_{g(t)}\eta (t))_{j}\\= & {} -\blacktriangle _{g(t)}R_{g(t)}-\frac{4}{3}\blacktriangle _{g(t)}|\varvec{T}(t)|^{2}_{g(t)} -4\nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij}, \end{aligned}$$

where the symmetric 2-tensor $\widehat{\varvec{T}}(t)$ is given by

$$\begin{aligned} \widehat{\varvec{T}}_{ij}:=\varvec{T}_{ik}\varvec{T}^{k}{}_{j}. \end{aligned}$$

(3.13)

Plugging those identities into the above evolution equation for $R_{g(t)}$, we get

$$\begin{aligned} \partial _{t}R_{g(t)}= & {} -4\blacktriangle _{g(t)}|\varvec{T}(t)|^{2}_{g(t)}-\blacktriangle _{g(t)} R_{g(t)} -4\nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij}\\&- \ R^{ij}\left( -2R_{ij}- \frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij} -4\widehat{\varvec{T}}_{ij}\right) \\= & {} \blacktriangle _{g(t)}R_{g(t)}-4\nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij} +2||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}+\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}R_{g(t)} +4R^{ij}\widehat{\varvec{T}}_{ij} \end{aligned}$$

which implies

$$\begin{aligned} \blacksquare _{g(t)}R_{g(t)}=2||\mathrm{Ric}_{g(t)}||^{2}_{g(t)} -\frac{2}{3}R^{2}_{g(t)}-4\nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij}+4\langle \langle \mathrm{Ric}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}. \end{aligned}$$

(3.14)

Observe that the last two terms on the right-hand side of (3.22) are not determined of their signs. In the following, we shall use the identity

$$\begin{aligned} \nabla ^{i}\varvec{T}_{ij} =0 \end{aligned}$$

(3.15)

follows from from (2.29) and (2.30), to simplify those two terms. Using the identity (3.15), the term $\nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij}$ can be simplified as follows.

$$\begin{aligned} \nabla ^{j}\nabla ^{i}\widehat{\varvec{T}}_{ij}= & {} \nabla ^{j}\nabla ^{i}\left( \varvec{T}_{i}{}^{k}\varvec{T}_{kj}\right) \ \ = \ \ \nabla ^{j}\left[ (\nabla ^{i}\varvec{T}_{i}{}^{k})\varvec{T}_{kj}+\varvec{T}_{i}{}^{k} (\nabla ^{i}\varvec{T}_{kj}) \right] \\= & {} \varvec{T}^{ik}(\nabla _{j}\nabla _{i}\varvec{T}_{k}{}^{j})-(\nabla ^{j}\varvec{T}^{ik}) (\nabla _{i}\varvec{T}_{jk}). \end{aligned}$$

On the other hand, from the Ricci identity

$$\begin{aligned} \nabla _{j}\nabla _{i}\varvec{T}_{k}{}^{j}=\nabla _{i}\nabla _{j}\varvec{T}_{k}{}^{j} -R_{jik\ell }\varvec{T}^{\ell j}-R_{ji}{}^{j\ell }\varvec{T}_{k\ell } =R_{ijk\ell }\varvec{T}^{\ell j}+R_{i\ell }\varvec{T}_{k}{}^{\ell }, \end{aligned}$$

we see that the evolution Eq. (3.14) is equivalent to

$$\begin{aligned} \blacksquare _{g(t)}R_{g(t)}=2||\mathrm{Ric}_{g(t)}||^{2}_{g(t)} -\frac{2}{3}R^{2}_{g(t)}+4R_{ijk\ell }\varvec{T}^{ik}\varvec{T}^{j\ell }+4(\nabla ^{j}\varvec{T}^{ik}) (\nabla _{i}\varvec{T}_{jk}). \end{aligned}$$

(3.16)

From (3.7) and (3.13) we can rewrite the term $||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}$ in (3.16) in terms of $\mathrm{Sic}_{g(t)}$ according to the following relation:

$$\begin{aligned} ||\mathrm{Sic}_{g(t)}||^{2}_{g(t)}= & {} \left( R_{ij}+\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij} +2\widehat{\varvec{T}}_{ij}\right) \left( R^{ij}+\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)}g^{ij} +2\widehat{\varvec{T}}^{ij}\right) \\= & {} \ \ ||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}+\frac{4}{3}|\varvec{T}(t)|^{2}_{g(t)}R_{g(t)} +4\langle \langle \mathrm{Ric}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}\\&+ \ \frac{28}{9}|\varvec{T}(t)|^{4}_{g(t)}+\frac{8}{3}|\varvec{T}(t)|^{2}_{g(t)} \mathrm{tr}_{g(t)}\widehat{\varvec{T}}(t)+4||\widehat{\varvec{T}}(t)||^{2}_{g(t)}\\= & {} \ \ ||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}-\frac{2}{3}R^{2}_{g(t)} +4\langle \langle \mathrm{Ric}_{g(t)}, \widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}\\&+\frac{7}{9}R^{2}_{g(t)}-\frac{4}{3}R^{2}_{g(t)}+4||\widehat{\varvec{T}}(t)||^{ 2}_{g(t)}\\= & {} \ \ ||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}+4||\widehat{\varvec{T}}(t)||^{2}_{g(t)} +4\langle \langle \mathrm{Ric}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)} -\frac{11}{9}R^{2}_{g(t)}, \end{aligned}$$

where we used the identities $\mathrm{tr}_{g(t)}\widehat{\varvec{T}}(t)=g^{ij}\varvec{T}_{ik}\varvec{T}^{k}{}_{j} =\varvec{T}_{ik}\varvec{T}^{ki}=-2|\varvec{T}(t)|^{2}_{g(t)}$ and $R_{g(t)}=-2|\varvec{T}(t)|^{2}_{g(t)}$. Replacing $R_{g(t)}$ by $S_{g(t)}$ according to the identity (3.9), we can rewrite (3.16) as

$$\begin{aligned} \blacksquare _{g(t)}S_{g(t)}= & {} \frac{4}{3}||\mathrm{Sic}_{g(t)}||^{2}_{g(t)} -\frac{16}{3}||\widehat{\varvec{T}}(t)||^{2}_{g(t)}-\frac{16}{3}\langle \langle \mathrm{Ric}_{g(t)}, \widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}+\frac{32}{27}R^{2}_{g(t)}\\&+ \ \frac{8}{3}R_{ijk\ell }\varvec{T}^{ik}\varvec{T}^{j\ell } +\frac{8}{3}(\nabla ^{j}\varvec{T}^{ik})(\nabla _{i}\varvec{T}_{jk}). \end{aligned}$$

Similarly, replacing $\langle \langle \mathrm{Ric}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}$ by $\langle \langle \mathrm{Sic}_{g(t)}, \widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}$ with respect to the identity

$$\begin{aligned} \langle \langle \mathrm{Sic}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)}= & {} \left( R_{ij}+\frac{2}{3}|\varvec{T}(t)|^{2}_{g(t)}g_{ij}+2\widehat{\varvec{T}}_{ij} \right) \widehat{\varvec{T}}^{ij}\\= & {} \langle \langle \mathrm{Ric}_{g(t)},\widehat{\varvec{T}}(t)\rangle \rangle _{g(t)} -\frac{1}{3}R^{2}_{g(t)}+2||\widehat{\varvec{T}}(t)||^{2}_{g(t)}, \end{aligned}$$

we obtain the following evolution equation for $S_{g(t)}$,

$$\begin{aligned} \blacksquare _{g(t)}S_{g(t)}=\frac{4}{3}\left[ \left| \left| \mathrm{Sic}_{g(t)}-2\widehat{\varvec{T}}(t) \right| \right| ^{2}_{g(t)}-S^{2}_{g(t)}\right] +\frac{8}{3}\left[ R_{ijk\ell }\varvec{T}^{ik}\varvec{T}^{j\ell } +(\nabla ^{j}\varvec{T}^{ik})(\nabla _{i}\varvec{T}_{jk})\right] . \end{aligned}$$

(3.17)

Next, we try to deal with the last bracket in (3.17), which contains two terms $R_{ijk\ell }\varvec{T}^{ik}\varvec{T}^{j\ell }$ and $(\nabla ^{j}\varvec{T}^{ik})(\nabla _{i} \varvec{T}_{jk})$. Using (2.27) and (2.33), the term $(\nabla ^{j}\varvec{T}^{ik})(\nabla _{i} \varvec{T}_{jk})$ is equal to

$$\begin{aligned} (\nabla ^{j}\varvec{T}^{ik})(\nabla _{i}\varvec{T}_{jk})= & {} \left[ \nabla ^{i}\varvec{T}^{jk}+\left( \frac{1}{2}R^{ij}{}_{ab} +\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \varphi ^{kab}\right] \nabla _{i}\varvec{T}_{jk}\\= & {} \ \ ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)} +\frac{1}{2}\left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \\&\bigg [ -\frac{1}{2}R_{ijmn}\varphi ^{mn}{}_{k}\varphi ^{kab} -\frac{1}{2}R_{kjmn}\varphi _{i}{}^{mn}\varphi ^{kab} \\&+ \ \frac{1}{2}R_{ikmn}\varphi _{j}{}^{mn}\varphi ^{kab} -\varvec{T}_{im}\varvec{T}_{jn}\varphi ^{mn}{}_{k}\varphi ^{kab}\\&-\varvec{T}_{km}\varvec{T}_{jn}\varphi _{i}{}^{mn}\varphi ^{kab} +\varvec{T}_{im}\varvec{T}_{kn}\varphi _{j}{}^{mn}\varphi ^{kab}\bigg ]. \end{aligned}$$

By symmetry the term

$$\begin{aligned} \left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \left( -\frac{1}{2}R_{kjmn}\varphi _{i}{}^{mn}\varphi ^{kab}+ \frac{1}{2}R_{ikmn}\varphi _{j}{}^{mn}\varphi ^{kab}\right) \end{aligned}$$

is equal to, interchanging $i\leftrightarrow j$ and $a\leftrightarrow b$ in the second term,

$$\begin{aligned} \left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \left( -\frac{1}{2}R_{kjmn}\varphi _{i}{}^{mn}\varphi ^{kab}\right) +\left( \frac{1}{2}R^{ji}{}_{ba}+\varvec{T}^{j}{}_{b}\varvec{T}^{i}{}_{a}\right) \left( \frac{1}{2}R_{jkmn}\varphi _{i}{}^{mn}\varphi ^{kba}\right) \end{aligned}$$

which is zero. Similarly, we have, by interchanging $m\leftrightarrow n$ and then $i\leftrightarrow j$, $a\leftrightarrow b$ in the first term,

$$\begin{aligned}&\left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \left( -\varvec{T}_{km}\varvec{T}_{jn}\varphi _{i}{}^{mn}\varphi ^{kab} +\varvec{T}_{im}\varvec{T}_{kn}\varphi _{j}{}^{mn}\varphi ^{kab}\right) \\&\quad = \ \ \left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \left( -\varvec{T}_{kn}\varvec{T}_{jm}\varphi _{i}{}^{nm}\varphi ^{kab} +\varvec{T}_{im}\varvec{T}_{kn}\varphi _{j}{}^{mn}\varphi ^{kab}\right) \\&\quad = \ \ \left( \frac{1}{2}R^{ij}{}_{ab}+\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \left( -\varvec{T}_{kn}\varvec{T}_{im}\varphi _{j}{}^{nm}\varphi ^{kba} +\varvec{T}_{im}\varvec{T}_{kn}\varphi _{j}{}^{mn}\varphi ^{kab}\right) \ \ = \ \ 0. \end{aligned}$$

