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A gradient flow of Spin(7)-structures

Shubham Dwivedi

(March 15, 2025)

Abstract

We formulate and study the negative gradient flow of an energy functional of Spin(7)-structures on compact $8$ -manifolds. The energy functional is the $L^{2}$ -norm of the torsion of the Spin(7)-structure. Our main result is the short-time existence and uniqueness of solutions to the flow. We also explain how this negative gradient flow is the most general flow of Spin(7)-structures. We also study solitons of the flow and prove a non-existence result for compact expanding solitons.

1. Introduction

Given an $8$ -dimensional smooth manifold $M$ , a Spin(7)-structure on $M$ is a reduction of the structure group of the frame bundle $\text{Fr}(M)$ to the Lie group Spin(7) $\subset\mathrm{SO}(8)$ . While many $8$ -manifolds admit Spin(7)-structures, the existence of torsion-free Spin(7)-structures is a difficult problem. Given the success of geometric flows in proving the existence of special geometric structures on manifolds it is natural to attempt to tackle such questions using a suitable geometric flow of Spin(7)-structures.

Unlike the case of geometric flows of $\mathrm{G}_{2}$ -structures, there have been fewer studies of flows of Spin(7)-structures. A flow of isometric/harmonic Spin(7)-structures was introduced and studied in detail in [DLE24]. The authors proved various analytic and geometric results for the flow. This flow can be seen as an instance of the more general theory of harmonic $H$ -structures which has been studied in a fairly detailed manner in [LS23] and [FLME22]. One disadvantage of the isometric flow is that it runs in an isometry class of a given Spin(7)-structure and hence the metric remains fixed throughout the flow. We remedy this here by studying the most general flow of Spin(7)-structures (in particular, the metric is also changing) which include the isometric flow as a special case. On a fixed oriented spin Riemannian $8$ -manifold, a Spin(7)-structure is equivalent (up to sign) to a unit spinor field. Using a spinorial approach, Ammann–Weiss–Witt [AWW16] also studied the negative gradient flow of a Dirichlet energy, thought of as a function of a unit spinor field. They proved general short-time existence and uniqueness. In contrast to their work, our approach is more direct and uses the Riemannian geometric properties of a Spin(7)-structure with the flow equation and related quantities all given in terms of the curvature of the underlying metric and the torsion of the Spin(7)-structure. We also describe all possible second order quasilinear flows of Spin(7)-structures. A study of some natural functionals of Spin(7)-structures and the corresponding Euler-Lagrange equations has also been done recently by Krasnov [Kra24].

Our point of view of studying general flows of Spin(7)-structures is that of [DGK23] where the authors study the most general second order flows of $\mathrm{G}_{2}$ -structures by classifying all linearly independent second order (in the $\mathrm{G}_{2}$ -structure) tensors coming from the Riemann curvature tensor and the covariant derivative of the torsion which can be made into a $3$ -form.

We now describe the energy functional and its negative gradient flow. More details are given in § 3.

Definition 1.1.

Let $M^{8}$ be a compact manifold which admits Spin(7)-structures. The energy functional $E$ on the set of Spin(7)-structures on $M$ is

\displaystyle E(\Phi)=\frac{1}{2}\int_{M}|T_{\Phi}|^{2}\operatorname{vol}_{\Phi}

(1.1)

where $T_{\Phi}$ is the torsion of the Spin(7)-structure $\Phi$ and the norm and the volume form are with respect to the metric induced by $\Phi$ . The critical points of the functional $E$ are torsion-free Spin(7)-structures which are also absolute minimizers.

We compute the first variation of $E$ in Lemma 3.3 which also gives the negative gradient of the energy functional. We describe the negative gradient flow below. See § 2 and § 3 for the definition of the terms and more details.

Definition 1.2 (Negative gradient flow).

Let $(M^{8},\Phi_{0})$ be a compact manifold with a Spin(7)-structure. Let $(T*T)_{ij}=8T_{b;il}T_{j;lb}-8T_{j;il}T_{b;lb}+2T_{i;lb}T_{j;lb}$ . The negative gradient flow of the energy functional $E$ is the flow of Spin(7)-structures which is the following initial value problem

\displaystyle\left\{\begin{array}[]{rl}&\dfrac{\partial\Phi}{\partial t}=\left(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+(T*T)-|T|^{2}g+2\operatorname{div}T\right)\diamond\Phi,\\ &\Phi(0)=\Phi_{0}.\end{array}\right.

(GF)

where $\mathrm{Ric}$ is the Ricci curvature of the underlying metric, $T_{8}$ is the $8$ -dimensional component of the torsion tensor $T$ and $\diamond$ is the operator described in (2.13).

Given any geometric flow, one of the main questions is whether there exist a solution to the flow for a short time and if any two solutions starting with the same initial conditions are unique. The main theorem of the paper is as follows.

Theorem 4.11. Let $(M^{8},\Phi_{0})$ be a compact $8$ -manifold with a Spin(7)-structure $\Phi_{0}$ and consider the negative gradient flow of the natural energy functional $E$ in (1.1). This is the flow (GF). Then there exists $\varepsilon>0$ and a unique smooth solution $\Phi(t)$ of (GF) for $t\in[0,\varepsilon)$ with $\varepsilon=\varepsilon(\Phi_{0})$ .

We prove the main theorem by explicitly calculating the principal symbols of the operators involved in the flow and then using a modified DeTurck’s trick.

As discussed at the end of § 2, the most general flow of Spin(7)-structures which is second order can be written as

\displaystyle\frac{\partial}{\partial t}\Phi(t)=(-\mathrm{Ric}+a\mathcal{L}_{T_{8}}g+b\operatorname{div}T+\text{l.o.t})\diamond\Phi

(1.3)

where $a,b\in\mathbb{R}$ and l.o.t are the terms which are lower order in $\Phi$ . The situation in the Spin(7)-case is a bit simpler than the corresponding case of the $\mathrm{G}_{2}$ -structures where there are more terms which could appear in a flow of $\mathrm{G}_{2}$ -structures [DGK23, §1]. This is because of the fact that for $p\in M$ , the subbundle $A_{p}M$ of admissible 4-forms in $\Lambda^{4}T^{*}_{p}M$ has codimension $27$ and thus any variation of a Spin(7)-structure is given only by $4$ -forms in $\Omega^{4}_{1+7+35}$ . Consequently, many naturally occurring tensors, for example the $27$ -dimensional Weyl curvature tensor cannot be a candidate for a flow of Spin(7)-structures (see § 2 for details) even though they can be made into a $4$ -form.

The methods in § 4 can be used to establish short-time existence and uniqueness results for the general flow in (1.3) subjected to conditions on the coefficients $a,b$ which will be sufficient conditions to apply the modified DeTurck’s trick described in § 4.4. The advantage of looking at a specific flow in the family (1.3), which for us is (GF) which corresponds to $a,b=2$ is that it also provides us with the lower order terms which are important when one wants to study further analytic properties of the flow. An example of the importance of lower order terms is our result on the non-existence of compact expanding solitons for the flow in § 5.

Solitons of a geometric flow are special solutions which move only by scalings and diffeomorphisms. These are also called self-similar solutions. A soliton can be viewed as a generalized fixed point of the flow modulo scalings and diffeomorphisms. These special solutions are important because they serve as singularity models for the flow. We study solitons of the flow (GF). We derive equations for the underlying metric and the divergence of the torsion of a soliton solution. We expect that if we have a Cheeger–Gromov–Hamilton type compactness theorem for the solutions of the flow then the solitons will play a role in understanding the behaviour of the flow near a singularity.

The paper is organized as follows. We start by reviewing the notion of Spin(7)-structures, the decomposition of space of forms and the torsion tensor in § 2. The first variation of the energy functional is computed in § 3. We give a brief review of elliptic and parabolic differential operators in § 4.1 which is followed by a modified DeTurck’s trick in § 4.4. The main theorem, Theorem 4.11, is proved in § 4.5. Finally, in § 5 we study solitons of the flow and prove Proposition 5.2 which is a non-existence result for compact expanding solitons of the flow.

Throughout the paper, we compute in a local orthonormal frame, so all indices are subscripts and any repeated indices are summed over all values from $1$ to $8$ . Our convention for labelling the Riemann curvature tensor is

\displaystyle R_{ijkl}\frac{\partial}{\partial x^{l}}=(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i})\frac{\partial}{\partial x^{k}}

in terms of coordinate vector fields and as a result, the Ricci tensor is $R_{ij}=R_{lijl}$ . We also have the Ricci identity which, for instance, for a $2$ -tensor $S$ reads as

\displaystyle\nabla_{i}\nabla_{j}S_{kl}-\nabla_{j}\nabla_{i}S_{kl}=-R_{ijkm}S_{ml}-R_{ijlm}S_{km}.

(1.4)

Acknowledgements. We are grateful to Panagiotis Gianniotis and Spiro Karigiannis for various discussions on this project in particular and on flows of special geometric structures in general. We would like to thank Thomas Walpuski for discussions on the torsion decomposition of Spin(7)-structures and Ragini Singhal for explaining the representation theoretic aspects of Spin(7)-structures. A part of this work was done when the author is a Simons-CRM Scholar-in-Residence at the CRM, Montreal. We would like to thank the Simons Foundation and the CRM for funding and CRM for providing a wonderful environment to work in.

2. Preliminaries on Spin(7)-structures

In this section, we briefly review the notion of a Spin(7)-structure on an $8$ -dimensional manifold $M$ and the associated decomposition of differential forms. We also discuss the torsion tensor of a Spin(7)-structure. More details can be found in [Joy00, Chapter 10] and [Kar08].

A Spin(7)-structure on $M$ is a $4$ -form $\Phi$ on $M$ . The existence of such a structure is a topological condition. The space of $4$ -forms which determine a Spin(7)-structure on $M$ is a subbundle $A$ of $\Omega^{4}(M)$ , called the bundle of admissible $4$ -forms. This is not a vector subbundle and it is not even an open subbundle, unlike the case for $\mathrm{G}_{2}$ -structures. For $p\in M$ , the subbundle $A_{p}(M)$ is of codimension $27$ in $\Lambda^{4}(T^{*}_{p}M)$ .

A Spin(7)-structure $\Phi$ determines a Riemannian metric and an orientation on $M$ in a nonlinear way which can be described in local coordinates as follows. Let $p\in M$ and extend a non-zero tangent vector $v\in T_{p}M$ to a local frame $\{v,e_{1},\cdots,e_{7}\}$ near $p$ . Define

	$\displaystyle B_{ij}(v)$	$\displaystyle=((e_{i}\lrcorner v\lrcorner\Phi)\wedge(e_{j}\lrcorner v\lrcorner\Phi)\wedge(v\lrcorner\Phi))(e_{1},\cdots,e_{7}),$
	$\displaystyle A(v)$	$\displaystyle=((v\lrcorner\Phi)\wedge\Phi)(e_{1},\cdots,e_{7}).$

Then the metric induced by $\Phi$ is given by

\displaystyle(g_{\Phi}(v,v))^{2}=-\frac{7^{3}}{6^{\frac{7}{3}}}\frac{(\textup{det}\ B_{ij}(v))^{\frac{1}{3}}}{A(v)^{3}}.

(2.1)

The metric and the orientation determine a Hodge star operator $\star$ , and the $4$ -form is self-dual, i.e., $\star\Phi=\Phi$ .

Definition 2.1.

Let $\nabla$ be the Levi-Civita connection of the metric $g_{\Phi}$ . The pair $(M^{8},\Phi)$ is a Spin(7)-manifold if $\nabla\Phi=0$ . This is a non-linear partial differential equation for $\Phi$ , since $\nabla$ depends on $g$ , which in turn depends non-linearly on $\Phi$ . A Spin(7)-manifold has Riemannian holonomy contained in the subgroup $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$ . Such a parallel Spin(7)-structure is also called torsion free.

2.1 Decomposition of the space of forms

The existence of a Spin(7)-structure $\Phi$ induces a decomposition of the space of differential forms on $M$ into irreducible Spin(7)-representations. We have the following orthogonal decomposition, with respect to $g_{\Phi}$ :

\displaystyle\Omega^{2}=\Omega^{2}_{7}\oplus\Omega^{2}_{21},\ \ \ \ \ \ \ \Omega^{3}=\Omega^{3}_{8}\oplus\Omega^{3}_{48},\ \ \ \ \ \ \ \ \ \Omega^{4}=\Omega^{4}_{1}\oplus\Omega^{4}_{7}\oplus\Omega^{4}_{27}\oplus\Omega^{4}_{35},

where $\Omega^{k}_{l}$ has pointwise dimension $l$ . Explicitly, $\Omega^{2}$ and $\Omega^{3}$ are described as follows:

	$\displaystyle\Omega^{2}_{7}$	$\displaystyle=\{\beta\in\Omega^{2}\mid\star(\Phi\wedge\beta)=-3\beta\},$		(2.2)
	$\displaystyle\Omega^{2}_{21}$	$\displaystyle=\{\beta\in\Omega^{2}\mid\star(\Phi\wedge\beta)=\beta\},$		(2.3)

and

	$\displaystyle\Omega^{3}_{8}$	$\displaystyle=\{X\lrcorner\Phi\mid X\in\Gamma(TM)\},$		(2.4)
	$\displaystyle\Omega^{3}_{48}$	$\displaystyle=\{\gamma\in\Omega^{3}\mid\gamma\wedge\Phi=0\}.$		(2.5)

For our computations, it is useful to describe the spaces of forms in local coordinates. For $\beta\in\Omega^{2}(M)$ ,

	$\displaystyle\beta_{ij}\in\Omega^{2}_{7}\iff\beta_{ab}\Phi_{abij}$	$\displaystyle=-6\beta_{ij},$		(2.6)
	$\displaystyle\beta_{ij}\in\Omega^{2}_{21}\iff\beta_{ab}\Phi_{abij}$	$\displaystyle=2\beta_{ij}.$		(2.7)

Remark 2.2.

A convention different than ours is also prevalent in the literature. In this convention, $\Omega^{2}_{7}$ and $\Omega^{2}_{21}$ are the $+3$ and $-1$ eigenspaces of the map $\beta\mapsto*(\Phi\wedge\beta),$ respectively. As a result, the constants on the right hand sides of (2.6) and (2.7) are $+6$ and $-2$ respectively.

For $\gamma\in\Omega^{3}(M)$ ,

	$\displaystyle\gamma_{ijk}\in\Omega^{3}_{8}$	$\displaystyle\iff\gamma_{ijk}=X_{l}\Phi_{ijkl}\ \ \ \ \textup{for\ some}\ X\in\Gamma(TM),$		(2.8)
	$\displaystyle\gamma_{ijk}\in\Omega^{3}_{48}$	$\displaystyle\iff\gamma_{ijk}\Phi_{ijkl}=0.$		(2.9)

If $\pi_{7}$ and $\pi_{21}$ are the projection operators on $\Omega^{2}$ , it follows from (2.6) and (2.7) that

	$\displaystyle\pi_{7}(\beta)_{ij}$	$\displaystyle=\frac{1}{4}\beta_{ij}-\frac{1}{8}\beta_{ab}\Phi_{abij},$		(2.10)
	$\displaystyle\pi_{21}(\beta)_{ij}$	$\displaystyle=\frac{3}{4}\beta_{ij}+\frac{1}{8}\beta_{ab}\Phi_{abij}.$		(2.11)

We will be using these equations throughout the paper. Finally, for $\beta_{ij}\in\Omega^{2}_{21}$ ,

\displaystyle\beta_{ab}\Phi_{bpqr}

\displaystyle=\beta_{pi}\Phi_{iqra}+\beta_{qi}\Phi_{irpa}+\beta_{ri}\Phi_{ipqa},

(2.12)

which can be used to show that $\Omega^{2}_{21}\equiv\mathfrak{so}(7)$ is the Lie algebra of Spin(7), with the commutator of matrices

\displaystyle[\alpha,\beta]_{ij}=\alpha_{il}\beta_{lj}-\alpha_{jl}\beta_{li}.

