[datatype=bibtex] [fieldsource=pmid, fieldtarget=pubmed]

#  Harmonic flow of $\mathrm{Spin}(7)$-structures

Shubham Dwivedi, Eric Loubeau & Henrique Sá Earp

(August 27, 2024)

###### Abstract

We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on
compact $8$-manifolds, as an instance of the harmonic flow of geometric
structures. Starting from a general perspective, we establish Shi-type
estimates and a correspondence between harmonic solitons and self-similar
solutions for arbitrary isometric flows of $H$-structures. We then specialise
to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for long-
time existence, via a monotonicity formula along the flow, which leads to an
$\varepsilon$-regularity theorem. Moreover, we prove Cheeger-Gromov and
Hamilton-type compactness theorems for the solutions of the harmonic flow, and
we characterise Type-$\mathrm{I}$ singularities as being modelled on shrinking
solitons. We also establish a Bryant-type description of isometric
$\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of
independent interest.

toc

## 1\. Introduction

We formulate and study the harmonic flow of $\mathrm{Spin}(7)$-structures on a
compact, oriented and spinnable $8$-manifold. Since the flow runs among
isometric $\mathrm{Spin}(7)$-structures, this problem fits naturally in the
recent thread of activity on isometric/harmonic flows of
$\mathrm{G}_{2}$-structures Grigorian2017,Grigorian2019,Bagaglini2019,dgk-
isometric and of almost complex structures He2019,He2019a. These can be
understood as instances of an abstract Dirichlet gradient flow of sections of
a natural homogeneous fibre bundle of $H$-structures, according to the general
theory developed in [LS19].

By contrast, to our knowledge, there have yet been no studies exploring _any_
flow of $\mathrm{Spin}(7)$-structures, ever since the fundamental reference
[Kar08]. We organise therefore the exposition around the harmonic
$\mathrm{Spin}(7)$-flow, even though several analytic results are actually
obtained for arbitrary $H$-structures, and can thus be seen as advances in the
general theory of _harmonic section flows_ formulated in [LS19]. Namely, in
§2.1, we describe deformations of $H$-structures, along the lines of [Kar08].
This leads to a general definition of harmonic $H$-solitons, as well as their
correspondence to self-similar solutions in Proposition 2.7, which generalises
[DGK21, §2.5]. Moreover, we compute Weitzenböck and Bochner formulas for
harmonic sections [§2.2.2], and use them to establish Shi-type estimates
[Proposition 2.16] for the gradient flow of the Dirichlet energy functional in
(2.1), which are instrumental in the study of the long-time behaviour of the
harmonic $\mathrm{Spin}(7)$-flow. While our approach builds upon previous
broad-ranging results on the harmonic flow of geometric structures from
[LS19], at times it may be effortlessly formulated in the even ampler class of
sections of Riemannian submersions with totally geodesic fibres.

Readers especially interested in the $\mathrm{Spin}(7)$ narrative may wish to
skip directly to §3. Throughout the text, a _$\mathrm{Spin}(7)$ -structure
manifold $(M^{8},\Phi,g)$_ consists of a smooth oriented and spinnable
$8$-manifold $M$, endowed with a $\mathrm{Spin}(7)$-structure
$\Phi\in\Omega^{4}(M)$ which induces the Riemannian metric $g$.

We begin by reviewing a number of well-known facts about irreducible
$\mathrm{Spin}(7)$-representations within differential forms and spinors. In
§3.3, we establish a Bryant-type formula for isometric
$\mathrm{Spin}(7)$-structures as sections of a $\mathbb{RP}^{7}$-bundle; this
formula emulates a well-known result for isometric $\mathrm{G}_{2}$-structures
on a $7$-manifold [Bry06, (3.6)], yet it does not seem to appear anywhere in
the literature in the following explicit form:

######  Theorem A.

Let $(M^{8},\Phi_{0},g_{0})$ be a $\mathrm{Spin}(7)$-structure manifold. Then
any other $\mathrm{Spin}(7)$-structure inducing the same Riemannian metric
$g_{0}$ can be parametrised by a function $f\in C^{\infty}(M)$ and a vector
field $X\in\mathscr{X}(M)$ such that $f^{2}+|X|^{2}=1$:

$\Phi_{(f,X)}=(f^{2}-|X|^{2})\Phi_{0}+2f\Theta_{0,X}+8(X\wedge(X\lrcorner\Phi_{0})),$

where the $4$-form $\Theta_{0,X}(a,b,c,d):=\Phi_{0}(a\cdot X,b,c,d)$ is
defined by spinorial multiplication.