Therefore, using the identity $\varphi _{ijk}\varphi ^{k}{}_{ab} =g_{ia}g_{jb}-g_{ib}g_{ja}+\psi _{ijab}$ (see [23]), we arrive at

$$\begin{aligned} (\nabla ^{j}\varvec{T}^{ik})(\nabla _{i}\varvec{T}_{jk})= & {} ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}\\&- \ \frac{1}{2}\left( \frac{1}{2}R^{ij}{}_{ab} +\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b} \right) \left( \frac{1}{2}R_{ij}{}^{mn}+\varvec{T}_{i}{}^{m}\varvec{T}_{j}{}^{n} \right) \varphi _{mnk}\varphi ^{kab}\\= & {} ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}-\frac{1}{2}\left( \frac{1}{2}R^{ij}{}_{ab} +\varvec{T}^{i}{}_{a}\varvec{T}^{j}{}_{b}\right) \\&\cdot \left( \frac{1}{2}R_{ij}{}^{mn}+\varvec{T}_{i}{}^{m}\varvec{T}_{j}{}^{n}\right) \left( \delta ^{a}_{m}\delta ^{b}_{n}-\delta ^{b}_{m}\delta ^{a}_{n} +\psi _{mn}{}^{ab}\right) \\= & {} ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)} -\frac{1}{8}\left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb} \right) \bigg [\left( R^{ijab}+2\varvec{T}^{ia}\varvec{T}^{jb}\right) \\&- \ \left( R^{ijba}+2\varvec{T}^{ib}\varvec{T}^{ja}\right) +\left( R^{ijmn}+2\varvec{T}^{im}\varvec{T}^{jn}\right) \psi _{mn}{}^{ab}\bigg ]. \end{aligned}$$

Since, by our convention,

$$\begin{aligned} \left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb}\right) \left( R^{ijab}+2\varvec{T}^{ia}\varvec{T}^{jb}\right) =||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}+4R_{ijab}\varvec{T}^{ia}\varvec{T}^{jb} +4||\varvec{T}(t)||^{4}_{g(t)} \end{aligned}$$

and

$$\begin{aligned} \left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb}\right) \left( R^{ijba}+2\varvec{T}^{ib}\varvec{T}^{ja}\right) = -||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}-4R_{ijab}\varvec{T}^{ia}\varvec{T}^{jb} +4||\widehat{\varvec{T}}(t)||^{2}_{g(t)}, \end{aligned}$$

it follows that

$$\begin{aligned} (\nabla ^{j}\varvec{T}^{ik})(\nabla _{i}\varvec{T}_{jk})= & {} ||\nabla _{t}\varvec{T}(t)||^{2}_{g(t)} +\frac{1}{8}\bigg [-2||\mathrm{Rm}_{t}||^{2}_{t} -8R_{ijab}\varvec{T}^{ia}\varvec{T}^{jb}-4||\varvec{T}(t)||^{4}_{g(t)}\\&+ \ 4||\widehat{\varvec{T}}(t)||^{2}_{g(t)}-\left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb}\right) \left( R^{ijmn}+2\varvec{T}^{im}\varvec{T}^{jn}\right) \psi _{mn}{}^{ab}\bigg ] \end{aligned}$$

and (3.17) can be written as

$$\begin{aligned} \blacksquare _{g(t)}S_{g(t)}= & {} \frac{4}{3}\left| \left| \mathrm{Sic}_{g(t)} -2\widehat{\varvec{T}}(t)\right| \right| ^{2}_{g(t)}+\frac{8}{3}||\nabla _{g(t)}\varvec{T}(t) ||^{2}_{g(t)}+\frac{4}{3}||\widehat{\varvec{T}}(t)||^{2}_{g(t)}\nonumber \\&- \ \frac{2}{3} ||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}-\frac{13}{3}S^{2}_{g(t)}\nonumber \\&- \ \frac{1}{3}\left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb}\right) \left( R^{ijmn}+2\varvec{T}^{im}\varvec{T}^{jn}\right) \psi _{mn}{}^{ab}. \end{aligned}$$

(3.18)

Finally, we deal with the last term J on the right-hand side of (3.18). From the identity $\psi _{ijk\ell } \psi ^{ijk\ell }=168$, we find that

$$\begin{aligned} J:= & {} -\frac{1}{3}\left( R_{ijab}+2\varvec{T}_{ia}\varvec{T}_{jb}\right) \left( R^{ijmn}+2\varvec{T}^{im}\varvec{T}^{jn}\right) \psi _{mn}{}^{ab} \\= & {} \ \ \frac{1}{3}\left( -R_{ij}{}^{ab}R^{ijmn}\psi _{mnab}-4\varvec{T}_{i}{}^{a} \varvec{T}_{j}{}^{b}R^{ijmn}\psi _{mnab} -4\varvec{T}^{a}{}_{i}\varvec{T}^{im}\varvec{T}^{b}{}_{j}\varvec{T}^{jn}\psi _{mnab}\right) \\= & {} \ \ \frac{1}{3}\bigg [\left| \left| R_{ij}{}^{ab}R^{ijmn}-\frac{1}{2} \psi ^{abmn}\right| \right| ^{2}_{g(t)}-\left| \left| R_{ij}{}^{ab}R^{ijmn} \right| \right| ^{2}_{g(t)} -\frac{168}{4} \\&+ \ \left| \left| 2\varvec{T}_{i}{}^{a}\varvec{T}_{j}{}^{b}R^{ijmn}-\psi ^{abmn}\right| \right| ^{2}_{g(t)} -4\left| \left| \varvec{T}_{i}{}^{a}\varvec{T}_{j}{}^{b}R^{ijmn}\right| \right| ^{2}_{g(t)} -168 \\&+ \ \left| \left| 2\widehat{\varvec{T}}^{am}\widehat{\varvec{T}}^{bn}-\psi ^{mnab} \right| \right| ^{2}_{g(t)}-4||\widehat{\varvec{T}}(t)||^{4}_{g(t)}-168\bigg ]. \end{aligned}$$

Plugging the expression for J into (3.18), we obtain

Proposition 3.2

The scalar curvature $R_{g(t)}$ or $S_{g(t)}$ evolves by

$$\begin{aligned} \blacksquare _{g(t)}S_{g(t)}= & {} \frac{4}{3}\left| \left| \mathrm{Sic}_{g(t)}-2\widehat{\varvec{T}}(t)\right| \right| ^{2}_{g(t)} +\frac{8}{3}||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}-\frac{13}{3}S^{2}_{g(t)} -126\nonumber \\&+ \ \frac{1}{3} \left| \left| R_{ijab}R^{ij}{}_{mn}-\psi _{abmn} \right| \right| ^{2}_{g(t)}+\frac{4}{3}||\widehat{\varvec{T}}(t)||^{2}_{g(t)}-\frac{4}{3} ||\widehat{\varvec{T}}(t)||^{4}_{g(t)}\nonumber \\&+ \ \frac{1}{3}\left| \left| 2\varvec{T}_{ia}\varvec{T}_{jb}R^{ij}{}_{mn} -\psi _{abmn}\right| \right| ^{2}_{g(t)} +\frac{1}{3}\left| \left| 2\widehat{\varvec{T}}_{am}\widehat{\varvec{T}}_{bn} -\psi _{abmn}\right| \right| ^{2}_{g(t)}\nonumber \\&- \ \frac{2}{3}||\mathrm{Rm}_{g(t)}||^{2}_{g(t)} -\frac{1}{3}\left| \left| R_{ijab}R^{ij}{}_{mn} \right| \right| ^{2}_{g(t)}-\frac{4}{3}\left| \left| \varvec{T}_{ia}\varvec{T}_{jb} R^{ij}{}_{mn}\right| \right| ^{2}_{g(t)}.\nonumber \\ \end{aligned}$$

(3.19)

Since $S_{g(t)}=\frac{2}{3}R_{g(t)}$, it follows from the above theorem that (1.8) holds true.

Before giving local curvature estimates for Laplacian flow in the next subsection, we derive evolution equations for $\mathrm{Ric}_{g(t)}$, $\mathrm{Rm}_{g(t)}$, and $\varvec{T}(t)$ in different forms. Using the Lichnerowicz Laplacian

$$\begin{aligned} \blacktriangle _{L,g(t)}\eta _{jk}:=\blacktriangle _{g(t)} \eta _{jk}-R_{j}{}^{p}\eta _{pk}-R_{k}{}^{p}\eta _{jp} +2R_{pjkq}h^{qp}, \end{aligned}$$

we see that the evolution equation for $R_{ij}$ can be written as

$$\begin{aligned} \partial _{t}R_{jk}=-\frac{1}{2} \left[ \blacktriangle _{L,g(t)}\eta _{jk} +\nabla _{j}\nabla _{k}\mathrm{tr}_{g(t)}\eta (t)+\nabla _{j}(d^{*}_{g(t)}\eta _{t})_{k} +\nabla _{k}(d^{*}_{g(t)}\eta _{t})_{j}\right] , \end{aligned}$$

where $(d^{*}_{g(t)}\eta (t))_{k}:=-\nabla ^{j}\eta _{jk}$. For $\eta _{ij} =-2R_{ij}-\frac{4}{3}||\varvec{T}(t)||^{2}_{g(t)}g_{ij} -4\varvec{T}_{i}{}^{k}\varvec{T}_{kj}$ we have proved $\mathrm{tr}_{g(t)}\eta (t)=\frac{8}{3}||\varvec{T}(t)||^{2}_{g(t)}$ and $(d^{*}_{g(t)}\eta (t))_{j}=\nabla _{j}R_{g(t)}+\frac{4}{3}\nabla _{j} ||\varvec{T}(t)||^{2}_{g(t)} +4\nabla ^{i}\widehat{\varvec{T}}_{ij}$ with $\widehat{\varvec{T}}_{ij}= \varvec{T}_{i}{}^{k}\varvec{T}_{kj}$. Then

$$\begin{aligned} \partial _{t}R_{jk}= & {} \blacktriangle _{L,g(t)} \left( R_{jk}+\frac{2}{3}||\varvec{T}(t)||^{2}_{g(t)}g_{jk}+2\widehat{ \varvec{T}}_{jk}\right) -\frac{1}{2}\nabla _{j}\bigg (\nabla _{k}R_{g(t)}+\frac{4}{3}\nabla _{k} ||\varvec{T}(t)||^{2}_{g(t)}\\&+ \ 4\nabla ^{i}\widehat{\varvec{T}}_{ik}\bigg )-\frac{4}{3}\nabla _{j}\nabla _{k}||\varvec{T}(t)||^{2}_{g(t)} -\frac{1}{2}\nabla _{k} \left( \nabla _{j}R_{t}+\frac{4}{3}\nabla _{j}||\varvec{T}(t)||^{2}_{g(t)} +4\nabla ^{i}\widehat{\varvec{T}}_{ij}\right) \\= & {} \ \ \blacktriangle _{L,g(t)}\left( R_{jk}+\frac{2}{3}||\varvec{T}(t)||^{2}_{g(t)}g_{jk} +2\widehat{\varvec{T}}_{jk}\right) -2\nabla _{j}\nabla ^{i}\widehat{\varvec{T}}_{ik} \\&- \ 2\nabla _{k}\nabla ^{i}\widehat{\varvec{T}}_{ij}-\frac{2}{3}\nabla _{j}\nabla _{k} ||\varvec{T}_{g(t)}||^{2}_{g(t)}. \end{aligned}$$

But the first term is equal to

$$\begin{aligned}&\blacktriangle _{L,g(t)}\left( R_{jk}+\frac{2}{3}||\varvec{T}(t)||^{2}_{g(t)}g_{jk} +2\widehat{\varvec{T}}_{jk}\right) \ \ = \ \ \blacktriangle _{g(t)}R_{jk}-2R_{j}{}^{p}R_{pk} +2R_{pjkq}R^{pq} \\&\quad + \ \bigg [\frac{2}{3}\left( \blacktriangle _{g(t)}||\varvec{T}(t)||^{2}_{g(t)} \right) g_{jk}+2\blacktriangle _{g(t)}\widehat{\varvec{T}}_{jk} -2R_{j}{}^{p}\widehat{\varvec{T}}_{pk}-2\widehat{\varvec{T}}_{j}{}^{p}R^{p}{}_{k} +4R_{pjkq}\widehat{\varvec{T}}^{pq}\bigg ], \end{aligned}$$

we have

$$\begin{aligned} \blacksquare _{g(t)}R_{ij}= & {} -2R_{i}{}^{p}R_{pj} +2R_{pijq}R^{pq}+\bigg [\frac{2}{3}\left( \blacktriangle _{g(t)}||\varvec{T}(t)||^{2}_{g(t)} \right) g_{ij}+2\blacktriangle _{g(t)}\widehat{\varvec{T}}_{ij}\nonumber \\&- \ 2R_{i}{}^{p}\widehat{\varvec{T}}_{pj}-2\widehat{\varvec{T}}_{i}{}^{p}R_{pj} +4R_{pijq}\widehat{T}^{pq} -2\nabla _{i}\nabla ^{p}\widehat{\varvec{T}}_{pj}\nonumber \\&- \ 2\nabla _{j}\nabla ^{p}\widehat{\varvec{T}}_{pi} -\frac{2}{3}\nabla _{i}\nabla _{j}||\varvec{T}(t)||^{2}_{g(t)}\bigg ]. \end{aligned}$$