To describe $\Omega^{4}$ in local coordinates, we use the $\diamond$ operator which was first described in [DGK21] for the $\mathrm{G}_{2}$ case and in [DLE24] for the Spin(7) (see also [FLME22, eq. (1.14)] for the general case of $H$ -structures). Given $A\in\Gamma(T^{*}M\otimes TM)$ , define

\displaystyle A\diamond\Phi=\frac{1}{24}(A_{ip}\Phi_{pjkl}+A_{jp}\Phi_{ipkl}+A_{kp}\Phi_{ijpl}+A_{lp}\Phi_{ijkp})dx^{i}\wedge dx^{j}\wedge dx^{k}\wedge dx^{l},

(2.13)

and hence

\displaystyle(A\diamond\Phi)_{ijkl}=A_{ip}\Phi_{pjkl}+A_{jp}\Phi_{ipkl}+A_{kp}\Phi_{ijpl}+A_{lp}\Phi_{ijkp}.

(2.14)

Recall that $\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus S_{0}\oplus\Omega^{2}$ , and $\Omega^{2}$ further splits orthogonally into (2.2) and (2.3), so

\displaystyle\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus S_{0}\oplus\Omega^{2}_{7}\oplus\Omega^{2}_{21}.

(2.15)

With respect to this splitting, we can write $A=\frac{1}{8}(\operatorname{tr}A)g+A_{35}+A_{7}+A_{21}$ where $A_{35}$ is a symmetric traceless $2$ -tensor. The diamond contraction (2.14) defines a linear map $A\mapsto A\diamond\Phi$ , from $\Omega^{0}\oplus S_{0}\oplus\Omega^{2}_{7}\oplus\Omega^{2}_{21}$ to $\Omega^{4}(M)$ . We record the following properties of the $\diamond$ operator whose proof can be found in [Kar08] or [DLE24].

Proposition 2.3.

Let $(M,\Phi)$ be a manifold with a Spin(7)-structure. Then

The Hodge star of $A\diamond\Phi$ is

\displaystyle\star(A\diamond\Phi)

\displaystyle=(\bar{A}\diamond\Phi)\ \ \textup{where}\ \ \bar{A}=\frac{1}{4}(\operatorname{tr}A)g_{ij}-A_{ji}.

(2.16)

If $B\in\Gamma(T^{*}M\otimes TM)$ and $A_{35},\ B_{35}$ denote the traceless symmetric parts of $A$ and $B$ respectively and $A_{7},\ B_{7}$ denote their $\Omega^{2}_{7}$ component. Then

\displaystyle\langle A\diamond\Phi,B\diamond\Phi\rangle

\displaystyle=24(\operatorname{tr}A)(\operatorname{tr}B)+96\langle A_{35},B_{35}\rangle+384\langle A_{7},B_{7}\rangle.

(2.17)

∎

More properties of the $\diamond$ operator for general $H$ -structures have been proved in [FLME22, Lemma 1.4].

The following proposition was originally proved in [Kar08, Prop. 2.3].

Proposition 2.4.

The kernel of the map $A\mapsto A\diamond\Phi$ is isomorphic to the subspace $\Omega^{2}_{21}$ . The remaining three summands $\Omega^{0},\ S_{0}$ and $\Omega^{2}_{7}$ are mapped isomorphically onto the subspaces $\Omega^{4}_{1},\ \Omega^{4}_{35}$ and $\Omega^{4}_{7}$ respectively.

Proof.

Using (2.17) with $A=B$ gives

\displaystyle|A\diamond\Phi|^{2}

\displaystyle=24(\operatorname{tr}A)^{2}+96|A_{35}|^{2}+384|A_{7}|^{2}

and hence $A\diamond\Phi=0\iff A=A_{21}$ . If $A_{21}=0$ then since all the coefficients in $|A\diamond\Phi|^{2}$ are positive hence the map $A\mapsto A\diamond\Phi$ is injective on $(\Omega^{2}_{21})^{\perp}$ and by dimension count, the map is a linear isomorphism. ∎

To understand $\Omega^{4}_{27}$ , we need another characterization of the space of $4$ -forms using the Spin(7)-structure. Following [Kar08], we adopt the following:

Definition 2.5.

On $(M,\Phi)$ , define a Spin(7)-equivariant linear operator $\Lambda_{\Phi}$ on $\Omega^{4}$ as follows. Let $\sigma\in\Omega^{4}(M)$ and let $(\sigma\cdot\Phi)_{ijkl}=\sigma_{ijmn}\Phi_{mnkl}$ . Then

\displaystyle(\Lambda_{\Phi}(\sigma))_{ijkl}=(\sigma\cdot\Phi)_{ijkl}+(\sigma\cdot\Phi)_{iklj}+(\sigma\cdot\Phi)_{iljk}+(\sigma\cdot\Phi)_{jkil}+(\sigma\cdot\Phi)_{jlki}+(\sigma\cdot\Phi)_{klij}.

(2.18)

Proposition 2.6.

The spaces $\Omega^{4}_{1}$ , $\Omega^{4}_{7},\ \Omega^{4}_{27}$ and $\Omega^{4}_{35}$ are all eigenspaces of $\Lambda_{\Phi}$ with distinct eigenvalues:

\displaystyle\begin{aligned} \Omega^{4}_{1}&=\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=-24\sigma\},\\ \Omega^{4}_{27}&=\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=4\sigma\},\end{aligned}

\displaystyle\begin{aligned} \Omega^{4}_{7}&=\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=-12\sigma\},\\ \Omega^{4}_{35}&=\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=0\}.\end{aligned}

(2.19)

Moreover, the decomposition of $\Omega^{4}(M)$ into self-dual and anti-self-dual parts is

\displaystyle\Omega^{4}_{+}=\{\sigma\in\Omega^{4}\mid\star\sigma=\sigma\}=\Omega^{4}_{1}\oplus\Omega^{4}_{7}\oplus\Omega^{4}_{27},\ \ \ \ \ \Omega^{4}_{-}=\{\sigma\in\Omega^{4}\mid\star\sigma=-\sigma\}=\Omega^{4}_{35}.

(2.20)

Before we discuss the torsion of a Spin(7)-structure, we note some contraction identities involving the $4$ -form $\Phi$ . In local coordinates $\{x^{1},\cdots,x^{8}\}$ , the $4$ -form $\Phi$ is

\displaystyle\Phi=\frac{1}{24}\Phi_{ijkl}\ dx^{i}\wedge dx^{j}\wedge dx^{k}\wedge dx^{l}

where $\Phi_{ijkl}$ is totally skew-symmetric. We have the following identities

$\displaystyle\Phi_{ijkl}\Phi_{abcl}$	$\displaystyle=g_{ia}g_{jb}g_{kc}+g_{ib}g_{jc}g_{ka}+g_{ic}g_{ja}g_{kb}$
	$\displaystyle\quad-g_{ia}g_{jc}g_{kb}-g_{ib}g_{ja}g_{kc}-g_{ic}g_{jb}g_{ka}$
	$\displaystyle\quad-g_{ia}\Phi_{jkbc}-g_{ib}\Phi_{jkca}-g_{ic}\Phi_{jkab}$
	$\displaystyle\quad-g_{ja}\Phi_{kibc}-g_{jb}\Phi_{kica}-g_{jc}\Phi_{kiab}$
	$\displaystyle\quad-g_{ka}\Phi_{ijbc}-g_{kb}\Phi_{ijca}-g_{kc}\Phi_{ijab}$	(2.21)
$\displaystyle\Phi_{ijkl}\Phi_{abkl}$	$\displaystyle=6g_{ia}g_{jb}-6g_{ib}g_{ja}-4\Phi_{ijab}$	(2.22)
$\displaystyle\Phi_{ijkl}\Phi_{ajkl}$	$\displaystyle=42g_{ia}$	(2.23)
$\displaystyle\Phi_{ijkl}\Phi_{ijkl}$	$\displaystyle=336.$	(2.24)

We also have contraction identities involving $\nabla\Phi$ and $\Phi$

$\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{abkl}$	$\displaystyle=-\Phi_{ijkl}(\nabla_{m}\Phi_{abkl})-4\nabla_{m}\Phi_{ijab}$	(2.25)
$\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{ajkl}$	$\displaystyle=-\Phi_{ijkl}(\nabla_{m}\Phi_{ajkl})$	(2.26)
$\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{ijkl}$	$\displaystyle=0.$	(2.27)

We now describe the torsion of a Spin(7)-structure. Given $X\in\Gamma(TM)$ , we know from [Kar08, Lemma 2.10] that $\nabla_{X}\Phi$ lies in the subbundle $\Omega^{4}_{7}\subset\Omega^{4}$ .

Definition 2.7.

The torsion tensor of a Spin(7)-structure $\Phi$ is the element of $\Omega^{1}_{8}\otimes\Omega^{2}_{7}$ defined by expressing $\nabla\Phi$ Since $\nabla_{X}\Phi\in\Omega^{4}_{7}$ , by Proposition 2.4, $\nabla\Phi$ can be written as

\displaystyle\nabla_{m}\Phi_{ijkl}=(T_{m}\diamond\Phi)_{ijkl}=T_{m;ip}\Phi_{pjkl}+T_{m;jp}\Phi_{ipkl}+T_{m;kp}\Phi_{ijpl}+T_{m;lp}\Phi_{ijkp}

(2.28)

where $T_{m;ab}\in\Omega^{2}_{7}$ , for each fixed $m$ . This defines the torsion tensor $T$ of a Spin(7)-structure, which is an element of $\Omega^{1}_{8}\otimes\Omega^{2}_{7}$ .

In terms of $\nabla\Phi$ , the torsion $T$ is given by

\displaystyle T_{m;ab}=\frac{1}{96}(\nabla_{m}\Phi_{ajkl})\Phi_{bjkl}

(2.29)

since $T$ is an element of $\Omega^{1}_{8}\otimes\Omega^{2}_{7}$ . Thus we have the following result, originally due to Fernández [Fer86].

Theorem 2.8.

[Fer86] The Spin(7)-structure $\Phi$ is torsion free if and only if $d\Phi=0$ . Since $\star\Phi=\Phi$ , this is equivalent to $d^{*}\Phi=0$ .

Remark 2.9.

The notation $T_{m;ab}$ should not be confused with taking two covariant derivatives of $T_{m}$ . The torsion tensor T is an element of $\Omega^{1}_{8}\otimes\Omega^{2}_{7}$ and thus, for each fixed index $m$ , $T_{m;ab}\in\Omega^{2}_{7}$ .

Finally, an important bit of information which we need about the torsion is that it satisfies a “Bianchi-type identity”. This was first proved by Karigiannis [Kar08, Theorem 4.2] using the diffeomorphism invariance of the torsion tensor and a different proof using the Ricci identity (1.4) was given in [DLE24, Theorem 3.9].

Theorem 2.10.

The torsion tensor $T$ satisfies the following “Bianchi-type” identity

\displaystyle\nabla_{i}T_{j;ab}-\nabla_{j}T_{i;ab}=2T_{i;am}T_{j;mb}-2T_{j;am}T_{i;mb}+\frac{1}{4}R_{jiab}-\frac{1}{8}R_{jimn}\Phi_{mnab}.

(2.30)

∎

Using the Riemannian Bianchi identity, we see that

\displaystyle R_{ijkl}\Phi_{ajkl}=-(R_{jkil}+R_{kijl})\Phi_{ajkl}=-R_{iljk}\Phi_{aljk}-R_{ikjl}\Phi_{akjl}

and hence we have the fact that

\displaystyle R_{ijkl}\Phi_{ajkl}=0.

(2.31)

Using this and contracting (2.30) on $j$ and $b$ gives the expression for the Ricci curvature of a metric induced by a Spin(7)-structure. Precisely,

\displaystyle R_{ij}=4\nabla_{i}T_{a;ja}-4\nabla_{a}T_{i;ja}-8T_{i;jb}T_{a;ba}+8T_{a;jb}T_{i;ba}

(2.32)

which also proves that the metric of a torsion free Spin(7)-structure is Ricci-flat, a result originally due to Bonan. Taking the trace of (2.32) gives the expression of the scalar curvature $R$

	$\displaystyle R$	$\displaystyle=4\nabla_{i}T_{a;ia}-4\nabla_{a}T_{i;ia}+8\|T\|^{2}+8T_{a;jb}T_{j;ba},$		(2.33)
		$\displaystyle\stackrel{{\scriptstyle\eqref{T8des}}}{{=}}8\operatorname{div}T_{8}+8\|T_{8}\|^{2}+8T_{a;jb}T_{j;ba}.$		(2.34)

Since $T=T_{8}+T_{48}$ , the $T_{8}$ component of torsion can be viewed as a vector field on $M$ and hence we want to look at alternate expression for the symmetric $2$ -tensor $\mathcal{L}_{T_{8}}g$ which will be used in § 3.1. We have the following linear algebra result whose proof can be found in [DGK23, Lemma 4.1].

Lemma 2.11.

Let $V$ and $W$ be finite-dimensional real vector spaces equipped with positive definite inner products, and suppose that

V=V_{1}\oplus_{\perp}\cdots\oplus_{\perp}V_{m}

is an orthogonal direct sum of subspaces. Let $\iota\colon V\to W$ and $\rho\colon W\to V$ be linear maps. Suppose that for every $1\leq k\leq m$ , there exist $b_{k},c_{k}$ both nonzero, such that for all $v_{k}\in V_{k}$ and $w\in W$ , we have

\mathrm{(i)}\,\,\rho\iota v_{k}=b_{k}v_{k}\quad\text{ and }\quad\mathrm{(ii)}\,\,\langle\rho w,v_{k}\rangle=c_{k}\langle w,\iota v_{k}\rangle.

(2.35)

Then in fact we have an isomorphism of $W$ with an orthogonal direct sum

W\cong(\ker\rho)\oplus_{\perp}V=(\ker\rho)\oplus_{\perp}V_{1}\oplus_{\perp}\cdots\oplus_{\perp}V_{k}.

(2.36)

∎

Let $V=\Lambda^{1}_{8}$ and $W=\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}$ in the previous lemma. Define $\iota:\Lambda^{1}_{8}\rightarrow\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}$ by

\displaystyle\iota(X)_{ijk}=X_{k}g_{ij}-X_{j}g_{ik}+X_{p}\Phi_{pijk},

and $\rho:\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}\rightarrow\Lambda^{1}_{8}$ by

\displaystyle\rho(\gamma)_{j}=\gamma_{i;ji}.