In §4, we deduce the specific form of the harmonic $\mathrm{Spin}(7)$-flow, as
an instance of the general harmonic flow of geometric structures. Consider the
negative gradient flow of the functional
$16E(\Phi)=8\|T_{\Phi}\|^{2}_{L^{2}(M)}$ [cf. (4.1)], restricted to the
isometry class $\llbracket\Phi_{0}\rrbracket$, in which $T_{\Phi}$ denotes the
full torsion tensor. By Lemma 4.4, the evolution of $\Phi$ is determined,
somewhat similarly to the $\mathrm{G}_{2}$ case, by the divergence of torsion:

######  Definition 1.1 (Harmonic $\mathrm{Spin}(7)$-flow).

Let $(M^{8},\Phi_{0},g)$ be a compact $\mathrm{Spin}(7)$-structure manifold.
The _harmonic $\mathrm{Spin}(7)$-flow_ is the following initial value problem:

$\displaystyle\left\\{\begin{array}[]{rl}&\dfrac{\partial\Phi}{\partial
t}=\operatorname{div}T\diamond\Phi\\\ &\Phi(0)=\Phi_{0}\end{array}\right.,$
(HF)

where $\diamond$ denotes the infinitesimal action of $\operatorname{End}{TM}$
on $\Omega^{4}(M)$ [Definition 2.2].

As a consequence, we obtain short-time existence and uniqueness of solutions
effortlessly from the general theory in [LS19, Theorem 1]. Moreover, after
some elementary inquiry into deformations of $H$-structures, we obtain a
general correspondence between harmonic $H$-solitons and self-similar
solutions [§4.2.1], which in the case of $\mathrm{Spin}(7)$ reads as follows:

######  Theorem B.

Suppose that a $\mathrm{Spin}(7)$-structure $\hat{\Phi}$ on $M$, a vector
field $\hat{X}\in\mathscr{X}(M)$ and $c\in\\{-1,0,1\\}$ define a harmonic
soliton [Definition 4.8]. Then there exist:

  * •

a function $\rho\in C^{\infty}(\mathbb{R},\mathbb{R}^{+})$ such that
$\rho(0)=1$,

  * •

a function $\alpha\in C^{\infty}(\operatorname{Dom}(\alpha),\mathbb{R})$, an
instant $\hat{t}\in\operatorname{Dom}(\alpha)$ such that $\alpha(\hat{t})=1$,
and an interval $\hat{I}\subset\operatorname{Dom}(\alpha)$ containing both
$\\{\hat{t}\\}$ and $\\{{\hat{t}}.\infty\\}$,

such that the $1$-parameter family $\\{f_{t}\\}\subset\operatorname{Diff}{M}$
defined by

$\displaystyle\left\\{\begin{array}[]{rl}\frac{d}{dt}f_{t}&=\alpha(t)\hat{X}\circ
f_{t}\\\ f_{\hat{t}}&=\operatorname{Id}_{M}\end{array}\right.,\quad
t\in\hat{I},$ (1.4)

induces a self-similar solution [Definition 4.6] of the (HF) by
$\Phi_{t}:=\rho^{4}(t)f_{t}^{*}\hat{\Phi}$. Moreover, the family of vector
fields $\\{X_{t}:=\alpha(t)\hat{X}\\}\subset\mathscr{X}(M)$ satisfies the
_harmonic $\mathrm{Spin}(7)$-soliton_ condition [Lemma 4.7]:

$\operatorname{div}{T_{t}}=X_{t}\lrcorner
T_{t}+\nabla_{7}(X_{t}),\quad\textup{for\ all}\ t\in\hat{I}.$

Finally, the data $(\rho,\alpha,\hat{t})$ can be explicitly constructed, cf.
Remark 2.12.

The analytic core of the paper is developed in §§4 and 5, following closely
the study of the harmonic $\mathrm{G}_{2}$-flow in [DGK21] and the methods
therein. We prove a general Cheeger–Gromov-type compactness for
$\mathrm{Spin}(7)$-structures in Theorem 4.18, and then use it to prove a
Hamilton-type compactness theorem for solutions of (HF) in Theorem 4.19.
Furthermore, we establish a monotonicity formula along the flow [Theorem 5.2],
which leads to $\varepsilon$-regularity [Theorem 5.5]. We introduce moreover
an _entropy_ functional for the solutions of (HF). Then indeed small initial
torsion implies long-time existence and convergence to a critical point and,
using Corollary 5.6, we show that small initial entropy eventually implies
pointwise small torsion. In summary, the following is proved in Theorems 5.9
and 5.10:

######  Theorem C (small torsion and entropy).

In the context of Definition 1.1, the following hold:

  1. (i)

For every $\varepsilon>0$ and $\sigma>0$, there exists a constant
$\lambda_{\varepsilon}=\lambda_{\varepsilon}(g,\sigma)>0$ such that, if the
entropy (5.3) satisfies

$\displaystyle\lambda(\Phi_{0},\sigma)<\lambda_{\varepsilon},$ (1.5)

then $|T_{\Phi(\tau)}|<\varepsilon$, where $\tau$ is the maximal existence
time of (HF).