(3.20)

Consequently, the norm of $\mathrm{Ric}_{g(t)}$ satisfies

$$\begin{aligned} \blacksquare _{g(t)}||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}= & {} -2||\nabla _{g(t)}\mathrm{Ric}_{g(t)}||^{2}_{g(t)} +\bigg [\frac{4}{3}R_{g(t)}\blacktriangle _{g(t)} ||\varvec{T}(t)||^{2}_{g(t)}\nonumber \\&+ \ 8R^{k}{}_{ij}{}^{\ell }\widehat{\varvec{T}}_{k\ell }R^{ij}+\frac{8}{3}||\mathrm{Ric}_{g(t)}||^{2}_{g(t)}||\varvec{T}(t)||^{2}_{g(t)}+4R_{kij\ell }R^{k\ell }R^{ij}\nonumber \\&\ +4R^{ij}\blacktriangle _{g(t)}\widehat{\varvec{T}}_{ij} -8 R^{ij}\nabla _{i}\nabla ^{k}\widehat{\varvec{T}}_{kj} -\frac{4}{3}R^{ij}\nabla _{i}\nabla _{j}||\varvec{T}(t)||^{2}_{g(t)}\bigg ].\nonumber \\ \end{aligned}$$

(3.21)

The general formula (e.g. formula (2.66) in [5]) for $R_{ijk}^{\ell }$ gives

$$\begin{aligned} \partial _{t}R^{\ell }_{ijk}= & {} -\nabla _{i}\nabla _{k}R_{j}{}^{\ell } -\nabla _{j}\nabla ^{\ell }R_{ik}+\nabla _{i}\nabla ^{\ell }R_{jk} +\nabla _{j}\nabla _{k}R_{i}{}^{\ell }+R_{ijk}{}^{q}R_{q}{}^{\ell }+R_{ij}{}^{\ell q}R_{kp}\nonumber \\&+ \ 2R_{ijk}{}^{q}\widehat{\varvec{T}}_{q}{}^{\ell }+2R_{ij}{}^{\ell q}\widehat{\varvec{T}}_{kp} -\frac{2}{3}\left( \nabla _{i}\nabla _{k}||\varvec{T}(t)||^{2}_{g(t)}\right) g_{j}{}^{\ell } -2\nabla _{i}\nabla _{k}\widehat{\varvec{T}}_{j}{}^{\ell }\nonumber \\&- \ 2\nabla _{j}\nabla ^{\ell }\widehat{\varvec{T}}_{ik} +2\nabla _{i}\nabla ^{\ell }\widehat{\varvec{T}}_{jk}+2\nabla _{j}\nabla _{k} \widehat{\varvec{T}}_{i}{}^{\ell }-\frac{2}{3}\left( \nabla _{j}\nabla ^{\ell }||\varvec{T}(t)||^{2}_{g(t)}\right) g_{ik}\nonumber \\&+ \ \frac{2}{3}\left( \nabla _{i}\nabla ^{\ell }||\varvec{T}(t)||^{2}_{g(t)}\right) g_{jk} +\frac{2}{3}\left( \nabla _{j}\nabla _{k}||\varvec{T}(t)||^{2}_{g(t)} \right) g_{i}{}^{\ell }. \end{aligned}$$

(3.22)

Hence, the evolution equation for $||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}$ is given by

$$\begin{aligned} \partial _{t}||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}= & {} \nabla ^{2}_{g(t)}\mathrm{Ric}_{g(t)}*\mathrm{Rm}_{g(t)}+\mathrm{Ric}_{g(t)}*\mathrm{Rm}_{g(t)} *\mathrm{Rm}_{g(t)}\nonumber \\&+ \ \mathrm{Rm}_{g(t)}*\mathrm{Rm}_{g(t)}*\widehat{\varvec{T}}(t)+\mathrm{Ric}_{g(t)}*\nabla ^{2}_{g(t)}||\varvec{T}(t)||^{2}_{g(t)}\nonumber \\&+ \ \mathrm{Rm}_{g(t)}*\nabla ^{2}_{g(t)}\widehat{\varvec{T}}(t) +\frac{8}{3}|\varvec{T}(t)|^{2}_{g(t)}||\mathrm{Rm}_{g(t)}||^{2}_{g(t)}. \end{aligned}$$

(3.23)

Moreover, it was proved in [32] that

$$\begin{aligned} ||\nabla _{g(t)}\mathrm{Rm}_{g(t)}||^{2}_{g(t)}\le & {} -\frac{1}{2}\blacksquare _{g(t)}||\mathrm{Rm}_{g(t)}||^{2}_{g(t)} +C_{1}||\mathrm{Rm}_{g(t)}||^{3}_{g(t)}+C_{1}||\mathrm{Rm}_{g(t)}||^{3/2}_{g(t)}\nonumber \\&\cdot \ ||\nabla ^{2}_{g(t)}\varvec{T}(t)||_{g(t)} +C_{1}||\mathrm{Rm}_{g(t)}||_{g(t)}||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)} \end{aligned}$$

(3.24)

where $C_{1}$ is some universal constant, and

$$\begin{aligned} \blacksquare _{g(t)}\varvec{T}(t)= & {} \mathrm{Rm}_{g(t)}*\varvec{T}(t) +\mathrm{Rm}_{g(t)}*\varvec{T}(t)*\psi (t)\nonumber \\&+ \ \nabla _{g(t)}\varvec{T}(t)*\varvec{T}(t) *\varphi (t)+\varvec{T}(t)*\varvec{T}(t)*\varvec{T}(t). \end{aligned}$$

(3.25)

Squaring (3.25) gives

$$\begin{aligned} ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}\le & {} -\frac{1}{2}\blacksquare _{g(t)} ||\varvec{T}(t)||^{2}_{g(t)}+C_{2}||\mathrm{Rm}_{g(t)}||_{g(t)}||\varvec{T}(t)||^{2}_{g(t)}\nonumber \\&+ \ C_{2}||\nabla _{g(t)}\varvec{T}(t)||_{g(t)} ||\varvec{T}(t)||^{2}_{g(t)}+C_{2}||\varvec{T}(t)||^{4}_{g(t)} \end{aligned}$$

(3.26)

for another universal constant $C_{2}$ which may differs from $C_{1}$. The Cauchy-Schwartz inequality shows $2C_{2}||\nabla _{g(t)}\varvec{T}(t)||_{g(t)} ||\varvec{T}(t)||^{2}_{g(t)}\le ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)} +C^{2}_{2}||\varvec{T}(t)||^{4}_{g(t)}$, so that the evolution inequality (3.26) becomes

$$\begin{aligned} ||\nabla _{g(t)}\varvec{T}(t)||^{2}_{g(t)}\le & {} -\blacksquare _{g(t)} ||\varvec{T}(t)||^{2}_{g(t)}\nonumber \\&+ \ C_{3}||\mathrm{Rm}_{g(t)}||_{g(t)}||\varvec{T}(t)||^{2}_{g(t)} +C_{3}||\varvec{T}(t)||^{4}_{g(t)}. \end{aligned}$$

(3.27)

Here $C_{3}$ is a universal constant.

3.2 Main idea of proving Theorem 1.4

In this section, we consider the Laplacian flow (3.1) on $\mathcal {M}\times [0,T]$, where $T\in (0,T_{\max })$. From now on we always omit the time subscripts from all considered quantities. From (3.7), (3.21), (3.23), (3.24), and (3.27) we have

$$\begin{aligned} ||\nabla \mathrm{Ric}||^{2}= & {} -\frac{1}{2}\blacksquare ||\mathrm{Ric}||^{2}+\mathrm{Ric}*\mathrm{Ric}*\mathrm{Rm} -\frac{1}{3}\left( \blacktriangle R\right) R -\frac{2}{3}||\mathrm{Ric}||^{2}R\\&+ \ 2\langle \langle \mathrm{Ric}, \blacktriangle \widehat{\varvec{T}} \rangle \rangle +\frac{1}{3}\langle \langle \mathrm{Ric}, \nabla ^{2}R\rangle \rangle +\mathrm{Ric}*\widehat{\varvec{T}}*\mathrm{Rm}+\mathrm{Ric}*\nabla ^{2} \widehat{\varvec{T}},\\ ||\nabla \mathrm{Rm}||^{2}\le & {} -\frac{1}{2}\blacksquare ||\mathrm{Rm}||^{2} +C||\mathrm{Rm}||^{3}+C||\mathrm{Rm}||^{3/2} ||\nabla ^{2}\varvec{T}||+C||\mathrm{Rm}||||\nabla \varvec{T}||^{2},\\ \partial _{t}||\mathrm{Rm}||^{2}= & {} \nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +\mathrm{Ric}*\mathrm{Rm}*\mathrm{Rm}+\mathrm{Rm}*\mathrm{Rm}*\widehat{\varvec{T}}\\&+ \ \mathrm{Ric}*\nabla ^{2}||\varvec{T}||^{2}+\mathrm{Rm}*\nabla ^{2} \widehat{\varvec{T}}+\frac{4}{3}||\varvec{T}||^{2}||\mathrm{Rm}||^{2},\\ ||\nabla \varvec{T}||^{2}\le & {} -\blacksquare ||\varvec{T}||^{2} +C||\mathrm{Rm}||||\varvec{T}||^{2}+C||\varvec{T}||^{4},\\ \partial _{t}dV= & {} \frac{2}{3}||\varvec{T}||^{2}dV, \ \ \ R \ \ = \ \ -||\varvec{T}||^{2}. \end{aligned}$$

Choose an open domain $\Omega$ of $\mathcal {M}$ and assume that

$$\begin{aligned} ||\mathrm{Ric}||\le K \end{aligned}$$

(3.28)

on $\Omega \times [0,T]$, Then the torsion $\varvec{T}$ satisfies²$||\varvec{T}||\lesssim K^{1/2}$ and metrics g(t) are all equivalent to g(0). We also observe from (2.25) and (3.11) that

$$\begin{aligned} ||\mathrm{Ric}||\lesssim 1\Longleftrightarrow |\Delta \varphi |\lesssim 1 \end{aligned}$$

(3.29)

and the following simple fact

$$\begin{aligned} \partial _{t}||A||^{2}=\frac{p}{2}||A||^{p-2}\partial _{t}||A||^{2} \end{aligned}$$

(3.30)

for any tensor A.

Choose a Lipschitz function $\eta$ with support in $\Omega$ (and independent of time t) and consider the quantity

$$\begin{aligned} \frac{d}{dt}\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV, \ \ \ \int :=\int _{\mathcal {M}}, \end{aligned}$$

where $p\ge 5$. As in [28], we introduce the following “good” quantities

$$\begin{aligned} A_{1}:= & {} \int ||\mathrm{Rm}||^{p}\eta ^{2p}dV, \ \ \ A_{2} \ \ := \ \ \int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV,\\ A_{3}:= & {} \int ||\mathrm{Rm}||^{p-1}||\nabla \eta ||^{2}\eta ^{2p-1}dV, \ \ \ A_{4} \ \ := \ \ \int ||\mathrm{Rm}||^{p-1} ||\nabla \eta ||^{2}\eta ^{2p-2}dV \end{aligned}$$

and also “bad” quantities

$$\begin{aligned} B_{1}:=\frac{1}{K}\int ||\nabla \mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV, \ \ \ B_{2}:= \int ||\nabla \mathrm{Rm}||^{2}||\mathrm{Rm}||^{p-3}\eta ^{2p}dV. \end{aligned}$$

We split the proof of Theorem 1.4 into four steps.