Clearly $\iota(X)_{ijk}=-\iota(X)_{ikj}$ . Consider

$\displaystyle\iota(X)_{ijk}\Phi_{abjk}$	$\displaystyle=\left(X_{k}g_{ij}-X_{j}g_{ik}+X_{p}\Phi_{pijk}\right)\Phi_{abjk}$
	$\displaystyle=X_{k}\Phi_{abik}-X_{j}\Phi_{abji}+X_{p}(6g_{pa}g_{ib}-6g_{pb}g_{ia}-4\Phi_{piab})$
	$\displaystyle=2X_{k}\Phi_{abik}+6X_{a}g_{ib}-6X_{b}g_{ia}+4X_{p}\Phi_{abip}$
	$\displaystyle=6X_{a}g_{ib}-6X_{b}g_{ia}+6X_{k}\Phi_{abik}=-6\iota(X)_{iab}$	(2.37)

where we have used (2.23) in the second equality. Hence (2.6) implies $\iota(X)\in\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}$ . We compute

\displaystyle(\rho\iota(X))_{j}

\displaystyle=\rho(X_{k}g_{ij}-X_{j}g_{ik}+X_{p}\Phi_{pijk})_{j}=-7X_{j}.

Also,

\displaystyle\langle\rho\gamma,X\rangle

\displaystyle=\gamma_{i;ji}X_{j},

and, using (2.6),

	$\displaystyle\langle\gamma,\iota X\rangle$	$\displaystyle=\gamma_{i;jk}(X_{k}g_{ij}-X_{j}g_{ik}+X_{p}\Phi_{pijk})$
		$\displaystyle=\gamma_{i;ik}X_{k}-X_{j}\gamma_{i;ji}+X_{p}\gamma_{i;jk}\Phi_{pijk}$
		$\displaystyle=-8X_{j}\gamma_{i;ji}.$

Thus, (i) and (ii) in (2.35) are satisfied with $b_{k}=-7$ and $c_{k}=-8$ respectively. We deduce from Lemma 2.11 that

\displaystyle\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}\cong(\ker\rho)\oplus_{\perp}\Lambda^{1}_{8}

(2.38)

with $\ker\rho=\{\gamma_{i;jk}\in\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}\mid\gamma_{i;ji}=0\}$ being $48$ -dimensional. In particular, since $T_{\Phi}\in\Lambda^{1}_{8}\otimes\Lambda^{2}_{7}$ , we see that

(T_{8})_{j}=T_{i;ji}

(2.39)

and thus

\displaystyle\left(\mathcal{L}_{T_{8}}g\right)_{ij}=\nabla_{i}(T_{8})_{j}+\nabla_{j}(T_{8})_{i}=\nabla_{i}(T_{k;jk})+\nabla_{j}(T_{k:ik}).

(2.40)

We will also need the expression for the Lie derivative of the Spin(7)-structure in § 4. Let $W\in\Gamma(TM)$ . Using (2.28) and (2.14), we have

	$\displaystyle(\mathcal{L}_{W}\Phi)_{ijkl}$	$\displaystyle=W_{p}\nabla_{p}\Phi_{ijkl}+\nabla_{i}W_{p}\Phi_{pjkl}+\nabla_{j}W_{p}\Phi_{ipkl}+\nabla_{k}W_{p}\Phi_{ijpl}+\nabla_{l}W_{p}\Phi_{ijkp}$
		$\displaystyle=W_{p}(T_{p}\diamond\Phi)_{ijkl}+(\nabla W\diamond\Phi)_{ijkl}$

and since $\nabla W=(\nabla W)_{\text{sym}}+(\nabla W)_{\text{skew}}=\frac{1}{2}\mathcal{L}_{W}g+(\nabla W)_{7}+(\nabla W)_{21}$ , where $(\nabla W)_{7}$ and $(\nabla W)_{21}$ denote the projection of the $2$ -form $(\nabla W)_{\text{skew}}$ onto the $\Omega^{2}_{7}$ and $\Omega^{2}_{21}$ components respectively. Using the fact that $\Omega^{2}_{21}\diamond\Phi=0$ , we get

\displaystyle(\mathcal{L}_{W}\Phi)_{ijkl}

\displaystyle=\left(\frac{1}{2}\mathcal{L}_{W}g+T(W)+(\nabla W)_{7}\diamond\Phi\right)_{ijkl}.

(2.41)

We record this observation in the following

Proposition 2.12.

A vector field $W$ is an infinitesimal symmetry of $\Phi$ , i.e., $\mathcal{L}_{W}\Phi=0$ if and only if

\displaystyle\mathcal{L}_{W}g=0\ \ \ \ \ \text{and}\ \ \ T(W)=-(\nabla W)_{7}.

(2.42)

∎

Remark 2.13.

Proposition 2.12 and (2.41) is essentially [DLE24, Lemma 2.6] where it was proved for arbitrary $H$ -structures.

Comparing with the case of manifolds with $\mathrm{G}_{2}$ -structures, the most general flow of $\mathrm{G}_{2}$ -structures which are second order quasilinear are described in [DGK23] and the authors prove the short-time existence and uniqueness theorem for a large class of flows. Such a family of flows was obtained by explicitly computing all independent second order differential invariants of $\mathrm{G}_{2}$ -structures which are $3$ -forms. Since the Riemann curvature tensor $\mathrm{Rm}$ and $\nabla T$ are the only two second order invariants of a $\mathrm{G}_{2}$ -structure, decomposing them into irreducible $\mathrm{G}_{2}$ -representations and then picking the linearly independent components (since $\mathrm{Rm}$ and $\nabla T$ are related by the so called $\mathrm{G}_{2}$ -Bianchi identity) which can be made into a $3$ -form provides a way to write the most general flow of $\mathrm{G}_{2}$ -structures. See [DGK23, §4] for more details. The existence of a Spin(7)-structure induces a decomposition of tensor bundles into irreducible Spin(7)-representations and one can similarly decompose $\mathrm{Rm}$ and $\nabla T$ into irreducible Spin(7)-representations and look at the linearly independent components which can be made into an admissible $4$ -form. We describe this briefly below without any proofs.

Let us denote the $k$ -dimensional irreducible Spin(7)-representation by $\mathbf{k}$ . Recall that the space $\mathcal{K}$ of curvature tensors on $(M,g)$ is the subspace of $S^{2}(\Lambda^{2})=\Gamma(\mathrm{S}^{2}(\Lambda^{2}T^{*}M))$ of elements satisfying the first Bianchi identity and hence we have the orthogonal decomposition

\displaystyle S^{2}(\Lambda^{2})

\displaystyle=\Omega^{4}\oplus_{\perp}\mathcal{K}.

Since for a manifold with a Spin(7)-structure $\Phi$ , we have $\Lambda^{2}=\mathbf{7}\oplus\mathbf{21}$ (see § 2.1), we have

\displaystyle S^{2}(\Lambda^{2})=S^{2}(\mathbf{7}\oplus\mathbf{21})=S^{2}(\mathbf{7})\oplus\mathbf{7}\otimes\mathbf{21}\oplus S^{2}(\mathbf{21}).

One can check the following decomposition of tensor bundles into irreducible Spin(7)-representations.

$\displaystyle\underbrace{\mathbf{7}\otimes\mathbf{7}}_{49}$	$\displaystyle=\mathbf{1}\oplus\mathbf{21}\oplus\mathbf{27}$	(2.43)
$\displaystyle\underbrace{\mathbf{7}\otimes\mathbf{21}}_{147}$	$\displaystyle=\mathbf{35}\oplus\mathbf{7}\oplus\mathbf{105}$	(2.44)
$\displaystyle\underbrace{\mathbf{21}\otimes\mathbf{21}}_{441}$	$\displaystyle=\mathbf{21}\oplus\mathbf{189}\oplus\mathbf{168}\oplus\mathbf{27}\oplus\mathbf{1}\oplus\mathbf{35}$	(2.45)
$\displaystyle\underbrace{S^{2}(21)}_{231}$	$\displaystyle=\mathbf{27}\oplus\mathbf{168}\oplus\mathbf{35}\oplus\mathbf{1}$	(2.46)

and hence we have

\displaystyle S^{2}(\Lambda^{2})

\displaystyle=(\mathbf{1}\oplus\mathbf{27})\oplus(\mathbf{35}\oplus\mathbf{7}\oplus\mathbf{105})\oplus(\mathbf{27}\oplus\mathbf{168}\oplus\mathbf{35}\oplus\mathbf{1}).

(2.47)

Since the space of curvature type tensors $\mathcal{K}$ is orthogonal to the space of $4$ -forms $\Omega^{4}=\mathbf{1}\oplus\mathbf{7}\oplus\mathbf{27}\oplus\mathbf{35}$ , we get the following decomposition

\displaystyle\mathcal{K}

\displaystyle=\underbrace{\mathbf{1}\oplus\mathbf{35}}_{\text{Ricci}}\oplus\underbrace{\mathbf{27}\oplus\mathbf{105}\oplus\mathbf{168}}_{\text{Weyl}}.

Thus, in the presence of a Spin(7)-structure, the Weyl tensor further decomposes as $W=W_{27}+W_{105}+W_{168}$ . Like the $\mathrm{G}_{2}$ -case there is again a $W_{27}$ , however unlike the $\mathrm{G}_{2}$ -case, this $W_{27}$ cannot be used for a flow of Spin(7)-structures as the space of admissible $4$ -forms is (pointwise) $63$ -dimensional and is $\mathbf{1}\oplus\mathbf{7}\oplus\mathbf{35}$ . Consequently, the only contribution from $\mathrm{Rm}$ for a flow of Spin(7)-structures are the traceless Ricci curvature $\mathrm{Ric}_{\circ}$ and the scalar curvature $Rg$ .

In a similar way we look at the possible contributions from $\nabla T$ for a flow of Spin(7)-structures. Since $T=T_{8}+T_{48}$ , hence at every point it lies in the representation $\mathbf{8}\oplus\mathbf{48}$ . Thus,

	$\displaystyle\nabla T$	$\displaystyle=\nabla T_{8}+\nabla T_{48}$
		$\displaystyle\in(\mathbf{8}\otimes\mathbf{8})\oplus(\mathbf{8}\otimes\mathbf{48})$
		$\displaystyle=(\mathbf{1}\oplus\mathbf{7}\oplus\mathbf{21}\oplus\mathbf{35})\oplus(\mathbf{35}\oplus\mathbf{27}\oplus\mathbf{189}\oplus\mathbf{21}\oplus\mathbf{105}\oplus\mathbf{7})$

and hence the components which can possibly contribute to a flow of Spin(7)-structures are $\mathbf{1},\mathbf{7}$ and $\mathbf{35}$ . The $\mathbf{1}$ component from $T$ is the $\operatorname{div}T_{8}$ term which can be written in terms of the scalar curvature $R$ and lower order terms as demonstrated in (2.34). Similarly, the two $\mathbf{35}$ components are a combination from $\mathcal{L}_{T_{8}}g$ and $\mathrm{Ric}$ because $\nabla T$ and $\mathrm{Rm}$ are related to each other by the Spin(7)-Bianchi identity (2.30). The contribution for the $\mathbf{7}$ component from $\nabla T$ is the $\operatorname{div}T$ term (since in the Spin(7)-case $\operatorname{div}T=-\operatorname{div}T^{t}$ ). One can do an explicit description of all the components above from $\mathrm{Rm}$ and $\nabla T$ as in [DGK23]. However, since we are only interested in flows of Spin(7)-structures for which we have all the relevant components, namely, $\mathrm{Ric},\mathcal{L}_{T_{8}}g$ and $\operatorname{div}T$ , we do not pursue the former here and we are currently doing this in collaboration with R. Singhal.

3. A gradient flow of Spin(7)-structures

3.1 Derivation of the flow

In this section we define the flow of Spin(7)-structures studied in this paper and prove that it is a negative gradient flow.

Definition 3.1.

Let $M^{8}$ be a compact manifold. The energy functional $E$ on the set of Spin(7)-structures on $M$ is

\displaystyle E(\Phi)=\frac{1}{2}\int_{M}|T_{\Phi}|^{2}\operatorname{vol}_{\Phi}

(3.1)

where $T_{\Phi}$ is the torsion of $\Phi$ and the norm and the volume form are with respect to the metric induced by $\Phi$ .

The natural question here is whether, given a Spin(7)-structure $\Phi_{0}$ , there is a “best” Spin(7)-structure. An obvious way to study this question is to consider the negative gradient flow of $E$ .

The most general flow of Spin(7)-structures [Kar08] is given by

\displaystyle\frac{\partial\Phi}{\partial t}=A\diamond\Phi=(h+X)\diamond\Phi

(3.2)

where $A_{ij}=h_{ij}+X_{ij}$ with $h_{ij}\in S^{2}$ and $X_{ij}\in\Omega^{2}_{7}$ . Thus, the most general flow of Spin(7)-structures is given by a time-dependent family of symmetric $2$ -tensors and a time-dependent family of $2$ -forms lying in the subspace $\Omega^{2}_{7}$ . In this case the evolution of the metric, the inverse of the metric and the volume form are given by

\displaystyle\frac{\partial g}{\partial t}=2h_{ij},\ \ \ \ \frac{\partial g^{-1}_{ij}}{\partial t}=-2h_{ij},\ \ \ \ \frac{\partial\operatorname{vol}}{\partial t}=\operatorname{tr}h\operatorname{vol}.

(3.3)

We start by considering the variation of the torsion $T$ with respect to variation of the Spin(7)-structures.

Proposition 3.2.

[Kar08, Thm. 3.4] Let $(\Phi_{t})_{t\in(-\varepsilon,\varepsilon)}$ be a smooth family of Spin(7)-structures. By (3.2), we can write $\left.\frac{\partial}{\partial t}\right|_{t=0}\Phi_{t}=A\diamond\Phi$ for some $h\in\Gamma(S^{2}T^{*}M)$ , $X\in\Omega^{2}_{7}$ . If $T_{t}$ is the torsion of $\Phi_{t}$ then

\displaystyle\left.\frac{\partial}{\partial t}\right|_{t=0}(T_{t})_{m;ab}=(h_{ap}T_{m;pb}-h_{bp}T_{m;pa})+\pi_{7}(\nabla_{b}h_{am}-\nabla_{a}h_{bm})+X_{ap}T_{m;pb}-X_{bp}T_{m;pa}+\pi_{7}(\nabla_{m}X_{ab}).

(3.4)

Let $E$ be the energy functional from (3.1). The following result motivates our formulation of the flow, cf. Definition 3.2.

Lemma 3.3.

Let $(\Phi_{t})_{t\in(-\varepsilon,\varepsilon)}$ be a smooth family of Spin(7)-structures and $\left.\frac{\partial}{\partial t}\right|_{t=0}\Phi_{t}=(h+X)\diamond\Phi$ for some $h\in\Gamma(S^{2}T^{*}M)$ , $X\in\Omega^{2}_{7}$ . The gradient of the energy functional $E(\Phi_{t})$ from (3.1) is given by

	$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}E(\Phi_{t})$	$\displaystyle=\int h_{am}\left(\frac{1}{2}R_{am}-(\mathcal{L}_{T_{8}}g)_{am}-4T_{b;al}T_{m;lb}+4T_{m;al}T_{b;lb}-T_{a;lb}T_{m;lb}\right.$
		$\displaystyle\qquad\qquad\quad\left.+\frac{1}{2}\|T\|^{2}g_{am}\right)\operatorname{vol}-\int X_{ab}\operatorname{div}T_{ab}\operatorname{vol}.$		(3.5)

Proof.