  2. (ii)

For every $\delta>0$, there exists $\varepsilon(\delta,g)>0$ such that, if
$|T_{\Phi_{0}}|<\varepsilon$, then (HF) exists for all time and converges
subsequentially smoothly to a $\mathrm{Spin}(7)$-structure $\Phi_{\infty}$
satisfying

$\operatorname{div}T_{\Phi_{\infty}}=0,\quad|T_{\Phi_{\infty}}|<\delta,\quad|\nabla^{k}T_{\Phi_{\infty}}|<C_{k},\quad\forall\
k\geq 1,$

for constants $C_{k}<+\infty$ depending only on $(M,g)$.

Moving on to the analysis of singularities for the flow in §5.3.2, we obtain
an upper bound on the “size” of the singular set $S$, which is closed:

######  Theorem D.

In the context of Definition 1.1, let

$\displaystyle
E(\Phi_{0})=\frac{1}{2}\int_{M}|T_{\Phi_{0}}|^{2}\operatorname{vol_{g}}\leq
E_{0}.$ (1.6)

Suppose that the maximal smooth harmonic $\mathrm{Spin}(7)$-flow
$\\{\Phi(t)\\}_{t\in[0,\tau)}$ starting at $\Phi_{0}$ blows up at time
$\tau<+\infty$. Then, as $t\rightarrow\tau$, (HF) converges smoothly to a
$\mathrm{Spin}(7)$-structure $\Phi_{\tau}$ away from a closed set $S$, with
finite $6$-dimensional Hausdorff measure satisfying

$\displaystyle\mathcal{H}^{6}(S)\leq CE_{0},$

for some constant $C<\infty$ depending on $g$. In particular, the Hausdorff
dimension of $S$ is at most 6.

Finally, in §5.3.3, we discuss Type-i@ singularities and prove that they are
modelled on shrinking harmonic solitons, cf. 4.2.1.

Notation and Conventions. We use the symbol $*$ to denote various tensor
contractions, the precise form of which is unimportant, and thus we use
$\star$ instead for the Hodge star operator. The symbol $C$ denotes some
positive real constant, which may change from line to line in the derivation
of an estimate. We frequently use Young’s inequality
$ab\leq\tfrac{1}{2\varepsilon}a^{2}+\tfrac{\varepsilon}{2}b^{2}$, for any
$\varepsilon,a,b>0$.

Throughout the paper, we compute in a local orthonormal frame
$\left\\{\frac{\partial}{\partial x^{k}}\right\\}$, all indices are subscripts
and repeated indices are summed from $1$ to $8$. The symbol $\Delta$ always
denotes the _analyst’s Laplacian_ $\Delta=\nabla_{k}\nabla_{k}$ which is the
opposite of the _rough Laplacian_ $\nabla^{*}\nabla$.

The Riemann curvature and the Ricci tensor are given respectively by

$\displaystyle R_{ijkm}\frac{\partial}{\partial x^{m}}$
$\displaystyle=(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i})\frac{\partial}{\partial
x^{k}},$ $\displaystyle R_{jk}$ $\displaystyle=R_{ljkl},$

and the Ricci identity is

$\nabla_{k}\nabla_{i}X_{l}-\nabla_{i}\nabla_{k}X_{l}=-R_{kilm}X_{m}.$ (1.7)

Schematically, for any tensor $S$, the Ricci identity implies that

$\nabla^{m}\Delta
S-\Delta\nabla^{m}S=\sum_{i=0}^{m}\nabla^{m-i}S*\nabla^{i}\mathrm{R}.$ (1.8)

The Riemannian second Bianchi identity is

$\nabla_{i}R_{jkab}+\nabla_{j}R_{kiab}+\nabla_{k}R_{ijab}=0,$

which contracts in $i,a$ to

$\nabla_{i}R_{ibjk}=\nabla_{k}R_{jb}-\nabla_{j}R_{kb}.$ (1.9)

Acknowledgements: We thank Robert Bryant for suggesting the approach via
squares of spinors, leading to Theorem A. We are also grateful to BIRS-CMO,
for providing the excellent working environment where this project originated.
SD would like to thank Spiro Karigiannis and Panagiotis Gianniotis, for
discussions on various aspects of geometric flows, and Thomas Walpuski, for
discussions on the spinorial description of $\mathrm{Spin}(7)$-structures.