(a):

In the first step, we can show that, see Lemma 3.3,

$$\begin{aligned} \frac{d}{dt}A_{1}\le & {} B_{1}+c K B_{2}+cK A_{4} +cK A_{1}+cK^{2}A_{2}\\&+ \ c\int \left( -\blacksquare ||\varvec{T}||^{2} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV. \end{aligned}$$

(b):

In the second step, we can prove that the term

$$\begin{aligned} c\int \left( -\blacksquare ||\varvec{T}||^{2} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \end{aligned}$$

is bounded from above by [see (3.42)]

$$\begin{aligned} B_{1}+cK B_{2}+cK^{2}A_{2}+cK A_{1} -\frac{d}{dt}\left[ \int c(-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

Observe that the above integral is nonnegative, since the scalar curvature R is nonpositive along the Laplacian flow on closed $G_{2}$-structures. Hence we obtain from the first step that, see Lemma 3.4,

$$\begin{aligned} \frac{d}{dt}A_{1}\le & {} 2B_{1}+cK B_{2}+cK A_{4}+cK A_{1} +cK^{2}A_{2}\\&- \ \frac{d}{dt}\left[ \int c(-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(c):

In the next two steps, we estimate the bad terms $B_{1}$ and $B_{2}$. In the third step, $B_{1}$ is estimated by [see (3.52)]

$$\begin{aligned} B_{1}\le & {} cK B_{2}+cK A_{4}+cK A_{1} +cK^{2}A_{2}\\&- \ \frac{d}{dt}\left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV+c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

Then the second step can be simplified as, see Lemma 3.5,

$$\begin{aligned} \frac{d}{dt}A_{1}\le & {} cK B_{2} +cK A_{4}+cK A_{1}+cK^{2}A_{2}\\&- \ \frac{d}{dt}\left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV+c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(d):

Finally, we estimate the term $B_{2}$. In this step we shall use the assumption that $p\ge 5$ (a technical assumption). Using the inequality $|| \nabla \varvec{T}||\lesssim ||\mathrm{Rm}||$ and $||\nabla ^{2}\varvec{T}|| \lesssim ||\nabla \mathrm{Rm}||+||\mathrm{Rm}||||\varvec{T}|| +||\nabla \varvec{T}|||\varvec{T}||+||\varvec{T}||^{3}$, we can prove [see (3.62)]

$$\begin{aligned} B_{2}\le cA_{4}+cA_{1}- \frac{d}{dt}\left[ \frac{1}{p-1} \int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\right] . \end{aligned}$$

Plugging it into the third step, we arrive at, see Lemma 3.6,

$$\begin{aligned} \frac{d}{dt}(A_{1}+cK A_{2})\le & {} cK(A_{1}+cK A_{2})+cK A_{4}\\&- \ \frac{d}{dt}\bigg [\frac{c}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV\\&+ \ c\int (-R)||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\bigg ]. \end{aligned}$$

The proof of Theorem 1.4

As in [25, 28], we choose a geodesic ball $\Omega :=B_{g(0)}(x_{0},\rho /\sqrt{K})$ and a cut-off function

$$\begin{aligned} \eta =\left( \frac{\rho /\sqrt{K}-d_{g(0)}(x_{0},\cdot )}{\rho /\sqrt{K}} \right) _{+}. \end{aligned}$$

Then, for all $t\in [0,T]$,

$$\begin{aligned} e^{-cK t}g(0)\le g(t)\le e^{cK t}g(0), \ \ \ ||\nabla _{g(t)}\phi ||_{g(t)}\le e^{cK T}||\nabla _{g(0)} \phi ||_{g(0)}\le \frac{\sqrt{K}e^{cKT}}{\rho }. \end{aligned}$$

Define

$$\begin{aligned} U:= & {} \int ||\mathrm{Rm}||^{p}\eta ^{2p} dV +cK\int ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&+ \ \frac{c}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV. \end{aligned}$$

(3.31)

Then (3.64) (see below) yields

$$\begin{aligned} U'\le cK U+cK A_{4}. \end{aligned}$$

(3.32)

For $A_{4}$, using the Young inequality, we have

$$\begin{aligned} A_{4}= & {} \int ||\mathrm{Rm}||^{p-1} ||\nabla \eta ||^{2}\eta ^{2p-2}dV \ \le \ \int _{B_{g(0)} (x_{0},\rho /\sqrt{K})} ||\mathrm{Rm}||^{p-1}\eta ^{2p-2}K\rho ^{-2}e^{cKT}dV \\\le & {} \ \ \int _{B_{g(0)}(x_{0},\rho /\sqrt{K})} \left[ \frac{(||\mathrm{Rm}||^{p-1} \eta ^{2p-2})^{p/(p-1)}}{\frac{p}{p-1}} +\frac{(K\rho ^{-2}e^{cKT})^{p}}{p}\right] dV \\\le & {} \ \ A_{1}+K^{p}\rho ^{-2p}p e^{cKT} \mathrm{Vol}_{g(t)}\left( B_{g(0)} \left( x_{0},\frac{\rho }{\sqrt{K}}\right) \right) \\\le & {} \ \ U+cK^{p}e^{cKT}\rho ^{-2p} \mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0},\frac{\rho }{\sqrt{K}}\right) \right) . \end{aligned}$$

Thus

$$\begin{aligned} U'\le cK U+cK^{p+1}e^{cKT} \rho ^{-2p}\mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0},\frac{\rho }{\sqrt{K}} \right) \right) . \end{aligned}$$

As in the proof of [25], one can easily deduce from above that

$$\begin{aligned}&\int _{B_{g(0)}(x_{0},\frac{\rho }{2\sqrt{K}})} ||\mathrm{Rm}_{g(t)}||^{p}_{g(t)} dV_{g(t)} \ \ \le \ \ c(1+K) e^{cKT} \int _{B_{g(0)}(x_{0},\frac{\rho }{\sqrt{K}})} ||\mathrm{Rm}_{g(0)}||^{p}_{g(0)} dV_{g(0)} \nonumber \\&\quad + \ c K^{p}\left( 1+\rho ^{-2p}\right) e^{cK T}\mathrm{Vol}_{g(t)}\left( B_{g(0)} \left( x_{0},\frac{\rho }{\sqrt{K}}\right) \right) . \end{aligned}$$

(3.33)

Indeed, writing $A:=cK$ and $B:=cK^{p+1}e^{cKT}\rho ^{-2p}$, we get

$$\begin{aligned} U'\le A U+B\!\ \mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) \end{aligned}$$

and then

$$\begin{aligned} e^{-At}U(t)\le U(0)+\int ^{t}_{0}B e^{-A\tau }\mathrm{Vol}_{g(\tau )} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) d\tau . \end{aligned}$$

On the other hand, the estimate $e^{-cKt}g(0)\le g(t)\le e^{cKt}g(0)$ yields

$$\begin{aligned} \mathrm{Vol}_{g(\tau )} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) \le e^{cKT}\mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) . \end{aligned}$$

Consequently,

$$\begin{aligned} U(t)\le e^{AT}\left[ U(0)+\frac{B}{A}e^{cKT}\mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) \right] , \ \ \ t\in [0,T]. \end{aligned}$$

At last, we estimate from (3.28) and Young’s inequality

$$\begin{aligned} U(0)= & {} \int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p}_{g(0)}\eta ^{2p}dV_{g(0)} +cK\int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p-1}_{g(0)}\eta ^{2p}dV_{g(0)} \\&+\frac{c}{K}\int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p-1}_{g(0)} ||\mathrm{Ric}_{g(0)}||^{2}_{g(0)}\eta ^{2p}dV_{g(0)}\\&+ c\int _{\mathcal {M}}(-R_{g(0)})||\mathrm{Rm}_{g(0)}||^{p-1}_{g(0)} \eta ^{2p}dV_{g(0)} \\\le & {} \int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p}_{g(0)}\eta ^{2p}dV_{g(0)} + cK\int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p-1}_{g(0)}\eta ^{2p}dV_{g(0)} \\\le & {} \int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p}_{g(0)} \eta ^{2p}dV_{g(0)}+C\int _{\mathcal {M}}\bigg [\left( ||\mathrm{Rm}_{g(0)}||^{p-1}_{g(0)}\eta ^{2(p-1)}\right) ^{\frac{p}{p-1}}dV_{g(0)} \\&+ \ \int _{\mathcal {M}}(K\eta ^{2})^{p}dV_{g(0)}\bigg ] \\\le & {} (1+K)\int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p}_{g(0)}\eta ^{2p}dV_{g(0)} + CK^{p}\mathrm{Vol}_{g(0)}\left( B_{g(0)}\left( x_{0}, \frac{\rho }{K}\right) \right) \\\le & {} C(1+K)\int _{\mathcal {M}}||\mathrm{Rm}_{g(0)}||^{p}_{g(0)}\eta ^{2p}dV_{g(0)} + CK^{p}e^{cKT}\mathrm{Vol}_{g(t)} \left( B_{g(0)}\left( x_{0}, \frac{\rho }{\sqrt{K}}\right) \right) \end{aligned}$$

which implies (3.33).

As an immediate consequence of the inequality (3.33) we give another proof of the part (a) in Theorem 1.2.

3.3 Proving four steps $(a)-(d)$

We are going to carry out the above mentioned four steps. From (3.23) and the above evolution equations, we have

$$\begin{aligned}&\frac{d}{dt}\int ||\mathrm{Rm}||^{p}\eta ^{2p} dV \nonumber \\&\quad = \int \left( \partial _{t}||\mathrm{Rm}||^{p}\right) \eta ^{2p}dV+\int ||\mathrm{Rm}||^{p}\eta ^{2p} \partial _{t}dV \nonumber \\&\quad = \ \ \int \frac{p}{2}||\mathrm{Rm}||^{p-2} \left( \partial _{t}||\mathrm{Rm}||^{2}\right) \eta ^{2p}dV+\int ||\mathrm{Rm}||^{p}\eta ^{2p} \left( -\frac{2}{3}R\right) dV \nonumber \\&\quad = \ \ \int \frac{p}{2} ||\mathrm{Rm}||^{p-2} \left[ \begin{array}{cc} \nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +\mathrm{Ric}*\mathrm{Rm}*\mathrm{Rm}\\ + \ \mathrm{Rm} *\mathrm{Rm}*\widehat{\varvec{T}}+\mathrm{Ric}*\nabla ^{2}||\varvec{T}||^{2}\\ + \ \mathrm{Rm}*\nabla ^{2}\widehat{\varvec{T}} +\frac{4}{3}||\varvec{T}||^{2}||\mathrm{Rm}||^{2} \end{array}\right] \eta ^{2p}dV \nonumber \\&\qquad - \ \ \frac{2}{3}\int R||\mathrm{Rm}||^{p} \eta ^{2p}dV\nonumber \\&\quad \le \ \ c\int ||\mathrm{Rm}||^{p-2}\bigg [\nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +K||\mathrm{Rm}||^{2}+K||\mathrm{Rm}||^{2} +\nabla ^{2}||\varvec{T}||^{2}*\mathrm{Ric} \nonumber \\&\qquad + \ \nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm}\bigg ] \eta ^{2p}dV+cK\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV \nonumber \\&\quad \le \ \ c\int ||\mathrm{Rm}||^{p-2} \left[ \nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +\nabla ^{2}||\varvec{T}||^{2}*\mathrm{Ric} +\nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm}\right] \eta ^{2p}dV \nonumber \\&\qquad + \ cK\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV. \end{aligned}$$

(3.34)

It was proved in [25] that the first integral in (3.34) is bounded by

$$\begin{aligned}&c\int ||\mathrm{Rm}||^{p-2} \left( \nabla ^{2}\mathrm{Ric}*\mathrm{Rm}\right) \eta ^{2p}dV \ \ \le \ \ \frac{1}{K}\int ||\nabla \mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&\quad + \ cK\int ||\nabla \mathrm{Rm}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}dV +cK\int ||\mathrm{Rm}||^{p-1}||\nabla \eta ||^{2}\eta ^{2p-2}dV. \end{aligned}$$

(3.35)

Since $||\varvec{T}||^{2}=-R$, the same inequality holds for the integral

$$\begin{aligned} c\int ||\mathrm{Rm}||^{p-2} \left( \nabla ^{2}||\varvec{T}||^{2}*\mathrm{Ric}\right) \eta ^{2p} dV. \end{aligned}$$