We use Proposition 3.2 and (3.3) to compute

$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}E(\Phi_{t})$	$\displaystyle=\left.\frac{d}{dt}\right\|_{t=0}\frac{1}{2}\int_{M}(T_{t})_{m;ab}(T_{t})_{n;cd}\ g^{mn}g^{ac}g^{bd}\operatorname{vol}$
	$\displaystyle=\int_{M}T_{m;ab}\partial_{t}T_{m;ab}\operatorname{vol}$
	$\displaystyle\quad-\int_{M}(h_{mn}T_{m;ab}T_{n;ab}+h_{ac}T_{m;ab}T_{m;cb}+h_{bd}T_{m;ab}T_{m;ad})\operatorname{vol}$
	$\displaystyle\quad+\frac{1}{2}\int_{M}\|T\|^{2}\operatorname{tr}h\operatorname{vol}.$	(3.6)

We calculate the first term on the right hand side of (3.6). Using (2.10) for the $\pi_{7}$ component and integration by parts, we have

	$\displaystyle\int T_{m;ab}\partial_{t}T_{m;ab}\operatorname{vol}$	$\displaystyle=\int T_{m;ab}[(h_{ap}T_{m;pb}-h_{bp}T_{m;pa})+\pi_{7}(\nabla_{b}h_{am}-\nabla_{a}h_{bm})]\operatorname{vol}$
		$\displaystyle\quad+\int T_{m;ab}[X_{ap}T_{m;pb}-X_{bp}T_{m;pa}+\pi_{7}(\nabla_{m}X_{ab})]\operatorname{vol}$
		$\displaystyle=\int[2h_{ap}T_{m;pb}T_{m;ab}+2\pi_{7}(\nabla_{b}h_{am})T_{m;ab}]\operatorname{vol}$
		$\displaystyle\quad+\int[2X_{ap}T_{m;pb}T_{m;ab}+\pi_{7}(\nabla_{m}X_{ab})T_{m;ab}]\operatorname{vol}$
		$\displaystyle=\int\left[2h_{ap}T_{m;pb}T_{m;ab}+\left(\frac{1}{2}\nabla_{b}h_{am}-\frac{1}{4}\nabla_{i}h_{jm}\Phi_{ijba}\right)T_{m;ab}\right]\operatorname{vol}$
		$\displaystyle\quad+\int\left(\frac{1}{4}\nabla_{m}X_{ab}T_{m;ab}-\frac{1}{8}\nabla_{m}X_{ij}\Phi_{ijab}T_{m;ab}\right)\operatorname{vol}$
		$\displaystyle=\int\left(2h_{ap}T_{m;pb}T_{m;ab}+\frac{1}{2}\nabla_{b}h_{am}T_{m;ab}-\frac{3}{2}\nabla_{i}h_{jm}T_{m;ij}\right)\operatorname{vol}$
		$\displaystyle\quad+\int\nabla_{m}X_{ab}T_{m;ab}\operatorname{vol}$
		$\displaystyle=\int\left(2h_{ap}T_{m;pb}T_{m;ab}+2\nabla_{b}h_{am}T_{m;ab}\right)\operatorname{vol}$
		$\displaystyle\quad-\int X_{ab}\operatorname{div}T_{ab}\operatorname{vol},$

where $(\operatorname{div}T)_{ab}=\nabla_{m}T_{m;ab}$ and we used the fact that $T_{m;pb}T_{m;ab}$ is symmetric in $a,b$ while $X_{ab}$ is skew-symmetric in the second equality, the fact that $T_{m;ab}\in\Omega^{1}\otimes\Omega^{2}_{7}$ and (2.6) in the second last equality and integration by parts in the last equality. Integrating by parts and then using the Spin(7)-Bianchi identity (2.30) gives

	$\displaystyle 2\int\nabla_{b}h_{am}T_{m;ab}\operatorname{vol}$	$\displaystyle=-2\int h_{am}\nabla_{b}T_{m;ab}\operatorname{vol}$
		$\displaystyle=-2\int h_{am}\left(\nabla_{m}T_{b;ab}+2T_{b;al}T_{m;lb}-2T_{m;al}T_{b;lb}+\frac{1}{4}R_{mbab}-\frac{1}{8}R_{mbkl}\Phi_{klab}\right)\operatorname{vol}$
		$\displaystyle=-2\int h_{am}\left(-\frac{1}{4}R_{am}+\nabla_{m}T_{b;ab}+2T_{b;al}T_{m;lb}-2T_{m;al}T_{b;lb}\right)\operatorname{vol}.$

from which we infer

	$\displaystyle\int T_{m;ab}\partial_{t}T_{m;ab}\operatorname{vol}$	$\displaystyle=\int h_{am}\left(\frac{1}{2}R_{am}-2\nabla_{m}T_{b;ab}-4T_{b;al}T_{m;lb}+4T_{m;al}T_{b;lb}-2T_{l;mb}T_{l;ab}\right)\operatorname{vol}$
		$\displaystyle\quad-\int X_{ab}\operatorname{div}T_{ab}\operatorname{vol}.$		(3.7)

Using (3.1) in (3.6) and the expression of $\mathcal{L}_{T_{8}}g$ from (2.40), we get

	$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}E(\Phi_{t})$	$\displaystyle=\int h_{am}\left(\frac{1}{2}R_{am}-2\nabla_{m}T_{b;ab}-4T_{b;al}T_{m;lb}+4T_{m;al}T_{b;lb}-2T_{l;mb}T_{l;ab}\right.$
		$\displaystyle\qquad\qquad\quad\left.-T_{m;lb}T_{a;lb}-T_{l;ab}T_{l;mb}-T_{l;ba}T_{l;bm}+\frac{1}{2}\|T\|^{2}g_{am}\right)\operatorname{vol}$
		$\displaystyle\quad-\int X_{ab}\operatorname{div}T_{ab}\operatorname{vol}$
		$\displaystyle=\int h_{am}\left(\frac{1}{2}R_{am}-(\mathcal{L}_{T_{8}}g)_{am}-4T_{b;al}T_{m;lb}+4T_{m;al}T_{b;lb}-T_{a;lb}T_{m;lb}\right.$
		$\displaystyle\qquad\qquad\quad\left.+\frac{1}{2}\|T\|^{2}g_{am}\right)\operatorname{vol}-\int X_{ab}\operatorname{div}T_{ab}\operatorname{vol},$

which completes the proof. ∎

Remark 3.4.

A similar calculation for $H$ -structures is done in [FLME22, Prop. 1.44]. The terms coming from $\Phi$ and $g$ are explicit for us as we are looking at a particular $H$ -structure.

We are interested in the negative gradient flow of $E(\Phi_{t})$ and based on the above computations, we propose the following flow of Spin(7)-structures.

Definition 3.5.

Let $(M^{8},\Phi_{0})$ be a compact manifold with a Spin(7)-structure. Let $(T*T)_{ij}=8T_{b;il}T_{j;lb}-8T_{j;il}T_{b;lb}+2T_{i;lb}T_{j;lb}$ . A gradient flow of Spin(7)-structures is the following initial value problem

\displaystyle\left\{\begin{array}[]{rl}&\dfrac{\partial\Phi}{\partial t}=\left(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+T*T-|T|^{2}g+2\operatorname{div}T\right)\diamond\Phi,\\ &\Phi(0)=\Phi_{0}.\end{array}\right.

(GF)

It follows from (3.3) and (2.33) that along (GF), the underlying metric and volume form evolve as

$\displaystyle\partial_{t}g_{ij}$	$\displaystyle=-2R_{ij}+4(\mathcal{L}_{T_{8}}g)_{ij}+16T_{b;il}T_{j;lb}-16T_{j;il}T_{b;lb}+4T_{i;lb}T_{j;lb}-2\|T\|^{2}g_{ij},$	(3.10)
$\displaystyle\partial_{t}(g^{-1})_{ij}$	$\displaystyle=2R_{ij}-4(\mathcal{L}_{T_{8}}g)_{ij}-16T_{b;il}T_{j;lb}+16T_{j;il}T_{b;lb}-4T_{i;lb}T_{j;lb}+2\|T\|^{2}g_{ij},$	(3.11)
$\displaystyle\partial_{t}\operatorname{vol}$	$\displaystyle=-(4\operatorname{div}T_{8}+6\|T\|^{2})\operatorname{vol}.$	(3.12)

We discuss the effect of scaling on tensors induced from a Spin(7)-structure. It follows from [DLE24, §4.3.1] that if $\Phi$ is a Spin(7)-structure then so is $\widetilde{\Phi}=c^{4}\Phi$ , $c>0$ constant. In this case, $\widetilde{g}=c^{2}g,\ \widetilde{g^{-1}}=c^{-2}g^{-1}$ and $\widetilde{\operatorname{vol}}=c^{8}\operatorname{vol}$ . Moreover,

\displaystyle\widetilde{\nabla}=\nabla,\ \ \widetilde{T}=c^{2}T,\ \text{and}\ \widetilde{\mathrm{Ric}}=\mathrm{Ric}.

The natural parabolic rescaling for a geometric evolution equation involves scaling the time variable t by $c^{2}t$ , when the space variable scales by $c$ and hence, keeping in mind that taking $\diamond$ with $\Phi$ involves contraction on one index, we see that each term on the right hand side of (GF) indeed has the correct scaling. We record for future reference that for $\widetilde{\Phi}=c^{4}\Phi$

\displaystyle|\widetilde{\nabla}^{j}\widetilde{\mathrm{Rm}}|_{\widetilde{g}}=c^{-(2+j)}|\nabla^{j}\mathrm{Rm}|_{g},\ \ \ |\widetilde{\nabla}^{j}\widetilde{T}|_{\widetilde{g}}=c^{-(1+j)}|\nabla^{j}T|_{g}.

(3.13)

In particular, $E(c^{4}\Phi)=c^{6}E(\Phi)$ is a positively homogeneous functional and hence an application of Euler’s theorem for positively homogeneous functional shows that the critical points of $E$ are torsion-free Spin(7)-structures, which in fact, are absolute minimizers and thus the flow (GF) is capable of detecting torsion-free Spin(7) structures.

4. Short-time existence and uniqueness

In this section we establish short-time existence and uniqueness of the flow (GF) of Spin(7)-structures, using a modification of the DeTurck’s trick and the explicit computation of the principal symbols of the second order linear differential operators defining the flow. We first calculate the principal symbols of the highest order terms on the right hand side of (GF), i.e., $\mathrm{Ric},\mathcal{L}_{T_{8}}g$ and $\operatorname{div}T$ in § 4.2. We use these to show that (GF) is a weakly parabolic PDE and the failure of parabolicity is only due to the diffeomorphism invariance of the tensors involved. We use a modification of the DeTurck’s trick from the Ricci flow to prove the short-time existence and uniqueness of solution in § 4.3.

4.1 Differential operators, ellipticity, and parabolicity

We give a brief review of parabolic PDEs and the existence and uniqueness of solutions of such equations. Other sources for the discussion below are [CK04, §3.2], [AH11, §5.1], [Top06, §4] and [DGK23, §6.1] (for the $\mathrm{G_{2}}$ case).

Let $E,F$ be vector bundles over a Riemannian manifold $(M,g)$ and let $L\colon\Gamma(E)\to\Gamma(F)$ be a linear differential operator of order $m$ . We write $L$ , for every $x\in M$ , in terms of local frames for $E$ and $M$ , as

L(\sigma)^{b}(x)=\sum_{l=0}^{m}[\hat{L}_{l}(x)]_{a}^{b,i_{1},\ldots,i_{l}}[\nabla^{l}_{i_{1},\ldots,i_{l}}\sigma(x)]^{a}=\sum_{l=0}^{m}[\hat{L}_{l}(x)]^{b}(\nabla^{l}\sigma(x))

(4.1)

where for each $l=0,1,\ldots,m$ , we write $\nabla^{l}\sigma\in\Gamma((T^{*}M)^{\otimes l}\otimes E)$ to denote the $l$ -th covariant derivative of $\sigma$ , and $\hat{L}_{l}\in\Gamma((TM)^{\otimes l}\otimes\mathrm{Hom}(E,F))$ . Here the index $a$ corresponds to a local frame for $E$ and the index $b$ corresponds to a local frame for $F$ .

For any such linear differential operator, we define its principal symbol so that for each $x\in M$ and $\xi\in T^{*}_{x}M$ , the map

\sigma_{\xi}(L)\colon E_{x}\to F_{x}

is the linear homomorphism

	^b	$\displaystyle=[\hat{L}_{m}(x)]^{b}(\xi,\ldots,\xi,\sigma),$		(4.2)
		$\displaystyle=[\hat{L}_{m}(x)]^{b,i_{1},\ldots,i_{m}}_{a}\xi_{i_{1}}\cdots\xi_{i_{m}}\sigma^{a}.$		(4.2)

The principal symbol satisfies the fundamental properties

\sigma_{\xi}(P+Q)=\sigma_{\xi}(P)+\sigma_{\xi}(Q),\qquad\sigma_{\xi}(P\circ Q)=\sigma_{\xi}(P)\circ\sigma_{\xi}(Q),

whenever $P$ , $Q$ are linear differential operators so that either $P+Q$ or $P\circ Q$ is well defined. We have the following

Definition 4.1.

A linear differential operator $L\colon\Gamma(E)\to\Gamma(F)$ is called elliptic if for any $x\in M$ , $\xi\in T^{*}_{x}M$ , $\xi\neq 0$ , the principal symbol $\sigma_{\xi}(L)\colon E_{x}\to F_{x}$ is a linear isomorphism.

Let $E$ be a vector bundle over $M$ with a fibre metric $\langle\cdot,\cdot\rangle$ . Consider a second order linear differential operator $L\colon\Gamma(E)\to\Gamma(E)$ . If there is a constant $c>0$ such that for any $\xi\in T^{*}_{x}M$ , $\xi\neq 0$ and $v\in E_{x}$ , we have

\langle\sigma_{\xi}(L)(v),v\rangle\geq c|\xi|^{2}|v|^{2},

then $L$ is called strongly elliptic.

Definition 4.2.

Let $E$ , $F$ be vector bundles over $M$ , let $\mathcal{U}\subseteq\Gamma(E)$ be open, and let $P\colon\mathcal{U}\to\Gamma(F)$ be a nonlinear differential operator. The operator $P$ is called elliptic at $v\in\mathcal{U}$ if the linearization

	$\displaystyle D_{v}P$	$\displaystyle\colon\Gamma(E)\to\Gamma(F),$
	$\displaystyle(D_{v}P)(w)$	$\displaystyle:={\left.{\frac{d}{ds}}\right\|}_{{s=0}}P(v+sw),$

is an elliptic linear differential operator.

Similarly, if $P\colon\mathcal{U}\to\Gamma(E)$ is a second order differential operator and $E$ is endowed with a bundle metric $\langle\cdot,\cdot\rangle$ , we say that $P$ is strongly elliptic at $\sigma\in\mathcal{U}$ if its linearization $D_{\sigma}P\colon\Gamma(E)\to\Gamma(E)$ is a strongly elliptic linear differential operator.

A nonlinear evolution equation of the form $\frac{\partial}{\partial t}\sigma=P(\sigma)$ , where $\sigma\in\mathcal{U}$ , is called parabolic at $\sigma$ if $P$ is strongly elliptic at $\sigma$ .

The importance of the above definition is due to the following standard result.

Theorem 4.3.

Let $M$ be a Riemannian manifold, let $E$ be a vector bundle over $M$ endowed with a fibre metric $\langle\cdot,\cdot\rangle$ , and let $\mathcal{U}\subseteq\Gamma(E)$ be open. Let $P\colon\mathcal{U}\to\Gamma(E)$ be a second order quasilinear differential operator, which is strongly elliptic at $\sigma_{0}\in\mathcal{U}$ . Then there exists $\epsilon>0$ and for any $t\in[0,\epsilon)$ a unique $\sigma(t)\in\mathcal{U}$ , such that

\frac{\partial\sigma(t)}{\partial t}=P(\sigma(t)),\qquad\sigma(0)=\sigma_{0}.

(4.3)

That is, a nonlinear evolution equation $\frac{\partial}{\partial t}\sigma=P(\sigma)$ which is parabolic at $\sigma_{0}$ has a unique short time smooth solution with initial condition $\sigma(0)=\sigma_{0}$ . ∎

4.2 Principal Symbols

Let $\Phi$ be a Spin(7)-structure and consider a variation $(\Phi_{t})_{t\in(\varepsilon,\varepsilon)}$ with $\Phi_{0}=\Phi$ and

\displaystyle\dot{\Phi}_{ijkl}=\left.\frac{\partial}{\partial t}\right|_{t=0}\Phi_{ijkl}=A_{ip}\Phi_{pjkl}+A_{jp}\Phi_{ipkl}+A_{kp}\Phi_{ijpl}+A_{lp}\Phi_{ijkp},

(4.4)

where $A_{ij}=h_{ij}+X_{ij}$ . Since the variation of the associated Riemannian metric is $\left.\frac{\partial}{\partial t}\right|_{t=0}g_{t}=2h$ hence the linearization $D\Gamma^{r}_{ia}$ of the Christoffel symbols satisfies

\delta_{qr}D\Gamma^{r}_{ia}=\nabla_{i}h_{aq}+\nabla_{a}h_{iq}-\nabla_{q}h_{ia},

(4.5)

where $\nabla$ is the Levi-Civita connection of the metric $g$ associated to the Spin(7)-structure $\Phi$ .

It follows that the principal symbol of $D\Gamma^{r}_{ia}$ , for any non-zero $\xi\in T_{x}^{*}M$ , is

\sigma(\delta_{rq}D\Gamma^{r}_{ia})(x,\xi)(\dot{\Phi})=(\xi_{i}h_{aq}+\xi_{a}h_{iq}-\xi_{q}h_{ia}).

(4.6)

Now we compute the linearization $D(\nabla\Phi)_{mijkl}$ of $\nabla_{m}\Phi_{ijkl}$ :

\displaystyle D(\nabla\Phi)(\dot{\Phi})_{mijkl}=-D\Gamma^{r}_{mi}(\dot{\Phi})\Phi_{rjkl}-D\Gamma^{r}_{mj}(\dot{\Phi})\Phi_{irkl}-D\Gamma^{r}_{mk}(\dot{\Phi})\Phi_{ijrl}-D\Gamma^{r}_{ml}(\dot{\Phi})\Phi_{ijkr}+\nabla_{m}\dot{\Phi}_{ijkl}.

The principal symbol of the differential operator $D(\nabla\Phi)$ is:

	$\displaystyle\sigma(D(\nabla\Phi))(x,\xi)(\dot{\Phi})_{mijkl}$	$\displaystyle=-(\xi_{m}h_{iq}+\xi_{i}h_{mq}-\xi_{q}h_{mi})\Phi_{qjkl}-(\xi_{m}h_{jq}+\xi_{j}h_{mq}-\xi_{q}h_{mj})\Phi_{iqkl}$
		$\displaystyle\quad-(\xi_{m}h_{kq}+\xi_{k}h_{mq}-\xi_{q}h_{mk})\Phi_{ijql}-(\xi_{m}h_{lq}+\xi_{l}h_{mq}-\xi_{q}h_{ml})\Phi_{ijkq}$
		$\displaystyle\quad+\xi_{m}(h_{ip}\Phi_{pjkl}+h_{jp}\Phi_{ipkl}+h_{kp}\Phi_{ijpl}+h_{lp}\Phi_{ijkp}$
		$\displaystyle\qquad\qquad+X_{ip}\Phi_{pjkl}+X_{jp}\Phi_{ipkl}+X_{kp}\Phi_{ijpl}+X_{lp}\Phi_{ijkp})$
		$\displaystyle=-(\xi_{i}h_{mq}-\xi_{q}h_{mi})\Phi_{qjkl}-(\xi_{j}h_{mq}-\xi_{q}h_{mj})\Phi_{iqkl}$
		$\displaystyle\quad-(\xi_{k}h_{mq}-\xi_{q}h_{mk})\Phi_{ijql}-(\xi_{l}h_{mq}-\xi_{q}h_{ml})\Phi_{ijkq}$
		$\displaystyle\quad+\xi_{m}(X_{ip}\Phi_{pjkl}+X_{jp}\Phi_{ipkl}+X_{kp}\Phi_{ijpl}+X_{lp}\Phi_{ijkp}).$

Namely

$\displaystyle\sigma(D(\nabla\Phi))(x,\xi)(\dot{\Phi})_{mijkl}$	$\displaystyle=-(\xi_{i}h_{mq}-\xi_{q}h_{mi})\Phi_{qjkl}-(\xi_{j}h_{mq}-\xi_{q}h_{mj})\Phi_{iqkl}$	(4.7)
	$\displaystyle\quad-(\xi_{k}h_{mq}-\xi_{q}h_{mk})\Phi_{ijql}-(\xi_{l}h_{mq}-\xi_{q}h_{ml})\Phi_{ijkq}$
	$\displaystyle\quad+\xi_{m}(X_{ip}\Phi_{pjkl}+X_{jp}\Phi_{ipkl}+X_{kp}\Phi_{ijpl}+X_{lp}\Phi_{ijkp}).$

The linearization $DT_{m;ib}$ of the torsion $T_{m;ib}=\frac{1}{96}\nabla_{m}\Phi_{ijkl}\Phi_{bjkl}$ is given by

DT(\dot{\Phi})_{m;ib}=\frac{1}{96}D(\nabla\Phi)_{mijkl}\Phi_{bjkl}+\textrm{l.o.t.}.

(4.8)

Hence, its principal symbol is given by

$\displaystyle\sigma(DT)(x,\xi)(\dot{\Phi})_{m;ib}$	$\displaystyle=\frac{1}{96}\big{[}-(\xi_{i}h_{mq}-\xi_{q}h_{mi})\Phi_{qjkl}-(\xi_{j}h_{mq}-\xi_{q}h_{mj})\Phi_{iqkl}$
	$\displaystyle\qquad\qquad-(\xi_{k}h_{mq}-\xi_{q}h_{mk})\Phi_{ijql}-(\xi_{l}h_{mq}-\xi_{q}h_{ml})\Phi_{ijkq}$
	$\displaystyle\qquad\qquad+\xi_{m}(X_{iq}\Phi_{qjkl}+X_{jq}\Phi_{iqkl}+X_{kq}\Phi_{ijql}+X_{lq}\Phi_{ijkq})\big{]}\Phi_{bjkl}$	(4.9)

We use the contraction identities in (2.21)-(2.24) and (2.10) to compute the principal symbol of $DT$ ,

\displaystyle\sigma(DT)(x,\xi)(\dot{\Phi})_{m;ib}=\left(\frac{1}{4}(\xi_{b}h_{im}-\xi_{i}h_{mb}+\xi_{j}h_{mq}\Phi_{ibjq})+\xi_{m}X_{ib}\right).

(4.10)

Using (4.10) and (2.39), the principal symbols of $\operatorname{div}T_{ib}$ and $(\mathcal{L}_{T_{8}}g)_{il}$ are

$\displaystyle\sigma(D\operatorname{div}T)(x,\xi)(\dot{\Phi})_{ib}$	$\displaystyle=\frac{1}{4}(\xi_{m}\xi_{b}h_{im}-\xi_{m}\xi_{i}h_{mb}+\xi_{m}\xi_{j}h_{mq}\Phi_{ibjq})+\|\xi\|^{2}X_{ib},$	(4.11)
$\displaystyle\sigma(D\mathcal{L}_{T_{8}}g)(x,\xi)(\dot{\Phi})_{li}$	$\displaystyle=\frac{1}{4}\left(\xi_{l}\xi_{m}h_{im}+\xi_{i}\xi_{m}h_{lm}-2\xi_{i}\xi_{l}\operatorname{tr}h+\xi_{l}\xi_{j}h_{mq}\Phi_{imjq}+\xi_{i}\xi_{j}h_{mq}\Phi_{lmjq}\right)$
	$\displaystyle\quad+\xi_{l}\xi_{m}X_{im}+\xi_{i}\xi_{m}X_{lm}$
	$\displaystyle=\frac{1}{4}\left(\xi_{l}\xi_{m}h_{im}+\xi_{i}\xi_{m}h_{lm}-2\xi_{i}\xi_{l}\operatorname{tr}h+\right)+\xi_{l}\xi_{m}X_{im}+\xi_{i}\xi_{m}X_{lm}.$	(4.12)

Recall that the Ricci curvature in terms of the torsion tensor is given by

\displaystyle R_{ij}=4\nabla_{i}T_{a;ja}-4\nabla_{a}T_{i;ja}-8T_{i;jb}T_{a;ba}+8T_{a;jb}T_{i;ba}.

From the computations in the previous section we obtain that the symbol of $D\mathrm{Ric}$ is given by

	$\displaystyle\sigma(D\mathrm{Ric})(x,\xi)(\dot{\Phi})_{ij}$	$\displaystyle=4\sigma(D\nabla T)_{ia;ja}-4\sigma(D\nabla T)_{ai;ja}$
		$\displaystyle=(\xi_{i}\xi_{a}h_{aj}-\xi_{i}\xi_{j}\operatorname{tr}h+4\xi_{i}\xi_{a}X_{ja})-(-\xi_{a}\xi_{j}h_{ia}+\|\xi\|^{2}h_{ij}+4\xi_{a}\xi_{i}X_{ja})$

which simplifies to

\displaystyle\sigma(D\mathrm{Ric})(x,\xi)(\dot{\Phi})_{jk}

\displaystyle=-|\xi|^{2}h_{ij}-\xi_{i}\xi_{j}\operatorname{tr}h+(\xi_{i}\xi_{a}h_{aj}+\xi_{a}\xi_{j}h_{ia}).

(4.13)

Similarly, the principal symbol of the scalar curvature is

\displaystyle\sigma(DR)(x,\xi)(\dot{\Phi})

\displaystyle=-2|\xi|^{2}\operatorname{tr}h+2\xi_{j}\xi_{a}h_{ja}.

(4.14)

4.3 Failure of parabolicity of the flow

Consider the differential operator $L:\Gamma(S^{2}T^{*}M)\oplus\Omega^{2}_{7}(M)\rightarrow\Omega^{4}(M)$

\displaystyle L(h,X)

\displaystyle=h\diamond\Phi+X\diamond\Phi.

Clearly, (GF) is $\partial_{t}\Phi(t)=L(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+(T*T)-|T|^{2}g,2\operatorname{div}T)$ . Since $\mathrm{Ric}$ and $T$ are diffeomorphism invariant tensors, $\varphi^{*}L(\Phi)=L(\varphi^{*}\Phi)$ for the operator in (GF). For the purposes of short time existence of the flow we are only interested in the highest order term so we instead consider the operator (which we still $L$ )

\displaystyle L(\Phi)=L(-\mathrm{Ric}+2\mathcal{L}_{T_{8}}g,2\operatorname{div}T)=(-\mathrm{Ric}+2\mathcal{L}_{T_{8}}g+2\operatorname{div}T)\diamond\Phi.

(4.15)

Moreover, using Proposition 2.4, we will view $L:\Gamma(S^{2}T^{*}M)\oplus\Omega^{2}_{7}\rightarrow\Gamma(S^{2}T^{*}M)\oplus\Omega^{2}_{7}$ . Since the bundle metric $\langle(h_{1},X_{1}),(h_{2},X_{2})\rangle=\langle h_{1},h_{2}\rangle+\langle X_{1},X_{2}\rangle$ on $\Gamma(S^{2}T^{*}M)\oplus\Omega^{2}_{7}(M)$ is uniformly equivalent to the natural inner product on $\Omega^{4}_{1\oplus 7\oplus 35}(M)$ , the operator $L$ is strongly elliptic if and only if there is a constant $c>0$ such that for any $x\in M$ , $\xi\in T^{*}_{x}M$ , $\xi\neq 0$ , and any $(h,X)\in\Gamma(S^{2}T^{*}M)\oplus\Omega^{2}_{7}(M)$ , we have

\displaystyle\langle\sigma_{\xi}(L)(h,X),(h,X)\rangle\geq c|(h,X)|^{2}=c(|h|^{2}+|X|^{2}).

(4.16)

We will see below that the operator $L$ is, in fact, not elliptic (and hence (GF) not parabolic) but the failure of ellipticity is only due to the diffeomorphism invariance of the tensors in the definition of $L$ . As a result, we use a modified DeTurck’s trick to prove short-time existence in Theorem 4.11.

We first note, using (4.13), (4.12) and (4.11), that

	$\displaystyle\sigma_{\xi}(L)(h,X)_{ij}$	$\displaystyle=\|\xi\|^{2}h_{ij}-\frac{1}{2}(\xi_{i}\xi_{a}h_{aj}+\xi_{a}\xi_{j}h_{ia})+2\xi_{i}\xi_{m}X_{jm}+2\xi_{j}\xi_{m}X_{im}$
		$\displaystyle\quad+\frac{1}{2}(\xi_{m}\xi_{j}h_{im}-\xi_{m}\xi_{i}h_{mj}+\xi_{m}\xi_{k}h_{mq}\Phi_{ijkq})+2\|\xi\|^{2}X_{ij}.$		(4.17)

In order to analyze the kernel of $\sigma_{\xi}(L)$ , we define the map

\delta^{*}\colon\Gamma(TM)\to\Omega^{4},\qquad\delta^{*}W=\mathcal{L}_{W}\Phi.

From (2.41) we have

\delta^{*}W=\mathcal{L}_{W}\Phi=\left(\tfrac{1}{2}(\mathcal{L}_{W}g)+T(W)+(\nabla W)_{7}\right)\diamond\Phi.

(4.18)

Proposition 4.4.