To deal with the last term in the bracket of (3.34), we use the same argument of [25] to conclude

$$\begin{aligned} c\int ||\mathrm{Rm}||^{p-2} \left( \nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm}\right) \eta ^{2p}dV= & {} \ \ c\int \left( \nabla ||\mathrm{Rm}||^{p-2}*\nabla \widehat{\varvec{T}}*\mathrm{Rm}\right) \eta ^{2p}dV\\&+ \ c\int \left( ||\mathrm{Rm}||^{p-2}*\nabla \widehat{\varvec{T}} *\nabla \mathrm{Rm}\right) \eta ^{2p}dV \\&+ \ c\int \left( ||\mathrm{Rm}||^{p-2}*\nabla \widehat{\varvec{T}}*\mathrm{Rm}*\nabla \eta \right) \eta ^{2p-1}dV \\\le & {} \ \ c\int ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}|| ||\nabla \widehat{\varvec{T}}||\eta ^{2p}dV\\&+\, c\int ||\mathrm{Rm}||^{p-2}||\nabla \widehat{\varvec{T}}|| ||\nabla \mathrm{Rm}||\eta ^{2p}dV \\&+ \ c\int ||\mathrm{Rm}||^{p-1}||\nabla \widehat{\varvec{T}}||||\nabla \eta ||\eta ^{2p-1}dV \\\le & {} \ \ c\int ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}|| ||\nabla \widehat{\varvec{T}}||\eta ^{2p}dV\\&+\, c\int ||\mathrm{Rm}||^{p-1} ||\nabla \widehat{\varvec{T}}||||\nabla \eta ||\eta ^{2p-1} dV. \end{aligned}$$

According to the Cauchy-Schwartz inequality, the first and second integrals are bounded by

$$\begin{aligned}&\int ||\mathrm{Rm}||^{p-2} ||\nabla \mathrm{Rm}||||\nabla \widehat{\varvec{T}}||\eta ^{2p}dV \\&\quad \le \ \ cK\int ||\nabla \mathrm{Rm}||^{2} ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV +\frac{1}{K}\int ||\nabla \widehat{\varvec{T}}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \end{aligned}$$

and

$$\begin{aligned}&\int ||\mathrm{Rm}||^{p-1}||\nabla \widehat{\varvec{T}}|| ||\nabla \eta ||\eta ^{2p-1}dV \\&\quad \le \ \ \frac{1}{K}\int ||\nabla \widehat{\varvec{T}}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK\int ||\mathrm{Rm}||^{p-1}||\nabla \eta ||^{2} \eta ^{2p-2}dV. \end{aligned}$$

Hence we obtain

$$\begin{aligned} c\int ||\mathrm{Rm}||^{p-2} \left( \nabla ^{2}\widehat{\varvec{T}} *\mathrm{Rm}\right) \eta ^{2p}dV\le & {} \frac{1}{K}\int ||\nabla \widehat{\varvec{T}}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\nonumber \\&+\, \ cK\int ||\nabla \mathrm{Rm}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}dV\nonumber \\&+\, cK\int ||\mathrm{Rm}||^{p-1} ||\nabla \eta ||^{2}\eta ^{2p-2}dV. \end{aligned}$$

(3.36)

Using $\widehat{\varvec{T}}=\varvec{T}*\varvec{T}$ and $R=-||\varvec{T}||^{2}$ yields

$$\begin{aligned}&\frac{1}{K}\int ||\nabla \widehat{\varvec{T}}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&\quad \le \ \ \frac{c}{K}\int ||\nabla \varvec{T}||^{2} ||\varvec{T}||^{2}||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \ \ \le \ \ c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&\quad \le \ \ c\int \left( -\frac{1}{4}\blacksquare ||\varvec{T}||^{2}+c||\mathrm{Rm}||||\varvec{T}||^{2}+c||\varvec{T}||^{4}\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&\quad = \ \ c\int \left( -\blacksquare ||\varvec{T}||^{2}\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&\qquad + \ cK\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV +cK^{2}\int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV. \end{aligned}$$

(3.37)

Hence, using (3.35), (3.36), and (3.37), we arrive at

Lemma 3.3

One has

$$\begin{aligned} A'_{1} \ \equiv \ \frac{d}{dt}A_{1}\le & {} B_{1}+cKB_{2} +cKA_{4}+cKA_{1}+cK^{2}A_{2}\nonumber \\&+ \ c\int \left( -\blacksquare ||\varvec{T}||^{2}\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV. \end{aligned}$$

(3.38)

In the following computations, we are mainly going to estimate or simplify the bad terms $B_{1}, B_{2}$, and also the term involving $-\blacksquare ||\varvec{T}||^{2}$. Integration by parts on the last integral in (3.38) and using $R=-||\varvec{T}||^{2}$, we obtain

$$\begin{aligned} c\int \left( -\blacksquare ||\varvec{T}||^{2} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV= & {} c\int \left( (\partial _{t}-\Delta )R\right) ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\\= & {} \ \ c\int \left( \partial _{t}R\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\\&+\, c\int \left\langle \nabla R,\nabla \left( ||\mathrm{Rm}||^{p-1} \eta ^{2p}\right) \right\rangle dV \\= & {} \ \ \frac{d}{dt}\left( c\int R||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right) \\&-\, c\int R\left( \partial _{t}||\mathrm{Rm}||^{p-1} \right) \eta ^{2p}dV \\&- \ c\int R||\mathrm{Rm}||^{p-1}\eta ^{2p}\partial _{t}dV\\&+ c\int \left\langle \nabla R, ||\mathrm{Rm}||^{p-3} \mathrm{Rm}*\nabla \mathrm{Rm}\right\rangle \eta ^{2p}dV \\&+\, \ c\int \left\langle \nabla R, ||\mathrm{Rm}||^{p-1} \eta ^{2p-1}\nabla \eta \right\rangle dV \\\le & {} \ \ c\int ||\mathrm{Rm}||^{p-2} \langle \nabla R,\nabla \mathrm{Rm}\rangle \eta ^{2p}dV\\&+ c\int ||\mathrm{Rm}||^{p-1}||\nabla R||||\nabla \eta ||\eta ^{2p-1}dV \\&+ \ c\int R^{2}||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\\&-\, c\int R\left( \partial _{t}||\mathrm{Rm}||^{p-1}\right) \eta ^{2p}dV \\&+ \ \frac{d}{dt}\left( c\int R||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \right) . \end{aligned}$$

The first two integrals can be simplified by using the Cauchy–Schwarz inequality as follows:

$$\begin{aligned}&c\int ||\mathrm{Rm}||^{p-2}\langle \nabla R,\nabla \mathrm{Rm}\rangle \eta ^{2p}dV \\&\quad \le c\int ||\nabla \mathrm{Ric}||||\nabla \mathrm{Rm}|| ||\mathrm{Rm}||^{p-2}\eta ^{2p}dV \\&\quad \le \ \ c\int \left( ||\nabla \mathrm{Rm}|| ||\mathrm{Rm}||^{\frac{p-3}{2}}\eta ^{p} \right) \left( ||\nabla \mathrm{Ric}||||\mathrm{Rm}||^{\frac{p-1}{2}} \eta ^{p}\right) dV \\&\quad \le \frac{1}{50}B_{1}+cK B_{2} \end{aligned}$$

and

$$\begin{aligned}&c\int ||\mathrm{Rm}||^{p-1} ||\nabla R||||\nabla \eta ||\eta ^{2p-1}dV \\&\quad \le \ \ c\int ||\mathrm{Rm}||^{p-1}||\nabla \mathrm{Ric}|| ||\nabla \eta ||\eta ^{2p-1}dV\\&\quad \le \ \ c\int \left( ||\mathrm{Rm}||^{\frac{p-1}{2}} ||\nabla \eta ||\eta ^{p-1}\right) \left( ||\mathrm{Rm}||^{\frac{p-1}{2}}||\nabla \mathrm{Ric}|| \eta ^{p}\right) dV \\&\quad \le \frac{1}{50}B_{1}+cK A_{4}. \end{aligned}$$

Therefore

$$\begin{aligned} c\int \left( -\blacksquare ||\varvec{T}||^{2} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\le & {} \frac{2}{50}B_{1}+cK B_{2}+cK A_{4}+cK^{2}A_{2}\nonumber \\&+ \ \frac{d}{dt}\left( c\int R||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \right) \nonumber \\&- c\int R\left( \partial _{t}||\mathrm{Rm}||^{p-1}\right) \eta ^{2p}dV. \end{aligned}$$

(3.39)

Now, the second integral in (3.39) is equal to

$$\begin{aligned}&- c\int R\left( \partial _{t}||\mathrm{Rm}||^{p-1}\right) \eta ^{2p}dV \ \ = \ \ c\int \left( -R\right) ||\mathrm{Rm}||^{p-3}\left( \partial _{t}||\mathrm{Rm}||^{2}\right) \eta ^{2p}dV \\&\quad = \ \ c\int (-R)||\mathrm{Rm}||^{p-3} \bigg [\nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +\mathrm{Ric}*\mathrm{Rm}*\mathrm{Rm} +\mathrm{Rm}*\mathrm{Rm}*\widehat{\varvec{T}} \\&\qquad + \ \mathrm{Ric}*\nabla ^{2}||\varvec{T}||^{2} +\mathrm{Rm}*\nabla ^{2}\widehat{\varvec{T}} +\frac{4}{3}||\varvec{T}||^{2}||\mathrm{Rm}||^{2}\bigg ]\eta ^{2p}dV \\&\quad \le \ \ c\int (-R)||\mathrm{Rm}||^{p-3} \left[ \nabla ^{2}\mathrm{Ric}*\mathrm{Rm} -\mathrm{Ric}*\nabla ^{2}R+\nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm} \right] \eta ^{2p}dV +cK^{2}A_{2}. \end{aligned}$$

Using the identity, where $p\ge 5$,

$$\begin{aligned} \nabla ||\mathrm{Rm}||^{p-3} =\frac{p-3}{2} \left( ||\mathrm{Rm}||^{2}\right) ^{\frac{p-3}{2}-1} \nabla ||\mathrm{Rm}||^{2}=||\mathrm{Rm}||^{p-5} \mathrm{Rm}*\nabla \mathrm{Rm} \end{aligned}$$

we obtain

$$\begin{aligned}&c\int (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p}(\nabla ^{2}\mathrm{Ric} *\mathrm{Rm})dV \nonumber \\&\quad = c\int (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p} (\nabla \mathrm{Ric}*\nabla \mathrm{Rm})dV \\&\qquad + \ c\int \left\{ \nabla \left[ (-R)||\mathrm{Rm}||^{p-3}\phi ^{2p}\right] *\nabla \mathrm{Ric}*\mathrm{Rm}\right\} dV \\&\qquad = c\int (-R)||\mathrm{Rm}||^{p-3} \eta ^{2p}(\nabla \mathrm{Ric}*\nabla \mathrm{Rm}) dV\\&\qquad +\, c\int ||\mathrm{Rm}||^{p-3}\eta ^{2p}(\nabla R*\nabla \mathrm{Ric} *\mathrm{Rm})dV \\&\qquad +\, \ c\int (-R)\eta ^{2p}\left( \nabla ||\mathrm{Rm}||^{p-3} *\nabla \mathrm{Ric}*\mathrm{Rm}\right) dV \\&\qquad +\, \ c\int (-R)||\mathrm{Rm}||^{p-3} \eta ^{2p-1}\left( \nabla \phi *\nabla \mathrm{Ric}*\mathrm{Rm} \right) dV \\&\quad \le \ \ c\int ||\mathrm{Rm}||^{p-2}\eta ^{2p} ||\nabla \mathrm{Ric}||||\nabla \mathrm{Rm}||dV\\&\qquad +\, c\int ||\nabla \mathrm{Ric}||||\nabla R||||\mathrm{Rm}||^{p-2}\eta ^{2p}dV \\&\qquad +\, \ c\int ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Ric}||||\nabla \mathrm{Rm}|| \eta ^{2p}dV\\&\qquad +\, c\int ||\mathrm{Rm}||^{p-1}\eta ^{2p-1} ||\nabla \eta ||||\nabla \mathrm{Ric}||dV \\&\quad \le \ \ c\int \left( ||\nabla \mathrm{Ric}|| ||\mathrm{Rm}||^{\frac{p-1}{2}}\eta ^{p}\right) \left( ||\nabla \mathrm{Rm}||||\mathrm{Rm}||^{\frac{p-3}{2}} \eta ^{p}\right) dV \\&\qquad +\, \ c\int \left( ||\nabla \mathrm{Ric}||||\mathrm{Rm}||^{\frac{p-1}{2}} \eta ^{p}\right) \left( ||\nabla \phi ||||\mathrm{Rm}||^{\frac{p-1}{2}} \eta ^{p-1}\right) dV \\&\qquad \le \frac{1}{50}B_{1}+cK B_{2} +cK A_{4}. \end{aligned}$$