Let $\delta^{*}\colon\Gamma(TM)\to\Omega^{4}$ be as in (4.18). For any nonzero $\xi\in T^{*}_{x}M$ , we have

	_jk	$\displaystyle=\tfrac{1}{2}(\xi_{j}W_{k}+\xi_{k}W_{j}),$		(4.19)
	$\displaystyle[\pi_{7}\circ\sigma_{\xi}(\delta^{*})(W)]_{jk}$	$\displaystyle=\frac{1}{8}\xi_{j}W_{k}-\frac{1}{8}\xi_{k}W_{j}-\frac{1}{8}\xi_{a}W_{b}\Phi_{abjk},$		(4.19)

and $\sigma_{\xi}(\delta^{*})\colon T^{*}_{x}M\to\Lambda^{4}(T^{*}_{x}M)$ is injective. Here $\pi_{7}$ denotes the identification of $\Omega^{4}_{7}$ component of $\sigma_{\xi}(\delta^{*})$ with an element in $\Omega^{2}_{7}$ .

Proof.

The expressions in (4.19) follow from (4.18), because $(\mathcal{L}_{W}g)_{jk}=\nabla{j}W_{k}+\nabla{k}W_{j}$ and $((\nabla W)_{7})_{jk}=\frac{1}{4}((\nabla W)_{\text{skew}})_{jk}-\frac{1}{8}((\nabla W)_{\text{skew}})_{ab}\Phi_{abjk}=\frac{1}{8}\nabla_{j}W_{k}-\frac{1}{8}\nabla_{k}W_{j}-\frac{1}{8}\nabla_{a}W_{b}\Phi_{abjk}$ . Suppose $W\in\ker\sigma_{\xi}(\delta^{*})$ . In particular we get $\xi_{j}W_{k}+\xi_{k}W_{j}=0$ . Multiplying by $\xi_{j}W_{k}$ and summing, we obtain

0=|\xi|^{2}|W|^{2}+\langle W,\xi\rangle^{2},

which implies that $W=0$ as $\xi\neq 0$ , so $\sigma_{\xi}(\delta^{*})$ is injective. ∎

Recall that $L(\Phi)$ is invariant under diffeomorphisms, that is,

L(\varphi^{*}\Phi)=\varphi^{*}(L(\Phi)),

for any diffeomorphism $\varphi\colon M\to M$ . It follows that for any vector field $W\in\Gamma(TM)$ , we have

\mathcal{L}_{W}(L(\Phi))=D_{\Phi}L(\mathcal{L}_{W}\Phi).

(4.20)

Since $W\mapsto\mathcal{L}_{W}(L(\Phi))$ is a first order linear differential operator on $W$ , whereas

W\mapsto D_{\Phi}L(\mathcal{L}_{W}\Phi)=(D_{\Phi}L\circ\delta^{*})(W)

is a priori a third order differential operator, it follows that

\sigma_{\xi}(D_{\Phi}L\circ\delta^{*})=\sigma_{\xi}(D_{\Phi}L)\circ\sigma_{\xi}(\delta^{*})=0.

and hence

\text{im}(\sigma_{\xi}(\delta^{*}))=\left\{\frac{1}{2}\xi_{i}V_{j}+\frac{1}{2}\xi_{j}V_{i}+\frac{1}{8}\xi_{i}V_{j}-\frac{1}{8}\xi_{j}V_{i}-\frac{1}{8}\xi_{a}V_{b}\Phi_{abij}:V\in T^{*}_{x}M\right\}\subseteq\ker\sigma_{\xi}(D_{\Phi}L).

(4.21)

Hence, by the injectivity of $\sigma_{\xi}(\delta^{*})$ , the principal symbol of $L$ has

\displaystyle\text{dim}\ker(\sigma(L))\geq 8=\text{dim}\ M

(4.22)

that is due to diffeomorphism invariance, so $L$ is never an elliptic differential operator.

Remark 4.5.

The relation in (4.21) can be checked directly by putting $h_{ij}=\frac{1}{2}(\xi_{i}V_{j}+\xi_{j}V_{i})$ and $X_{ij}=\frac{1}{8}(\xi_{i}V_{j}-\xi_{j}V_{i}-\xi_{a}V_{b}\Phi_{abij})$ in (4.3) (which is, of course, expected).

We now show that the dimension of kernel of $\sigma(L)$ is at most $8$ . To do so, we introduce the following two operators. Consider

	$\displaystyle B_{1}$	$\displaystyle\colon S^{2}(T^{}_{x}M)\to T^{}_{x}M,$	$\displaystyle B_{1}(h)_{k}$	$\displaystyle=\xi_{a}h_{ak}-\tfrac{1}{2}\xi_{k}\operatorname{tr}h,$		(4.23)
	$\displaystyle B_{2}$	$\displaystyle\colon T^{}_{x}M\to\Lambda^{2}_{7}(T^{}_{x}M),$	$\displaystyle B_{2}(W)_{ij}$	$\displaystyle=\frac{1}{8}(\xi_{i}W_{j}-\xi_{j}W_{i}-\xi_{a}W_{b}\Phi_{abij}).$		(4.23)

We recall that the map $B_{1}$ is the symbol of the Bianchi map $\mathcal{S}^{2}\to\Omega^{1}$ given by $h\mapsto\nabla_{i}h_{ik}-\frac{1}{2}\nabla_{k}(\operatorname{tr}h)$ . The Ricci curvature $\mathrm{Ric}$ lies in the kernel of the Bianchi map by the twice contracted Riemannian second Bianchi identity. The map $B_{2}$ is the $\pi_{7}\circ\sigma_{\xi}(\delta^{*})$ map in (4.19) and is precisely the symbol of the operator $W\mapsto(\nabla W)_{7}$ . The reason we need the map $B_{2}$ is because while doing a modified version of the DeTurck’s trick, we need to add a term of the form $\mathcal{L}_{W}\Phi$ to the right hand side of (GF), and the operator $(\nabla W)_{7},W\in\Gamma(TM)$ shows up in the $\Omega^{4}_{7}$ part of $\mathcal{L}_{W}\Phi$ , by equation (2.41).

It is convenient to define the operator $\widetilde{B}\colon S^{2}\oplus\Omega^{2}_{7}\to\Omega^{1}$ by

\tilde{B}(h,X)_{k}=B_{1}(h)_{k}-2\sigma_{\xi}(D_{\Phi}T_{8})(h,X)_{k}.

(4.24)

We can rewrite the components of $\sigma_{\xi}(DL)$ in terms of the map $\widetilde{B}$ .

Proposition 4.6.

Let $L$ be the operator as in (4.15). In terms of the maps $B_{1},B_{2}$ of (4.23) and $\widetilde{B}$ of (4.24), the $1+35$ and $7$ parts of the principal symbol of linearization $D_{\Phi}L$ can be expressed as

$\displaystyle\pi_{1+35}\circ\sigma_{\xi}(DL)(h,X)_{ij}$	$\displaystyle=\|\xi\|^{2}h_{ij}-\xi_{i}\widetilde{B}(h,X)_{j}-\xi_{j}\widetilde{B}(h,X)_{i},$	(4.25)
$\displaystyle\pi_{7}\circ\sigma_{\xi}(DL)(h,X)_{ij}$	$\displaystyle=2\|\xi\|^{2}X_{ij}-4(B_{2}(\widetilde{B}(h,X))_{ij}+\frac{1}{4}\left(\xi_{i}\xi_{m}h_{mj}-\xi_{j}\xi_{m}h_{im}-\xi_{a}\xi_{m}h_{mb}\Phi_{abij}\right)$
	$\displaystyle\quad+\xi_{i}\xi_{m}X_{jm}-\xi_{j}\xi_{m}X_{im}-\xi_{a}\xi_{m}X_{bm}\Phi_{abij}.$
	$\displaystyle=2\|\xi^{2}\|X_{ij}-8(B_{2}(\widetilde{B}(h,X)))_{ij}+2\xi_{j}\xi_{m}X_{im}-2\xi_{i}\xi_{m}X_{jm}+2\xi_{k}\xi_{m}X_{qm}\Phi_{kqij}$

Proof.

We note that (4.3) implies

\displaystyle\pi_{1+35}\circ\sigma_{\xi}(DL)(h,X)_{ij}

\displaystyle=|\xi|^{2}h_{ij}-\frac{1}{2}(\xi_{i}\xi_{a}h_{aj}+\xi_{a}\xi_{j}h_{ia})+2\xi_{i}\xi_{m}X_{jm}+2\xi_{j}\xi_{m}X_{im}.

(4.26)

Further, the definitions of the maps $B_{1}$ , $\widetilde{B}$ and equations (4.10), (2.39) give

	$\displaystyle\widetilde{B}(h,X)_{k}$	$\displaystyle=\xi_{i}h_{ik}-\frac{1}{2}\xi_{k}\operatorname{tr}h-2\left(\frac{1}{4}(\xi_{m}h_{km}-\xi_{k}\operatorname{tr}h+\xi_{j}h_{mq}\Phi_{kmjq})+\xi_{m}X_{km}\right)$
		$\displaystyle=\frac{1}{2}\xi_{i}h_{ik}-2\xi_{m}X_{km}$		(4.27)

and hence $\xi_{i}h_{ik}=2\widetilde{B}(h,X)_{k}+4\xi_{m}X_{km}$ . Substituting this in (4.26) gives the first equation in (4.25). We see from (4.3) that

	$\displaystyle\pi_{7}\circ\sigma_{\xi}(DL)(h,X)_{ij}$	$\displaystyle=2\|\xi\|^{2}X_{ij}+\frac{1}{2}(\xi_{j}\xi_{m}h_{mi}-\xi_{i}\xi_{m}h_{mj}+\xi_{k}\xi_{m}h_{mq}\Phi_{kqij}).$	(4.28)
Using the expression for $\xi_{i}h_{ik}=2\widetilde{B}(h,X)_{k}+4\xi_{m}X_{im}$ in (4.28) gives
		$\displaystyle=2\|\xi\|^{2}X_{ij}+\frac{1}{2}\left(\xi_{j}(2\widetilde{B}(h,X)_{i}+4\xi_{m}X_{im})-\xi_{i}(2\widetilde{B}(h,X)_{j}+4\xi_{m}X_{jm})\right.$
		$\displaystyle\qquad\qquad\qquad\qquad\left.+\xi_{k}(2\widetilde{B}(h,X)_{q}+4\xi_{m}X_{qm})\Phi_{kqij}\right),$	(4.29)
which on using the definition of the map $B_{2}$ from (4.23) give
		$\displaystyle=2\|\xi^{2}\|X_{ij}-8(B_{2}(\widetilde{B}(h,X)))_{ij}+2\xi_{j}\xi_{m}X_{im}-2\xi_{i}\xi_{m}X_{jm}+2\xi_{k}\xi_{m}X_{qm}\Phi_{kqij}$

which is the second equation in (4.25). ∎

Remark 4.7.

The operator $\widetilde{B}$ plays a role similar to the role of the Bianchi operator $B_{1}$ in the analysis of the principal symbol of the Ricci tensor, for instance in the Ricci flow and for the analysis of the symbol of large class of flows of $\mathrm{G_{2}}$ -structures as in [DGK23].

Our main goal is to prove that the dimension of kernel of the symbol of the operator $L$ is $8$ which will prove that the diffeomorphism invariance of $\mathrm{Ric}$ and $T$ are the only reason for the failure of parabolicity of (GF) and hence we can use a modified version of the DeTurck’s trick. To calculate an upper bound on $\text{dim}\ \ker(\sigma(DL))$ , we compute the adjoint of the map $\widetilde{B}$ .

Let $Y\in\Gamma(TM)$ . Then

	$\displaystyle\langle\widetilde{B}((h,X)),Y\rangle$	$\displaystyle=\left(\frac{1}{2}\xi_{i}h_{ik}-2\xi_{m}X_{km}\right)Y_{k}$
		$\displaystyle=\frac{1}{4}h_{ik}(\xi_{i}Y_{k}+\xi_{k}Y_{i})-2X_{km}\left(\frac{1}{8}\xi_{m}Y_{k}-\xi_{k}Y_{m}-\frac{1}{8}\xi_{a}Y_{b}\Phi_{abmk}\right),$

where we used (2.6) for $X\in\Omega^{2}_{7}(M)$ . Thus, $\widetilde{B}^{*}:\Omega^{1}\to S^{2}\oplus\Omega^{2}_{7}$ , $\widetilde{B}^{*}(Y)=(\widetilde{B}_{1}^{*}(Y),\widetilde{B}^{*}_{2}(Y))$ with

	$\displaystyle\widetilde{B}^{*}_{1}(Y)_{ij}$	$\displaystyle=\frac{1}{4}(\xi_{i}Y_{k}+\xi_{k}Y_{i})$		(4.30)
	$\displaystyle\widetilde{B}^{*}_{2}(Y)_{ij}$	$\displaystyle=-\frac{1}{4}(\xi_{i}Y_{j}-\xi_{j}Y_{i}-\xi_{a}Y_{b}\Phi_{abij}).$		(4.30)

Lemma 4.8.

The map $\widetilde{B}^{*}\colon\Omega^{1}\to S^{2}\oplus\Omega^{2}_{7}$ is injective. Consequently, $\dim(\ker\widetilde{B})=35$ .

Proof.

Let $Y\in\ker\widetilde{B}^{*}$ . Then $\widetilde{B}^{*}_{1}(Y)=0$ and hence

	$\displaystyle 0$	$\displaystyle=\|\widetilde{B}^{*}_{1}(Y)\|^{2}$
		$\displaystyle=\frac{1}{16}(\xi_{i}Y_{k}+\xi_{k}Y_{i})(\xi_{i}Y_{k}+\xi_{k}Y_{i})$
		$\displaystyle=\frac{1}{8}(\|\xi\|^{2}\|Y\|^{2}+\langle\xi,Y\rangle^{2})$

and since $\xi\neq 0$ , $Y=0$ . This proves that $\widetilde{B}^{*}$ is injective and hence $\text{dim}\ \text{im}(\widetilde{B}^{*})=8$ . Since $S^{2}\oplus\Omega^{2}_{7}=\ker\widetilde{B}^{*}\oplus\text{im}\ \widetilde{B}^{*}$ , we get that $\text{dim}(\ker\widetilde{B})=36+7-8=35$ . ∎

We prove our main result in this section on the dim $\ker(\sigma(DL))$ .

Proposition 4.9.

Consider the operator $L$ from (4.15). Then $\ker(\sigma_{\xi}(DL))=\text{im}(\sigma_{\xi}(\delta^{*}))$ and hence $\text{dim}\ker(\sigma(DL))=8=\text{dim}\ M$ and the failure of parabolicity of (GF) is only due to diffeomorphism invariance of the tensors involved.

Proof.

We observe from Proposition 4.6, equation (4.25) that

\displaystyle\sigma_{\xi}(DL)\mid_{\ker\widetilde{B}}(h,X)_{ij}

\displaystyle=|\xi|^{2}h_{ij}+2|\xi|^{2}X_{ij}+2\xi_{j}\xi_{m}X_{im}-2\xi_{i}\xi_{m}X_{jm}+2\xi_{k}\xi_{m}X_{qm}\Phi_{kqij}

(4.31)

and hence if $(h,X)\in\ker(\sigma_{\xi}(DL))\cap\ker\widetilde{B}$ then $h=0$ and

\displaystyle|\xi|^{2}X_{ij}+\xi_{j}\xi_{m}X_{im}-\xi_{i}\xi_{m}X_{jm}+\xi_{k}\xi_{m}X_{qm}\Phi_{kqij}

\displaystyle=0.