Similarly, we can prove

$$\begin{aligned} c\int (-R)||\mathrm{Rm}||^{p-3} \left( -\mathrm{Ric}*\nabla ^{2} R\right) \eta ^{2p}dV \le \frac{1}{50}B_{1}+cK B_{2}+cKA_{4}. \end{aligned}$$

Using $\nabla \widehat{\varvec{T}}=\nabla \varvec{T}*\varvec{T}\le c||\nabla \varvec{T}|||| \varvec{T}|| \le c K^{1/2}||\nabla \varvec{T}||$ yields

$$\begin{aligned}&c\int (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p} \left( \nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm}\right) dV \\&\quad = c\int (-R)||\mathrm{Rm}||^{p-3} \eta ^{2p}(\nabla \widehat{\varvec{T}}*\nabla \mathrm{Rm})dV \\&\qquad +\, \ c\int \left\{ \nabla \left[ (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p}\right] *\nabla \widehat{\varvec{T}}*\mathrm{Rm}\right\} dV \\&\quad = \ \ c\int (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p} (\nabla \widehat{T}*\nabla \mathrm{Rm}) dV\\&\qquad +\, c\int ||\mathrm{Rm}||^{p-3}\eta ^{2p} (\nabla R*\nabla \widehat{\varvec{T}}*\mathrm{Rm})dV \\&\qquad +\, \ c\int (-R)\eta ^{2p} \left( \nabla ||\mathrm{Rm}||^{p-3}*\nabla \widehat{\varvec{T}} *\mathrm{Rm}\right) dV \\&\qquad +\, \ c\int (-R)||\mathrm{Rm}||^{p-3}\eta ^{2p-1} \left( \nabla \eta *\nabla \widehat{\varvec{T}} *\mathrm{Rm}\right) dV \\&\quad \le \ \ c\int \left( ||\mathrm{Rm}||^{p-2} \eta ^{2p}||\nabla \mathrm{Rm}||\right. \\&\qquad \left. +||\mathrm{Rm}||^{p-1} \eta ^{2p-1}||\nabla \eta ||\right) \left( K^{1/2}||\nabla \varvec{T}||\right) dV \\&\quad \le \ c\int \left( ||\nabla \mathrm{Rm}|| ||\mathrm{Rm}||^{\frac{p-3}{2}}\eta \right) \left( ||\nabla \varvec{T}|| K^{1/2}||\mathrm{Rm}||^{\frac{p-1}{2}} \eta ^{p}\right) dV \\&\qquad + \ \int \left( ||\nabla \eta ||||\mathrm{Rm}||^{\frac{p-1}{2}} \eta ^{p-1}\right) \left( ||\nabla \varvec{T}|| K^{1/2} ||\mathrm{Rm}||^{\frac{p-1}{2}}\eta ^{p}\right) dV \\&\quad \le \ \ \epsilon c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV +\frac{cK}{\epsilon }B_{2}+\frac{cK}{\epsilon }A_{4}. \end{aligned}$$

According to (3.39) we get

$$\begin{aligned}&c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \\&\quad \le \ \ c\int \left( -\blacksquare ||\varvec{T}||^{2}\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cKA_{1}+cK^{2}A_{2} \\&\quad \le \ \ \frac{2}{50}B_{1}+cK B_{2}+cK A_{4}+cK^{2}A_{2} +cKA_{1} \\&\qquad + \ \frac{d}{dt}\left( c\int R||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right) -c\int R\left( \partial _{t}||\mathrm{Rm}||^{p-1}\right) \eta ^{2p}dV \\&\quad \le \ \ \frac{2}{50}B_{1}+cK B_{2} +cK A_{4}+cK^{2}A_{2}+cK A_{1} \\&\qquad + \ \frac{d}{dt}\left( \int cR||\mathrm{Rm}||^{p-1}\eta ^{2p} dV\right) +c\int (-R)||\mathrm{Rm}||^{p-3}\left( \partial _{t} ||\mathrm{Rm}||^{2}\right) \eta ^{2p}dV. \end{aligned}$$

Hence

$$\begin{aligned}&c\int (-R)||\mathrm{Rm}||^{p-3}\left( \partial _{t} ||\mathrm{Rm}||^{2}\right) \eta ^{2p}dV \ \\&\quad \le \ \ \frac{2}{50}B_{1}+cK B_{2} +cK A_{4}+\frac{cK}{\epsilon }B_{2}+\frac{cK}{\epsilon }A_{4} \\&\qquad + \ \epsilon \left[ \frac{2}{50}B_{1} +cK B_{2}+cK A_{4}+cK^{2}A_{2}+cK A_{1}\right. \\&\qquad \left. +\frac{d}{dt}\left( \int cR||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right) \right] \\&\qquad + \ \epsilon c\int (-R)||\mathrm{Rm}||^{p-3} \left( \partial _{t}||\mathrm{Rm}||^{2}\right) \eta ^{2p}dV. \end{aligned}$$

Choosing $\epsilon =\frac{1}{2}$ yields

$$\begin{aligned}&\frac{c}{2}\int (-R)||\mathrm{Rm}||^{p-3}\left( \partial _{t}||\mathrm{Rm}||^{2} \right) \eta ^{2p}dV \\&\quad \le \ \ \frac{3}{50}B_{1}+cK B_{2} +cK A_{4}+cK^{2} A_{2}+cK A_{1}+\frac{d}{dt} \left( \int cR||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \right) \end{aligned}$$

and

$$\begin{aligned}&c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \\&\quad \le \ \ \frac{8}{50}B_{1} +cK B_{2}+c K A_{4}+c K^{2}A_{2}+c KA_{1} +\frac{d}{dt}\left( \int 2cR||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \right) . \end{aligned}$$

Thus

$$\begin{aligned}&c\int (-R)||\mathrm{Rm}||^{p-3} \left( \partial _{t}||\mathrm{Rm}||^{2} \right) \eta ^{2p}dV \ \ \le \ \ \frac{3}{50}B_{1}+cK B_{2}\nonumber \\&\quad + \ cK A_{4}+c K^{2}A_{2}+c KA_{1}+\frac{d}{dt} \left( \int cR||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \right) \end{aligned}$$

(3.40)

and

$$\begin{aligned}&c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \ \ \le \ \ \frac{8}{50}B_{1}+c K B_{2}\nonumber \\&\quad + \ cK A_{4}+c K^{2}A_{2}+cK A_{1} +\frac{d}{dt}\left( \int cR||\mathrm{Rm}||^{p-1}\eta ^{2p} dV\right) \end{aligned}$$

(3.41)

and

$$\begin{aligned}&c\int \left( -\blacksquare ||\varvec{T}||^{2}\right) ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \ \ \le \ \ \frac{5}{50}B_{1} +c K B_{2}\nonumber \\&\quad + \ c K^{2}A_{2}+cK A_{1}+\frac{d}{dt} \left( \int cR||\mathrm{Rm}||^{p-1}\eta ^{2p} dV\right) . \end{aligned}$$

(3.42)

From (3.38) and (3.42) we arrive at

Lemma 3.4

One has

$$\begin{aligned} A'_{1}\le & {} 2B_{1}+c K B_{2}+cK A_{4} +c K^{2}A_{2}+c K A_{1}\nonumber \\&+ \ \frac{d}{dt}\left( \int cR||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right) . \end{aligned}$$

(3.43)

We next estimate $B_{1}$ and $B_{2}$. Actually, we shall see that $B_{1}$ can be estimated in terms of $B_{2}$. Hence the key step is to estimate $B_{2}$. For $B_{1}$, using

$$\begin{aligned} ||\nabla \mathrm{Ric}||^{2}= & {} -\frac{1}{2}\blacksquare ||\mathrm{Ric}||^{2} +\mathrm{Ric}*\mathrm{Ric}*\mathrm{Rm} -\frac{1}{3}(\blacktriangle R)\varvec{T}-\frac{2}{3}R||\mathrm{Ric}||^{2}\\&+ \ 2\langle \langle \mathrm{Ric}, \blacktriangle \widehat{\varvec{T}}\rangle \rangle +\frac{1}{3}\langle \langle \mathrm{Ric},\nabla ^{2}R\rangle \rangle +\mathrm{Ric}*\widehat{\varvec{T}}*\mathrm{Rm}+\mathrm{Ric}*\nabla ^{2} \widehat{\varvec{T}}. \end{aligned}$$

we obtain

$$\begin{aligned} B_{1}\le & {} \frac{1}{2K}\int ||\mathrm{Rm}||^{p-1} \eta ^{2p} \left( \blacktriangle -\partial _{t}\right) ||\mathrm{Ric}||^{2}dV +c K A_{1} \nonumber \\&+ \ \frac{1}{3K}\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}\Delta R\!\ dV +\frac{2}{K}\int \langle \langle \mathrm{Ric},\blacktriangle \widehat{\varvec{T}} \rangle \rangle ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&+ \ \frac{1}{3K}\int \langle \langle \mathrm{Ric}, \nabla ^{2}R\rangle \rangle ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV+\frac{1}{K}\int ||\mathrm{Rm}||^{p-1}\left( \mathrm{Ric} *\nabla ^{2}\widehat{\varvec{T}}\right) \eta ^{2p}dV.\nonumber \\ \end{aligned}$$

(3.44)

From the estimates $\nabla ||\mathrm{Ric}||^{2} \lesssim ||\mathrm{Ric}||||\nabla \mathrm{Ric}||$, $\nabla ||\mathrm{Rm}||^{p-1} \lesssim ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}||$, and $\partial _{t}||\mathrm{Rm}||^{p-1}=\frac{p-1}{2}||\mathrm{Rm}||^{p-3} \partial _{t}||\mathrm{Rm}||^{2}$, we have

$$\begin{aligned}&\int ||\mathrm{Rm}||^{p-1} \eta ^{2p} \left( \blacktriangle -\partial _{t}\right) ||\mathrm{Ric}||^{2}dV \\&\quad = \ \ \int \nabla ||\mathrm{Ric}||^{2}*\nabla \left( ||\mathrm{Rm}||^{p-1} \eta ^{2p}\right) dV -\int ||\mathrm{Rm}||^{p-1} \eta ^{2p}\left( \partial _{t}||\mathrm{Ric}||^{2} \right) dV \\&\quad = \ \ \int \left( \nabla ||\mathrm{Ric}||^{2}*\nabla ||\mathrm{Rm}||^{p-1}\right) \eta ^{2p}dV+\int \left( \nabla ||\mathrm{Ric}||^{2}*\nabla \eta \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p-1}dV \\&\qquad - \ \frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1} \eta ^{2p} ||\mathrm{Ric}||^{2}dV\right] +\int \left( \partial _{t}||\mathrm{Rm}||^{p-1} \right) \eta ^{2p}||\mathrm{Ric}||^{2}dV \\&\qquad + \ \int ||\mathrm{Rm}||^{p-1}\eta ^{2p} ||\mathrm{Ric}||^{2}(\partial _{t}dV) \\&\quad \le \ \ cK\int ||\nabla \mathrm{Ric}||||\nabla \mathrm{Rm}|| ||\mathrm{Rm}||^{p-2}\eta ^{2p}dV +cK\int ||\nabla \mathrm{Ric}||||\nabla \eta ||||\mathrm{Rm}||^{p-1}\eta ^{2p-1}dV \\&\qquad + \ c\int ||\mathrm{Rm}||^{p-3} \left( \partial _{t}||\mathrm{Rm}||^{2}\right) \eta ^{2p}||\mathrm{Ric}||^{2}dV +cK^{2}A_{1} \\&\qquad - \ \frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1}||\mathrm{Ric}||^{2} \eta ^{2p}dV\right] \\&\quad \le \ \ cK\left( \frac{1}{50c}B_{1}+c K B_{2}\right) +cK\left( \frac{1}{50 c}B_{1}+c K A_{4}\right) +cK^{2}A_{1} \\&\qquad + \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \partial _{t} ||\mathrm{Rm}||^{2}\right) dV -\frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1}||\mathrm{Ric}||^{2} \eta ^{2p}dV\right] \\&\quad \le \ \ \frac{2}{50}K B_{1} +cK^{2}B_{2}+c K^{2}A_{4}+cK^{2}A_{1} \\&\qquad + \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \partial _{t} ||\mathrm{Rm}||^{2}\right) dV -\frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1}||\mathrm{Ric}||^{2} \eta ^{2p}dV\right] . \end{aligned}$$