(4.32)

Multiplying both sides by $\xi_{i}$ gives

		$\displaystyle\|\xi\|^{2}\xi_{i}X_{ij}+\xi_{i}\xi_{j}\xi_{m}X_{im}-\|\xi\|^{2}\xi_{m}X_{jm}+\xi_{i}\xi_{k}\xi_{m}X_{qm}\Phi_{kqij}=0$
which on using the fact that $\xi_{i}\xi_{j}$ is symmetric in $i$ and $j$ whereas $X_{ij}$ and $\Phi_{ijkl}$ are skew-symmetric in $i,j$ and $\xi\neq 0$ gives
		$\displaystyle\xi_{i}X_{ij}=0,$

and so $X(\xi)=0$ . Therefore, from (4.32), we get $|\xi|^{2}X_{ij}=0$ and hence $X=0$ . Thus,

\displaystyle\ker(\sigma_{\xi}(DL))\cap\ker\widetilde{B}=\{0\}.

(4.33)

Since $\text{dim}\ker(\widetilde{B})=35$ by Lemma 4.8, we get

\displaystyle\text{dim}\ker(\widetilde{B})\leq 36+7-35=8,

(4.34)

which in combination with (4.22) gives $\text{dim}\ker(\sigma_{\xi}(DL))=8$ . ∎

4.4 A modified DeTurck’s trick

In this section we prove that, given a background Spin(7)-structure $\widetilde{\Phi}$ , it is always possible to modify the operator $L$ from (4.15) to an operator which is strongly elliptic at $\widetilde{\Phi}$ , that is an operator $\widetilde{L}$ whose symbol satisfies

\langle[\sigma_{\xi}(D_{\tilde{\Phi}}\widetilde{L})](h,X),(h,X)\rangle\geq c|\xi|^{2}|(h,X)|^{2}=c|\xi|^{2}(|h|^{2}+|X|^{2})

for some constant $c>0$ . We do this by doing a modification of the DeTurck’s which was originally formulated to give an alternative proof of the short-time existence and uniqueness of solutions to the Ricci flow by DeTurck [DeT83].

Let $\widetilde{\Phi}$ be a fixed Spin(7)-structure on $M$ , for instance, one can take the initial Spin(7)-structure when running the flow. Motivated by the definition of the map $\widetilde{B}$ in (4.24), we define the vector field $W(\Phi,\widetilde{\Phi})$ on $M$ by

\displaystyle W^{k}

\displaystyle=g^{ij}\left(\Gamma^{k}_{ij}-\widetilde{\Gamma}^{k}_{ij}\right)-4(T_{8})^{k}=\widetilde{W}^{k}-4(T_{8})^{k},

(4.35)

where $\widetilde{g}$ is the Riemannian metric induced by $\widetilde{\Phi}$ and $\widetilde{\Gamma}$ is its Christoffel symbols. The vector field $\widetilde{W}$ is the same vector field which is used in the DeTurck’s trick for the Ricci flow and the vector field $-4T_{8}$ is the extra term which we need for the DeTurck’s trick in the Spin(7)-case. We define the modified operator $\widetilde{L}$ as

\displaystyle\widetilde{L}(\Phi)

\displaystyle=L(\Phi)+\mathcal{L}_{W(\Phi,\widetilde{\Phi})}\Phi

(4.36)

where $L$ is the operator in (4.15). Using (2.41) we have

\displaystyle\mathcal{L}_{W}\Phi=\mathcal{L}_{\widetilde{W}-4T_{8}}\Phi=\left(\frac{1}{2}\mathcal{L}_{\widetilde{W}}g-2\mathcal{L}_{T_{8}}g+T(\widetilde{W}-4T_{8})+(\nabla\widetilde{W}-4\nabla T_{8})_{7}\right)\diamond\Phi.

Since we only need the highest order terms for the purpose of short-time existence and uniqueness, we neglect the $T(\widetilde{W}-4T_{8})$ term above and consequently, the operator $\widetilde{L}$ is given by

\displaystyle\widetilde{L}(\Phi)=\left(-\mathrm{Ric}+\frac{1}{2}\mathcal{L}_{\widetilde{W}}g+2\operatorname{div}T+(\nabla\widetilde{W}-4\nabla T_{8})_{7}\right)\diamond\Phi.

(4.37)

We claim that the operator $\widetilde{L}$ is elliptic. We need to calculate the principal symbol of the linearization of $\widetilde{L}$ . It is well known (for example, see [CK04, §3.2]) that the linearization of $\widetilde{W}$ , upto lower order terms, is

\displaystyle(D_{\widetilde{\Phi}}\widetilde{W})(h,X)=2\left(\operatorname{div}_{\widetilde{g}}h-\frac{1}{2}\nabla\operatorname{tr}_{\widetilde{g}}h\right).

The factor of 2 is here because the variation of metric in the Spin(7) case is given by $2h$ . Thus,

	$\displaystyle\sigma_{\xi}\left(\frac{1}{2}D\mathcal{L}_{\widetilde{W}}g\right)(h,X)_{ij}$	$\displaystyle=\sigma_{\xi}\left(\frac{1}{2}\left(\nabla_{i}(2\operatorname{div}h-\nabla\operatorname{tr}h)_{j}+\nabla_{j}(2\operatorname{div}h-\nabla\operatorname{tr}h)_{i}\right)\right)$
		$\displaystyle=\xi_{i}\xi_{m}h_{mj}+\xi_{j}\xi_{m}h_{mj}-\xi_{i}\xi_{j}\operatorname{tr}h,$

which on using (4.13) gives

\displaystyle\sigma_{\xi}\left(D\left(-\mathrm{Ric}+\frac{1}{2}\mathcal{L}_{\widetilde{W}}g\right)\right)

\displaystyle=|\xi|^{2}h_{ij}.

(4.38)

We now calculate the symbol of the remaining terms in (4.37). Since

\displaystyle(\nabla\widetilde{W}_{7})_{ij}

\displaystyle=\frac{1}{8}\nabla_{i}\widetilde{W}_{j}-\frac{1}{8}\nabla_{j}\widetilde{W}_{i}-\frac{1}{8}\nabla_{a}\widetilde{W}_{b}\Phi_{abij}

\displaystyle(D_{\widetilde{\Phi}}\nabla\widetilde{W}_{7})_{ij}

\displaystyle=\frac{1}{8}\nabla_{i}(2\operatorname{div}h-\operatorname{tr}h)_{j}-\frac{1}{8}\nabla_{j}(2\operatorname{div}h-\operatorname{tr}h)_{i}-\frac{1}{8}\nabla_{a}(2\operatorname{div}h-\operatorname{tr}h)_{b}\Phi_{abij}

and hence

	$\displaystyle\sigma_{\xi}(D_{\widetilde{\Phi}}\nabla\widetilde{W}_{7})_{ij}$	$\displaystyle=\frac{1}{8}\xi_{i}(2\xi_{m}h_{mj}-\xi_{j}\operatorname{tr}h)-\frac{1}{8}\xi_{j}(2\xi_{m}h_{mi}-\xi_{i}\operatorname{tr}h)-\frac{1}{8}\xi_{a}(2\xi_{m}h_{mb}-\xi_{b}\operatorname{tr}h)\Phi_{abij}$
		$\displaystyle=\frac{1}{4}\xi_{i}\xi_{m}h_{mj}-\frac{1}{4}\xi_{j}\xi_{m}h_{mi}-\frac{1}{4}\xi_{a}\xi_{m}h_{mb}\Phi_{abij}.$		(4.39)

Similarly,

\displaystyle-4((\nabla T_{8})_{7})_{ij}

\displaystyle=-\frac{1}{2}\nabla_{i}(T_{m;jm})+\frac{1}{2}\nabla_{j}(T_{m;im})+\frac{1}{2}\nabla_{a}(T_{m;bm})\Phi_{abij}

which on using (4.10) gives

$\displaystyle-4(\sigma_{\xi}(D_{\widetilde{\Phi}}(\nabla T_{8})_{7}))_{ij}$	$\displaystyle=-\frac{1}{2}\xi_{i}\left(\frac{1}{4}(\xi_{m}h_{jm}-\xi_{j}\operatorname{tr}h+\xi_{k}h_{mq}\Phi_{jmkq})+\xi_{m}X_{jm}\right)$
	$\displaystyle\quad+\frac{1}{2}\xi_{j}\left(\frac{1}{4}(\xi_{m}h_{im}-\xi_{i}\operatorname{tr}h+\xi_{k}h_{mq}\Phi_{imkq})+\xi_{m}X_{im}\right)$
	$\displaystyle\quad+\frac{1}{2}\xi_{a}\left(\frac{1}{4}(\xi_{m}h_{bm}-\xi_{b}\operatorname{tr}h+\xi_{k}h_{mq}\Phi_{bmkq})+\xi_{m}X_{bm}\right)\Phi_{abij}$
	$\displaystyle=-\frac{1}{8}\xi_{i}\xi_{m}h_{jm}+\frac{1}{8}\xi_{j}\xi_{m}h_{im}+\frac{1}{8}\xi_{a}\xi_{m}h_{bm}\Phi_{abij}$
	$\displaystyle\quad-\frac{1}{2}\xi_{i}\xi_{m}X_{jm}+\frac{1}{2}\xi_{j}\xi_{m}X_{im}+\frac{1}{2}\xi_{a}\xi_{m}X_{bm}\Phi_{abij}.$	(4.40)

Using (4.38), (4.11), (4.4) and (4.40) we compute the symbol of $\widetilde{L}$ as

$\displaystyle\sigma_{\xi}(D\widetilde{L}(\Phi)(h,X))_{ij}$	$\displaystyle=\|\xi\|^{2}h_{ij}+\frac{1}{2}(\xi_{m}\xi_{j}h_{im}-\xi_{m}\xi_{i}h_{mj}+\xi_{m}\xi_{a}h_{mb}\Phi_{ijab})+2\|\xi\|^{2}X_{ij}$
	$\displaystyle\quad+\frac{1}{4}\xi_{i}\xi_{m}h_{mj}-\frac{1}{4}\xi_{j}\xi_{m}h_{mi}-\frac{1}{4}\xi_{a}\xi_{m}h_{mb}\Phi_{abij}-\frac{1}{8}\xi_{i}\xi_{m}h_{jm}+\frac{1}{8}\xi_{j}\xi_{m}h_{im}+\frac{1}{8}\xi_{a}\xi_{m}h_{bm}\Phi_{abij}$
	$\displaystyle\quad-\frac{1}{2}\xi_{i}\xi_{m}X_{jm}+\frac{1}{2}\xi_{j}\xi_{m}X_{im}+\frac{1}{2}\xi_{a}\xi_{m}X_{bm}\Phi_{abij}$
	$\displaystyle=\|\xi\|^{2}h_{ij}+2\|\xi\|^{2}X_{ij}-\frac{3}{8}\xi_{i}\xi_{m}h_{mj}+\frac{3}{8}\xi_{j}\xi_{m}h_{mi}+\frac{3}{8}\xi_{a}\xi_{m}h_{mb}\Phi_{abij}$
	$\displaystyle\quad-\frac{1}{2}\xi_{i}\xi_{m}X_{jm}+\frac{1}{2}\xi_{j}\xi_{m}X_{im}+\frac{1}{2}\xi_{a}\xi_{m}X_{bm}\Phi_{abij}$	(4.41)

Using (4.41) we calculate

	$\displaystyle\left\langle\sigma_{\xi}(D\widetilde{L}(\Phi)(h,X)),(h,X)\right\rangle$	$\displaystyle=\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\xi_{i}\xi_{m}h_{mj}X_{ij}-4\xi_{i}\xi_{m}X_{jm}X_{ij}$
		$\displaystyle=\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\langle h(\xi),X(\xi)\rangle+4\|X(\xi)\|^{2}$
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\|h(\xi)\|\|X(\xi)\|+4\|X(\xi)\|^{2}$
which on using Young’s inequality on the 3rd term gives
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-\frac{1}{2}\|h(\xi)\|^{2}-\frac{9}{2}\|X(\xi)\|^{2}+4\|X(\xi)\|^{2}$
and we use $\|h(\xi)\|^{2}\leq\|h\|^{2}\|\xi\|^{2}$ and $\|X(\xi)\|^{2}\leq\|X\|^{2}\|\xi\|^{2}$ to get
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-\frac{1}{2}\|h\|^{2}\|\xi\|^{2}-\frac{1}{2}\|X\|^{2}\|\xi\|^{2}$
		$\displaystyle\geq\frac{1}{2}\|\xi\|^{2}(\|h\|^{2}+\|X\|^{2})=\frac{1}{2}\|(h,X)\|^{2}.$	(4.42)

The above computations leading to (4.42) and (4.16) prove the following

Proposition 4.10.

Let $(M^{8},\widetilde{\Phi})$ be an $8$ -manifold with a Spin(7)-structure $\widetilde{\Phi}$ and let $\widetilde{L}$ be the operator

\displaystyle\widetilde{L}(\Phi)=L(\Phi)+\mathcal{L}_{W(\Phi,\widetilde{\Phi})}\Phi=\left(-\mathrm{Ric}+2\mathcal{L}_{T_{8}}g+\frac{1}{2}\mathcal{L}_{W}g+2\operatorname{div}T+(\nabla\widetilde{W}-4\nabla T_{8})_{7}\right)\diamond\Phi,

(4.43)

where $W(\Phi,\widetilde{\Phi})^{k}=g^{ij}\left(\Gamma^{k}_{ij}-\widetilde{\Gamma}^{k}_{ij}-4(T_{8})^{k}\right)$ . Then $\widetilde{L}(\Phi)$ is strongly elliptic at $\widetilde{\Phi}$ . ∎

4.5 Short-time existence and uniqueness

In this section, we prove the main theorem of the paper by using the modified DeTurck’s trick whose details were given in § 4.4.

Theorem 4.11.

Let $(M^{8},\Phi_{0})$ be a compact $8$ -manifold with a Spin(7)-structure $\Phi_{0}$ and consider the negative gradient flow (GF) of the natural energy functional $E$ in (3.1). Then there exists $\varepsilon>0$ and a unique smooth solution $\Phi(t)$ of (GF) for $t\in[0,\varepsilon)$ with $\varepsilon=\varepsilon(\Phi_{0})$ .

Proof.