Thus

$$\begin{aligned} \int ||\mathrm{Rm}||^{p-1} \eta ^{2p}\blacksquare ||\mathrm{Ric}||^{2}dV \ \\le & {} \ \ \frac{2}{50}KB_{1}+c K^{2}B_{2}+c K^{2}A_{4}+cK^{2}A_{1} \nonumber \\&+ \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \partial _{t} ||\mathrm{Rm}||^{2}\right) dV\nonumber \\&-\frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1}||\mathrm{Ric}||^{2} \eta ^{2p}dV\right] . \end{aligned}$$

(3.45)

Consider the term

$$\begin{aligned}&c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \partial _{t}||\mathrm{Rm}||^{2}\right) dV \ \ = \ \ c\int ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3}\eta ^{2p} \\&\quad \bigg [\nabla ^{2}\mathrm{Ric}*\mathrm{Rm} +\mathrm{Ric}*\mathrm{Rm}*\mathrm{Rm}+\mathrm{Rm} *\mathrm{Rm}*\widehat{\varvec{T}}+\mathrm{Ric} *\nabla ^{2}||\varvec{T}||^{2}+\mathrm{Rm}*\nabla ^{2} \widehat{\varvec{T}} \\&\qquad + \ \frac{4}{3}||\varvec{T}||^{2}||\mathrm{Rm}||^{2}\bigg ]dV \ \ \le \ \ c\int ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3} \eta ^{2p}\bigg [\nabla ^{2}\mathrm{Ric}*\mathrm{Rm} -\nabla ^{2}R*\mathrm{Ric} \\&\qquad + \ \nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm} \bigg ]dV+cK^{2}A_{2}. \end{aligned}$$

The three terms in the bracket can be estimated as follows. Firstly

$$\begin{aligned}&c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \nabla ^{2}\mathrm{Ric}*\mathrm{Rm}\right) dV \\&\quad = \ \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3}\eta ^{2p} \left( \nabla \mathrm{Ric}*\nabla \mathrm{Rm}\right) dV \\&\qquad + \ c\int \left\{ \nabla \left[ ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\right] *\nabla \mathrm{Ric}*\mathrm{Rm}\right\} dV \\&\quad = \ \ c\int ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \nabla \mathrm{Ric}*\nabla \mathrm{Rm}\right) dV \\&\qquad + \ c\int ||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \nabla ||\mathrm{Ric}||^{2}*\nabla \mathrm{Ric} *\mathrm{Rm}\right) dV \\&\qquad + \ c\int ||\mathrm{Ric}||^{2} \eta ^{2p}\left( \nabla ||\mathrm{Rm}||^{p-3}*\nabla \mathrm{Ric} *\mathrm{Rm}\right) dV \\&\qquad + \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p-1}\left( \nabla \eta *\nabla \mathrm{Ric}*\mathrm{Rm}\right) dV \\&\quad \le \ cK\int ||\mathrm{Rm}||^{p-2} \eta ^{2p}||\nabla \mathrm{Ric}||||\nabla \mathrm{Rm}||dV +cK\int ||\mathrm{Rm}||^{p-1}\eta ^{2p-1} ||\nabla \mathrm{Ric}||||\nabla \eta ||dV \\&\quad \le \ \ cK\left( \epsilon B_{1}+\frac{K}{\epsilon }B_{2} \right) +cK\left( \epsilon B_{1}+\frac{K}{\epsilon }A_{4} \right) \ \ \le \ \ \frac{1}{50}K B_{1} +cK^{2}B_{2}+cK^{2}A_{4}. \end{aligned}$$

The same estimate holds for

$$\begin{aligned} c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( -\nabla ^{2}R*\mathrm{Ric}\right) dV. \end{aligned}$$

Finally,

$$\begin{aligned}&c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\left( \nabla ^{2}\widehat{\varvec{T}}*\mathrm{Rm}\right) dV \ \ = \ \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p}\\&\quad \left( \nabla \widehat{\varvec{T}}*\nabla \mathrm{Rm}\right) dV+c\int \left\{ \nabla \left( ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3}\eta ^{2p}\right) *\nabla \widehat{\varvec{T}} *\mathrm{Rm}\right\} dV \\&\quad \le \ \ c\int ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3}\eta ^{2p} \left( K^{1/2}||\nabla \varvec{T}||||\nabla \mathrm{Rm}||\right) dV \\&\qquad + \ c\int \left( \nabla ||\mathrm{Ric}||^{2}\right) ||\mathrm{Rm}||^{p-3}\eta ^{2p} ||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||dV \\&\qquad + \ c\int ||\mathrm{Rm}||^{2} \left( \nabla ||\mathrm{Rm}||^{p-3}\right) \eta ^{2p}||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||dV \\&\qquad + \ c\int ||\mathrm{Ric}||^{2}||\mathrm{Rm}||^{p-3} \eta ^{2p-1} ||\nabla \eta ||||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||dV \\&\quad \le \ \ c K\int ||\mathrm{Rm}||^{p-2} \eta ^{2p}\left( K^{1/2}||\nabla \varvec{T}||||\nabla \mathrm{Rm}||\right) dV \\&\qquad + \ cK\int ||\mathrm{Rm}||^{p-1}\eta ^{2p-1} \left( K^{1/2}||\nabla \eta ||||\nabla \varvec{T}||\right) dV \\&\quad \le \ \ K\left[ cK B_{2} +\frac{cK}{\epsilon }A_{4} +\epsilon c\int ||\nabla \varvec{T}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\right] \\&\quad \le \ \ \frac{8}{50}K B_{1}+cK^{2}B_{2} +cK^{2}A_{4}+cK^{3}A_{2}+cK^{2}A_{1} +\frac{d}{dt}\left[ cK\int R||\mathrm{Rm}||^{p-1} \eta ^{2p} dV\right] \end{aligned}$$

Therefore

$$\begin{aligned}&c\int ||\mathrm{Ric}||^{2} ||\mathrm{Rm}||^{p-3} \eta ^{2p} \left( \partial _{t}||\mathrm{Rm}||^{2} \right) dV \ \ \le \ \ \frac{10}{50}K B_{1}+cK^{2}B_{2} +cK^{2}A_{4}+cK^{3}A_{2} \nonumber \\&+ \ cK^{2}A_{1} +cK\frac{d}{dt}\left[ \int R||\mathrm{Rm}||^{p-1}\eta ^{2p} dV\right] \end{aligned}$$

(3.46)

and

$$\begin{aligned}&\frac{1}{2K}\int ||\mathrm{Rm}||^{p-1} \eta ^{2p}(\blacktriangle -\partial _{t}) ||\mathrm{Ric}||^{2}dV \ \ \le \ \ \frac{6}{50}B_{1}+cK B_{2}+cK A_{4} +cK^{2}A_{2}+cK A_{1} \nonumber \\&\qquad - \ \frac{1}{K}\frac{d}{dt} \left[ \int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV \right] +c\frac{d}{dt}\left[ \int R||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] \nonumber \\&\quad \le \ \ \frac{6}{50}B_{1}+cKB_{2} +cKA_{4}+cK^{2}A_{2}+cK A_{1} \nonumber \\&\qquad - \ \frac{d}{dt} \left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(3.47)

In the following, we estimate the left four terms in (3.44). We start from terms involving the scalar curvature.

$$\begin{aligned}&\frac{1}{3K}\int (-R)||\mathrm{Rm}||^{p-1}\eta ^{2p} \Delta R\!\ dV \ \ = \ \ -\frac{1}{3K} \int \nabla R\cdot \nabla \left[ (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}\right] dV \nonumber \\&\quad = \ \ -\frac{1}{3K}\int \nabla R \cdot \bigg [-\nabla R||\mathrm{Rm}||^{p-1} \eta ^{2p}+(-R)\nabla ||\mathrm{Rm}||^{p-1} \eta ^{2p} \nonumber \\&\qquad + \ 2p(-R)||\mathrm{Rm}||^{p-1} \eta ^{2p-1}\nabla \eta \bigg ]dV \ \ \le \ \ \frac{1}{3K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int (-R)||\mathrm{Rm}||^{p-2}||\nabla R|| ||\nabla \mathrm{Rm}||\eta ^{2p}dV \nonumber \\&\qquad + \ \frac{c}{K}\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p-1}||\nabla R||||\nabla \eta ||dV \nonumber \\&\quad \le \ \ \frac{1}{3K} \int ||\nabla R||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&\qquad +\frac{1}{3K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK B_{2} \nonumber \\&\qquad + \ \frac{1}{3K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK A_{4} \nonumber \\&\quad \le \ \ \frac{1}{K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK B_{2}+cK A_{4}. \end{aligned}$$

(3.48)

The another term involving the scalar curvature can be estimated by

$$\begin{aligned}&\frac{1}{3K}\int \langle \langle \mathrm{Ric}, \nabla ^{2}R\rangle \rangle ||\mathrm{Rm}||^{p-1}\eta ^{2p} dV \ \ = \ \ -\frac{1}{3K}\int \nabla ^{j}R\nabla ^{i}\left[ R_{ij} ||\mathrm{Rm}||^{p-1}\eta ^{2p}\right] dV\nonumber \\&\quad = \ \ -\frac{1}{3K} \int \nabla ^{j}R\bigg [\frac{1}{2}\nabla _{j}R||\mathrm{Rm}||^{p-1} \eta ^{2p}+R_{ij}\nabla ^{i}||\mathrm{Rm}||^{p-1}\eta ^{2p} \nonumber \\&\qquad + \ R_{ij}||\mathrm{Rm}||^{p-1}2p\eta ^{2p-1} \nabla ^{i}\eta \bigg ]dV \ \ \le \ \ -\frac{1}{6K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int ||\mathrm{Ric}||||\nabla R|| ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}|| \eta ^{2p}dV \nonumber \\&\qquad + \ \frac{c}{K}\int ||\nabla R||||\mathrm{Ric}||||\mathrm{Rm}||^{p-1} \eta ^{2p-1}||\nabla \eta ||dV\nonumber \\&\qquad \le \ \ -\frac{1}{6K} \int ||\nabla R||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV+\frac{1}{18K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK B_{2} \nonumber \\&\quad + \ \frac{1}{18K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p} dV+cK A_{4} \ \ \le \ \ cK B_{2}+cK A_{4}. \end{aligned}$$

(3.49)

Using (3.41) we obtain

$$\begin{aligned}&\frac{2}{K}\int \langle \langle \mathrm{Ric},\blacktriangle \widehat{ \varvec{T}}\rangle \rangle ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \ \ = \ \ \frac{1}{K} \int \left( \mathrm{Ric}*\blacktriangle \widehat{\varvec{T}}\right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&\quad = \ \ \frac{1}{K}\int \left( \nabla \mathrm{Ric} *\nabla \widehat{T}\right) ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV +\frac{1}{K}\int \mathrm{Ric}*\nabla \widehat{\varvec{T}} *\nabla \left( ||\mathrm{Rm}||^{p-1} \eta ^{2p}\right) dV\nonumber \\&\quad \le \frac{c}{K}\int ||\nabla \mathrm{Ric}|| ||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||^{p-1} \eta ^{2p}dV+\frac{c}{K}\int ||\mathrm{Ric}|| ||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||^{p-2} ||\nabla \mathrm{Rm}||\eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int ||\mathrm{Ric}|| ||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||^{p-1} \eta ^{2p-1}||\nabla \eta ||dV\nonumber \\&\quad \le \ \ \frac{1}{50}B_{1} +c\int ||\nabla \varvec{T}||^{2} ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV +cK B_{2}\nonumber \\&\qquad + \ c\int ||\nabla \varvec{T}||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +cK A_{4}+c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \nonumber \\&\quad \le \ \ \frac{1}{50}B_{1} +cK B_{2}+cK A_{4} +c\int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-1} \eta ^{2p}dV \nonumber \\&\quad \le \ \ \frac{9}{50}B_{1} +cK B_{2}+cK A_{4}+cK^{2}A_{2} +cK A_{1} +\frac{d}{dt}\left[ \int cR||\mathrm{Rm}||^{p-1} \eta ^{2p}d\!\right] \!.\quad \quad \end{aligned}$$