Let $\widetilde{\Phi}=\Phi_{0}$ and define $W(\Phi,\widetilde{\Phi})$ as in (4.35). Since the terms $8T_{b;al}T_{m;lb}-8T_{m;al}T_{b;lb}+2T_{a;lb}T_{m;lb}-|T|^{2}g$ are at most first order in $\Phi$ , we deduce from Proposition 4.10 that the linearization of the operator

\displaystyle\widetilde{L}(\Phi)=\left(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+8T_{b;al}T_{m;lb}-8T_{m;al}T_{b;lb}+2T_{a;lb}T_{m;lb}-|T|^{2}g+2\operatorname{div}T\right)\diamond\Phi+\mathcal{L}_{W}\Phi

is strongly elliptic. We obtain from standard parabolic theory (Theorem 4.3) that there is a unique smooth solution $\widetilde{\Phi}(t)$ for $t\in[0,\varepsilon)$ of the initial value problem

	$\displaystyle\frac{\partial}{\partial t}\bar{\Phi}(t)$	$\displaystyle=\widetilde{L}(\bar{\Phi}(t)),$
	$\displaystyle\bar{\Phi}(0)$	$\displaystyle=\Phi_{0}.$

Let $\varphi_{t}:M\to M,\ t\in[0,\varepsilon)$ be the one-parameter family of diffeomorphisms defined by

	$\displaystyle\frac{\partial}{\partial t}\varphi_{t}$	$\displaystyle=-W(\bar{\Phi}(t),\widetilde{\Phi})\circ\varphi_{t},$
	$\displaystyle\varphi_{0}$	$\displaystyle=\operatorname{Id}_{M}.$

Since $M$ is compact, the family of diffeomorphisms $\varphi_{t}$ exists by [CK04, Lemma 3.15] as long as the solutions $\bar{\Phi}(t)$ exists. The family $\Phi(t)=\varphi^{*}_{t}(\bar{\Phi}(t))$ then satisfies

	$\displaystyle\dfrac{\partial\Phi(t)}{\partial t}=\partial_{t}(\varphi^{*}_{t}(\bar{\Phi}(t)))$	$\displaystyle=\varphi^{*}_{t}\left(\mathcal{L}_{-W(t)}\bar{\Phi}(t)+\partial_{t}\bar{\Phi}(t)\right)$
		$\displaystyle=\varphi^{*}_{t}\left(\left(\mathcal{L}_{-W(t)}\bar{\Phi}(t)-\overline{\mathrm{Ric}}+2\mathcal{L}_{\bar{T}_{8}}\bar{g}+\overline{\text{l.o.t.}}+2\overline{\operatorname{div}}\bar{T}\right)\diamond_{\bar{\Phi}(t)}\bar{\Phi}(t)+\mathcal{L}_{W(t)}\bar{\Phi}(t)\right)$
		$\displaystyle=\left(-\overline{\mathrm{Ric}}+2\mathcal{L}_{\bar{T}_{8}}\bar{g}+\overline{\text{l.o.t.}}+2\overline{\operatorname{div}}\bar{T}\right)\diamond_{(\varphi^{}_{t}(\bar{\Phi}(t)))}(\varphi^{}_{t}(\bar{\Phi}(t)))$
		$\displaystyle=\left(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+8T_{b;al}T_{m;lb}-8T_{m;al}T_{b;lb}+2T_{a;lb}T_{m;lb}-\|T\|^{2}g+2\operatorname{div}T\right)\diamond\Phi$

with $\Phi(0)=\operatorname{Id}^{*}(\bar{\Phi}(0))=\Phi_{0}$ and $\overline{\text{l.o.t}}=8\bar{T}_{b;al}\bar{T}_{m;lb}-8\bar{T}_{m;al}\bar{T}_{b;lb}+2\bar{T}_{a;lb}\bar{T}_{m;lb}-|\bar{T}|^{2}_{\bar{g}}\bar{g}$ . This proves the short-time existence of solutions to (GF).

We now prove uniqueness, by using the uniqueness of solutions to the harmonic map heat flow. Suppose $\Phi_{i}(t),i=1,2$ are solutions to (GF) with the same initial condition. Let $(F_{i})_{t}:M\to M$ be a one-parameter family of diffeomorphisms given as

	$\displaystyle\frac{\partial}{\partial t}(F_{i})_{t}$	$\displaystyle=-4(T_{8})_{\Phi_{i}(t)}\circ(F_{i})_{t},$		(4.44)
	$\displaystyle(F_{i})_{0}$	$\displaystyle=\operatorname{Id}_{M}.$		(4.44)

Using (2.41), we see that the family $\bar{\Phi}_{i}(t)=(F_{i})^{*}_{t}\Phi_{i}(t)$ solves

	$\displaystyle\frac{\partial}{\partial t}\bar{\Phi}_{i}(t)$	$\displaystyle=\left(-\mathrm{Ric}_{\bar{g}_{i}(t)}-4(\bar{\nabla}\bar{T}_{8})_{7}+2\operatorname{div}\bar{T}_{8}+\overline{\text{l.o.t}}\right)\diamond\bar{\Phi}_{i}(t)$		(4.45)
	$\displaystyle\bar{\Phi}_{i}(0)=\Phi_{0}.$			(4.45)

If we set $\hat{\Phi}_{i}(t)=((\varphi_{i})^{-1}_{t})^{*}\bar{\Phi}_{i}(t)$ by defining the diffeomorphisms $\varphi_{i}(t)$ as $\partial_{t}(\varphi_{i})_{t}=-\widetilde{W}(\hat{g}_{i}(t),\widetilde{g})$ then it is known [CK04, §4.3] that $\varphi_{i}(t):M\to M,\ t\in[0,\varepsilon_{i}^{{}^{\prime}})$ is a solution to

	$\displaystyle\frac{\partial}{\partial t}\varphi_{i}(t)$	$\displaystyle=\Delta_{g_{i}(t),g_{0}}(\varphi_{i})(t),$		(4.46)
	$\displaystyle\varphi_{i}(0)$	$\displaystyle=\operatorname{Id}_{M}$		(4.46)

which is the harmonic map heat flow from $(M^{8},g_{0})$ to $(M^{8},g_{i}(t))$ with the initial value being the identity map. Again using (2.41), we see that the maps $\hat{\Phi}_{i}(t)$ satisfy

	$\displaystyle\frac{\partial}{\partial t}\hat{\Phi}_{i}(t)$	$\displaystyle=\left(-\mathrm{Ric}_{\hat{g}_{i}(t)}+\frac{1}{2}\mathcal{L}_{\widetilde{W}(\hat{g}_{i}(t),\widetilde{g})}\hat{g}_{i}(t)+(\hat{\nabla}\widetilde{W}-4\hat{\nabla}\hat{T}_{8})_{7}+2\operatorname{div}\hat{T}_{8}+\widehat{\text{l.o.t.}}\right)\diamond\hat{\Phi}_{i}(t)$		(4.47)
	$\displaystyle\hat{\Phi}_{i}(0)$	$\displaystyle=\Phi_{0}.$		(4.47)

for $t\in[0,\varepsilon_{i}^{{}^{\prime}}).$ Since we have proved above that the operator in (4.47) is parabolic, we know from the standard theory of uniqueness of parabolic equations that $\hat{\Phi}_{1}(t)=\hat{\Phi}_{2}(t)$ for all $t\in[0,\varepsilon^{{}^{\prime}})$ with $\varepsilon^{{}^{\prime}}=\text{min}\{\varepsilon_{1}^{{}^{\prime}},\varepsilon_{2}^{{}^{\prime}}\}$ and as a result $\hat{g}_{1}(t)=\hat{g}_{2}(t)$ which gives $\varphi_{1}(t)=\varphi_{2}(t)$ . Consequently

\displaystyle\bar{\Phi}_{1}(t)=(\varphi_{1})^{*}_{t}\hat{\Phi}_{1}(t)=(\varphi_{2})^{*}_{t}\hat{\Phi}_{2}(t)=\bar{\Phi}_{2}(t),\ \ \ \ \ \ \ \ \text{for\ all}\ \ \ t\in[0,\varepsilon^{{}^{\prime}}).

Finally, we have

	$\displaystyle 0=\frac{\partial\operatorname{Id}_{M}}{\partial t}$	$\displaystyle=\frac{\partial}{\partial t}\left((F_{i})_{t}\circ(F_{i})^{-1}_{t}\right)$
		$\displaystyle=\frac{\partial(F_{i})_{t}}{\partial t}\circ(F_{i})^{-1}_{t}+((F_{i})_{t})_{*}\left(\frac{\partial(F_{i})^{-1}_{t}}{\partial t}\right)$
		$\displaystyle=-4(T_{8})_{\Phi_{i}(t)}+((F_{i})_{t})_{*}\left(\frac{\partial(F_{i})^{-1}_{t}}{\partial t}\right)$

which gives

\displaystyle\left(\frac{\partial(F_{i})^{-1}_{t}}{\partial t}\right)=4((F_{i})^{-1}_{t})_{*}(T_{8})_{\Phi_{i}(t)}=4(T_{8})_{(F_{i})^{*}_{t}\Phi_{i}(t)}\circ(F_{i})^{-1}_{t}=4(T_{8})_{\bar{\Phi}_{i}(t)}\circ(F_{i})^{-1}_{t}.

But since we proved above that $\bar{\Phi}_{1}(t)=\bar{\Phi}_{2}(t),$ we get $(F_{1})^{-1}_{t}=(F_{2})^{-1}_{t}$ and hence $\Phi_{1}(t)=\Phi_{2}(t)$ for all $t\in[0,\varepsilon^{{}^{\prime}})$ which proves the uniqueness of solutions to (GF) and completes the proof of the theorem. ∎

Remark 4.12.

A particularly interesting flow of Spin(7)-structures is the "coupling" of the Ricci flow of the metric and the isometric/harmonic flow of Spin(7)-structures

\frac{\partial}{\partial t}\Phi(t)=(-\mathrm{Ric}+\operatorname{div}T)\diamond\Phi.

This flow induces precisely the Ricci flow $\frac{\partial}{\partial t}g=-2\mathrm{Ric}$ on the metric, and the only other thing it does to the Spin(7)-structure $\Phi$ is to deform it by the isometric flow which was studied in detail in [DLE24]. This “coupling” of Ricci flow with the isometric flow has good short-time existence and uniqueness. This can be seen by going through the modified DeTurck’s trick in § 4.4 and choosing $W=\widetilde{W}$ and then checking that the resulting modified flow is strictly parabolic. The isometric flow of Spin(7)-structures has many good properties, in particular there is an almost monotonicity formula for the solutions to the flow (see [DLE24]). It would be interesting to study whether the Ricci flow coupled with the harmonic flow has similar analytic properties.

5. Solitons

Let $M^{8}$ be an $8$ -manifold. A soliton for (GF) is a triple $(\Phi,Y,\lambda)$ with $Y\in\Gamma(TM)$ and $\lambda\in\mathbb{R}$ such that

\displaystyle\left(-\mathrm{Ric}+2(\mathcal{L}_{T_{8}}g)+T*T-|T|^{2}g+2\operatorname{div}T\right)\diamond\Phi=\lambda\Phi+\mathcal{L}_{Y}\Phi

(5.1)

where $(T*T)_{ij}=8T_{b;il}T_{j;lb}-8T_{j;il}T_{b;lb}+2T_{i;lb}T_{j;lb}$ . Those Spin(7)-structures which satisfy (5.1) are special as they give self-similar solutions to (GF). Recall that a self-similar solution to (GF) is a solution of the form

\displaystyle\Phi(t)=\lambda(t)^{4}\varphi(t)^{*}\Phi(0)

where $\lambda(t)$ are time-dependent scalings and $\varphi(t):M\to M$ are a one-parameter family of diffeomorphisms. Note that the power of $4$ on $\lambda(t)$ is just for convenience in calculations. A straightforward calculation (for instance see [DGK21, Lemma 2.17], [DLE24, Prop. 2.11] or [FLME22, Prop. 1.55]) shows that the solitons for (GF) and self-similar solutions are in one-to-one correspondence.

We say a soliton $(\Phi,Y,\lambda)$ is expanding if $\lambda>0$ ; steady if $\lambda=0$ ; and shrinking if $\lambda<0$ .

We now derive the condition satisfied by the metric $g$ induced by $\Phi$ and $\operatorname{div}T$ when $(\Phi,Y,\lambda)$ is a soliton, which we expect to have further use.

Proposition 5.1.

Let $(\Phi,Y,\lambda)$ be a solitons as defined in (5.1). Then the induced metric $g$ satisfies

\displaystyle-R_{ij}+2(\mathcal{L}_{T_{8}}g)_{ij}+8T_{b;il}T_{j;lb}-8T_{j;il}T_{b;lb}+2T_{i;lb}T_{j;lb}-|T|^{2}g_{ij}=\frac{\lambda}{4}g_{ij}+\frac{1}{2}(\mathcal{L}_{Y}g)_{ij},

(5.2)

and $\operatorname{div}T$ satisfies

\displaystyle(\operatorname{div}T)_{ij}=\frac{1}{2}\left(T(Y)+(\nabla Y)_{7}\right)_{ij}.

(5.3)

Proof.

The definition of the $\diamond$ operator in (2.13) implies $\Phi=\left(\dfrac{g}{4}\diamond\Phi\right)$ . Moreover, (2.41) gives $\mathcal{L}_{Y}\Phi=\left(\frac{1}{2}\mathcal{L}_{Y}g+T(Y)+(\nabla Y)_{7}\right)\diamond\Phi$ . Putting these expressions in (5.1) and using the fact that $\Omega^{4}_{1+35}$ is orthogonal to $\Omega^{4}_{7}$ gives (5.2) and (5.3). ∎

We can prove the non-existence theorem for compact expanding solitons of (GF).

Proposition 5.2.

1.

There are no compact expanding solitons of (GF).
2.

The only compact steady solitons of (GF) are given by torsion-free Spin(7)-structures.

Proof.

Taking the trace of (5.2) gives

\displaystyle-R+4\operatorname{div}T_{8}+8T_{b;il}T_{i;lb}+8|T_{8}|^{2}+2|T|^{2}-8|T|^{2}=2\lambda+\operatorname{div}Y

which on using the expression for the scalar curvature (2.34) simplifies to

\displaystyle-4\operatorname{div}T_{8}-6|T|^{2}=2\lambda+\operatorname{div}Y.

(5.4)

Integrating (5.4) on compact $M$ gives

\displaystyle-3\int_{M}|T|^{2}\operatorname{vol}=\lambda\text{Vol}(M).

So $\lambda\leq 0$ and $\lambda=0$ if and only if $T\equiv 0$ . ∎

References

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Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin.
dwivedis@hu-berlin.de

	$\displaystyle\left\langle\sigma_{\xi}(D\widetilde{L}(\Phi)(h,X)),(h,X)\right\rangle$	$\displaystyle=\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\xi_{i}\xi_{m}h_{mj}X_{ij}-4\xi_{i}\xi_{m}X_{jm}X_{ij}$
		$\displaystyle=\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\langle h(\xi),X(\xi)\rangle+4\|X(\xi)\|^{2}$
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-3\|h(\xi)\|\|X(\xi)\|+4\|X(\xi)\|^{2}$
which on using Young’s inequality on the 3rd term gives
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-\frac{1}{2}\|h(\xi)\|^{2}-\frac{9}{2}\|X(\xi)\|^{2}+4\|X(\xi)\|^{2}$
and we use $\|h(\xi)\|^{2}\leq\|h\|^{2}\|\xi\|^{2}$ and $\|X(\xi)\|^{2}\leq\|X\|^{2}\|\xi\|^{2}$ to get
		$\displaystyle\geq\|\xi\|^{2}\|h\|^{2}+2\|\xi\|^{2}\|X\|^{2}-\frac{1}{2}\|h\|^{2}\|\xi\|^{2}-\frac{1}{2}\|X\|^{2}\|\xi\|^{2}$
		$\displaystyle\geq\frac{1}{2}\|\xi\|^{2}(\|h\|^{2}+\|X\|^{2})=\frac{1}{2}\|(h,X)\|^{2}.$	(4.42)