(3.50)

Similarly, we can prove

$$\begin{aligned}&\frac{1}{K}\int \left( \mathrm{Ric}*\nabla ^{2}\widehat{\varvec{T}} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \ \ = \ \ \frac{1}{K}\int \left( \nabla \mathrm{Ric}*\nabla \widehat{\varvec{T}} \right) ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV \nonumber \\&\qquad + \ \frac{1}{K}\int \mathrm{Ric}*\nabla \widehat{\varvec{T}} *\nabla \left( ||\mathrm{Rm}||^{p-1} \eta ^{2p}\right) dV\ \ \le \ \ \frac{1}{K}\int \left( \nabla \mathrm{Ric} *\nabla \widehat{\varvec{T}}\right) ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int ||\mathrm{Ric}||||\nabla \widehat{\varvec{T}}|| ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}||\eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int ||\mathrm{Ric}||||\nabla \widehat{\varvec{T}}|| ||\mathrm{Rm}||^{p-1}\eta ^{2p-1} ||\nabla \eta ||dV\nonumber \\&\quad \le \frac{c}{K}\int ||\nabla \mathrm{Ric}|| ||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||^{p-1} \eta ^{2p}dV+\frac{c}{K}\int ||\mathrm{Ric}||||\nabla \widehat{\varvec{T}}||||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}|| \eta ^{2p}dV\nonumber \\&\qquad + \ \frac{c}{K}\int ||\mathrm{Ric}||||\nabla \widehat{\varvec{T}}|| ||\mathrm{Rm}||^{p-1} \eta ^{2p-1}||\nabla \eta ||dV \nonumber \\&\quad \le \ \ \frac{9}{50}B_{1} +cK B_{2}+cK A_{4}+cK^{2}A_{2} +cK A_{1} +\frac{d}{dt}\left[ \int cR||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\!\right] \! .\quad \quad \end{aligned}$$

(3.51)

Plugging (3.45) and (3.48)–(3.51) into (3.44), and using (3.41) and $||\nabla R||^{2} \le cK||\nabla \varvec{T}||^{2}$, we obtain

$$\begin{aligned} B_{1}\le & {} \frac{6}{50}B_{1}+cK B_{2} +cK A_{4}+cK^{2}A_{2}+cK A_{1} \\&- \ \frac{d}{dt}\left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] \\&+ \ \frac{1}{K}\int ||\nabla R||^{2} ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV +\frac{18}{50} B_{1}-\frac{d}{dt}\left[ c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] \\\le & {} \ \ \frac{32}{50}B_{1}+cK B_{2}+cK A_{4} +cK^{2}A_{2}+cK A_{1} \\&- \ \frac{d}{dt}\left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

Thus

$$\begin{aligned} B_{1}\le & {} cK B_{2}+cK A_{4}+cK^{2}A_{2}+cK A_{1} \nonumber \\&- \ \frac{d}{dt}\left[ \frac{1}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] \end{aligned}$$

(3.52)

From (3.43) and (3.52), we can conclude that

Lemma 3.5

One has

$$\begin{aligned} A'_{1} \ \\le & {} \ \ cK B_{2}+cK A_{4}+cK^{2}A_{2}+cK A_{1} \nonumber \\&- \ \frac{d}{dt}\left[ \frac{c}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] .\qquad \end{aligned}$$

(3.53)

Observe that two terms in the bracket are both nonnegative, since $R =-||\varvec{T}||^{2}\le 0$.

Finally, we estimate the term $B_{2}$. Using the evolution inequality

$$\begin{aligned} ||\nabla \mathrm{Rm}||^{2} \le -\frac{1}{2}\blacksquare ||\mathrm{Rm}||^{2} +c||\mathrm{Rm}||^{3}+c||\nabla ^{2}\varvec{T}||||\mathrm{Rm}||^{3/2} +c||\mathrm{Rm}||||\nabla \varvec{T}||^{2} \end{aligned}$$

we obtain

$$\begin{aligned} B_{2}= & {} \ \ \int ||\nabla \mathrm{Rm}||^{2} ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \ \ \le \ \ \int \bigg [-\frac{1}{2}\blacksquare ||\mathrm{Rm}||^{2} +c||\mathrm{Rm}||^{3} \nonumber \\&+ \ c||\nabla ^{2}\varvec{T}||||\mathrm{Rm}||^{3/2} +c||\mathrm{Rm}||||\nabla \varvec{T}||^{2}\bigg ]||\mathrm{Rm}||^{p-3} \eta ^{2p}dV\nonumber \\\le & {} \ \ -\frac{1}{2}\int \left( \blacksquare ||\mathrm{Rm}||^{2} \right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV +cA_{1}\nonumber \\&+ \ c\int ||\nabla ^{2}\varvec{T}||||\mathrm{Rm}||^{p-3/2}\eta ^{2p}dV +c\int ||\nabla ^{2}\varvec{T}||^{2}||\mathrm{Rm}||^{p-2} \eta ^{2p}dV. \end{aligned}$$

(3.54)

For the first integral one has

$$\begin{aligned}&- \frac{1}{2}\int \left( \blacksquare ||\mathrm{Rm}||^{2} \right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \ \ = \ \ \frac{1}{2}\int \left( \blacktriangle ||\mathrm{Rm}||^{2}\right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \\&\qquad - \ \frac{1}{2}\int \left( \partial _{t}||\mathrm{Rm}||^{2}\right) ||\mathrm{Rm}||^{p-3} \eta ^{2p}dV \ \ = \ \ -\frac{1}{2}\int \left( \partial _{t}||\mathrm{Rm}||^{2} \right) ||\mathrm{Rm}||^{p-3} \eta ^{2p}dV \\&\qquad - \ \frac{1}{2}\int \nabla ||\mathrm{Rm}||^{2} \left[ \left( \nabla ||\mathrm{Rm}||^{p-3} \right) \eta ^{2p}+||\mathrm{Rm}||^{p-3} \left( \nabla \eta ^{2p}\right) \right] dV \\&\quad = \ \ -\frac{p-3}{4}\int \left( \nabla ||\mathrm{Rm}||^{2}\right) ^{2} ||\mathrm{Rm}||^{p-5}\eta ^{2p}dV \\&\qquad + \ c\int ||\mathrm{Rm}||^{p-2}||\nabla \mathrm{Rm}||||\nabla \eta ||\eta ^{2p-1}dV -\frac{1}{2}\int \left( \partial _{t}||\mathrm{Rm}||^{2} \right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \\&\quad \le \ \ \frac{1}{50} B_{2}+cA_{4}-\frac{1}{2}\int \left( \partial _{t} ||\mathrm{Rm}||^{2}\right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV. \end{aligned}$$

Here we used the assumption that $p\ge 5$. On the other hand,

$$\begin{aligned}&- \frac{1}{2}\int \left( \partial _{t}||\mathrm{Rm}||^{2} \right) ||\mathrm{Rm}||^{p-3} \eta ^{2p}dV \ \ = \ \ -\frac{1}{2}\frac{d}{dt} \left[ \int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\right] \\&\qquad + \ \frac{1}{2}\int ||\mathrm{Rm}||^{2} \left( \partial _{t}||\mathrm{Rm}||^{p-3}\right) \eta ^{2p}dV +\frac{1}{2}\int ||\mathrm{Rm}||^{p-1} \eta ^{2p}\left( \partial _{t}dV\right) \\&\quad \le \ \ \frac{p-3}{4}\int ||\mathrm{Rm}||^{p-3} \left( \partial _{t}||\mathrm{Rm}||^{2} \right) \eta ^{2p}dV+cA_{1}-\frac{1}{2}\frac{d}{dt} \left[ \int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\right] \end{aligned}$$

so that

$$\begin{aligned} -\frac{1}{2} \int \left( \partial _{t}||\mathrm{Rm}||^{2}\right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \le c A_{1}-\frac{1}{p-1}\frac{d}{dt} \left[ \int ||\mathrm{Rm}||^{p-1}\eta ^{2p}dV\right] . \end{aligned}$$

Therefore

$$\begin{aligned}&-\frac{1}{2}\int \left( \blacksquare ||\mathrm{Rm}||^{2}\right) ||\mathrm{Rm}||^{p-3}\eta ^{2p}dV \ \ \le \ \ \frac{1}{50}B_{2}+c A_{4}+cA_{1}\nonumber \\&\quad - \ \frac{1}{p-1}\frac{d}{dt}\left[ \int ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(3.55)

To estimate the remainder two integrals, we recall from (2.35) that

$$\begin{aligned} \nabla \varvec{T}=\mathrm{Rm}*\varphi +\varvec{T}*\varvec{T}*\varphi \end{aligned}$$

(3.56)

and from (2.14) that

$$\begin{aligned} \nabla \varphi =\varvec{T}*\psi . \end{aligned}$$

(3.57)

From (3.56) we get

$$\begin{aligned} ||\nabla \varvec{T}||\le c||\mathrm{Rm}||+c||\varvec{T}||^{2}\le c||\mathrm{Rm}||. \end{aligned}$$

(3.58)

In particular, the inequality (3.58) yields

$$\begin{aligned} \int ||\nabla \varvec{T}||^{2}||\mathrm{Rm}||^{p-2} \eta ^{2p}dV\le c\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV \le cA_{1}. \end{aligned}$$

(3.59)

Taking the derivative of (3.56) and using (3.57) we obtain

$$\begin{aligned} \nabla ^{2}\varvec{T}=\nabla \mathrm{Rm}*\varphi +\mathrm{Rm}*\varvec{T}*\psi +\nabla \varvec{T}*\varvec{T}*\varphi +\varvec{T}*\varvec{T}*\varvec{T}*\psi . \end{aligned}$$

(3.60)

The particular case $||\nabla ^{2}\varvec{T}||\le c||\nabla \mathrm{Rm}|| +c||\mathrm{Rm}||||\varvec{T}||+c||\nabla \varvec{T}||||\varvec{T}||+c||\varvec{T}||^{3}$ leads to

$$\begin{aligned}&c\int ||\nabla ^{2}\varvec{T}||||\mathrm{Rm}||^{p-3/2}\eta ^{2p}dV\ \ \le \ \ c\int \bigg [||\nabla \mathrm{Rm}||+||\mathrm{Rm}||||\varvec{T}|| +||\nabla \varvec{T}||||\varvec{T}|| \nonumber \\&\quad + \ ||\varvec{T}||^{3}\bigg ]||\mathrm{Rm}||^{p-3/2}\eta ^{2p}dV \ \ \le \ \ c\int \left( ||\nabla \mathrm{Rm}||||\mathrm{Rm}||^{p-3/2} \eta ^{p}\right) \left( ||\mathrm{Rm}||^{p/2}\eta ^{p} \right) dV\nonumber \\&\quad + \ c\int ||\mathrm{Rm}||^{p}\eta ^{2p}dV \ \ \le \ \ \frac{1}{50}B_{2}+cA_{1}. \end{aligned}$$

(3.61)

Plugging (3.55), (3.59), and (3.61) into (3.54) we arrive at

$$\begin{aligned} B_{2}\le cA_{4}+cA_{1}-\frac{d}{dt}\left[ \frac{1}{p-1} \int ||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(3.62)

Together with (3.53) and (3.62) we finally obtain

$$\begin{aligned}&(A_{1}+cK A_{2})' \ \ \le \ \ cK(A_{1}+cK A_{2}) +cK A_{4}\nonumber \\&\quad - \ \frac{d}{dt}\left[ \frac{c}{K}\int ||\mathrm{Rm}||^{p-1}||\mathrm{Ric}||^{2} \eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] . \end{aligned}$$

(3.63)

Equivalently,

Lemma 3.6

If $||\mathrm{Ric}||\le K$ and $p\ge 5$, one has

$$\begin{aligned}&\frac{d}{dt} \left[ A_{1}+cK A_{2}+\frac{c}{K}\int ||\mathrm{Rm}||^{p-1} ||\mathrm{Ric}||^{2}\eta ^{2p}dV +c\int (-R)||\mathrm{Rm}||^{p-1} \eta ^{2p}dV\right] \nonumber \\&\quad \le \ \ cK(A_{1}+cK A_{2})+cK A_{4}. \end{aligned}$$

(3.64)