EL and HSE benefited from an ongoing CAPES-COFECUB bilateral collaboration
(2018-2021), granted by the Brazilian Coordination for the Improvement of
Higher Education Personnel (CAPES) – Finance Code 001 [88881.143017/2017-01],
and COFECUB [MA 898/18], and from a CAPES-MathAmSud (2021-2022) grant
[88881.520221/2020-01].

HSE has also been funded by the São Paulo Research Foundation (Fapesp)
[2018/21391-1] and the Brazilian National Council for Scientific and
Technological Development (CNPq) [307217/2017-5].

## 2\. $H$-Structures, homogeneous sections and harmonic flow

In many respects, $\mathrm{Spin}(7)$-structures should be seen as a particular
occurrence of $H$-structures and some of the analytical properties of the
harmonic flow are best formulated in the general setting of sections of
Riemannian submersions with totally geodesic fibres.

In §2.1, we briefly review the unifying abstract framework required to define
the harmonicity of $H$-structures, by mediation of their corresponding
sections, describe their deformation theory and classify the solitons of the
harmonic section flow. For further details, the reader is kindly referred to
[LS19] and references therein. Subsequently, in §2.2, we adopt the wider
perspective of sections of Riemannian submersions in order to prove a very
general Shi-type estimate, yielding fractional-polynomial time-bounds on all
derivatives of sections along the flow.

###  2.1 $H$-Structures

A _geometric structure_ defined by an $H$-invariant tensor
$\xi\in\mathcal{T}^{p,q}(M)$ on a manifold $M$ can be viewed as a section of
the homogeneous fibre bundle $\pi:N:=P/H\to M$, by reduction of the frame
bundle $P:=\mathrm{Fr}(M)$ and a one-to-one correspondence [see (2.6)]:

$\\{\xi:M\to\mathcal{T}^{p,q}(M)\\}\quad\leftrightarrow\quad\\{\sigma:M\to
N\\}.$

Assuming $M$ compact, a Riemannian metric $g$ on $M$ defines a suitable fibre
metric $\eta$ on $N$, thus assigning a Dirichlet energy to such sections
$\sigma\in\Gamma(N)$:

$E(\sigma):=\frac{1}{2}\int_{M}|d^{\mathcal{V}}\sigma|_{\eta}^{2}\operatorname{vol_{g}},$
(2.1)

where the _vertical torsion_ $d^{\mathcal{V}}\sigma$ is the projection of
$d\sigma$ onto the distribution $\mathcal{V}=\ker\pi_{*}\subset TN$. An
$H$-structure is said to be _harmonic_ if $\sigma\in\operatorname{Crit}({E})$,
and _torsion-free_ if $d^{\mathcal{V}}\sigma=0$. The critical set is the
vanishing locus of the _vertical tension_ field
$\tau^{\mathcal{V}}(\sigma):=\operatorname{tr}_{g}\nabla
d^{\mathcal{V}}\sigma$, where $\nabla^{\mathcal{V}}$ is the vertical part of
the Levi-Civita connection of $(N,\eta)$. The explicit form of
$\tau^{\mathcal{V}}$, in each particular context, defines a natural geometric
PDE.

A sufficient setup for the development of this general theory consists of the
following data:

$\bullet$ $G$ a compact semi-simple Lie group, with natural bi-invariant
metric $\eta$; $\bullet$ $H\subset(G,\eta)$ a normal reductive Lie subgroup;
$\bullet$ $P$ a principal $G$-bundle over a closed Riemannian manifold
$(M^{n},g)$; $\bullet$ $\eta$ an $H$-invariant fibre metric on $P$,
constructed from any compatible metric on $G$. ($\mathcal{A}$)

######  Definition 2.1.

Under the assumptions ($\mathcal{A}$), the _harmonic section flow_ is the
negative gradient flow associated to the Dirichlet energy (2.1):

$\left\\{\begin{array}[]{rl}\displaystyle\partial_{t}\sigma_{t}&=\tau^{\mathcal{V}}(\sigma_{t})\\\
\sigma_{t}|_{t=0}&=\sigma\in\Gamma(\pi)\end{array}\right.,\quad\text{on}\quad
M_{{\rm T}}:=M\times\left[0,{\rm T}\right[.$ (HSF)

We will see below that, by mediation of the _universal section_ $\Xi$ [cf.
(2.6)], to each family $\\{\sigma_{t}\\}$ solving (HSF), there corresponds a
family of $H$-structures $\\{\xi_{t}:=\sigma_{t}^{*}\Xi\\}$.

####  2.1.1 The geometric setting of $H$-structures

We describe $H$-structures as sections of an ad-hoc homogeneous fibre bundle.
If $p:P\to M$ is a principal $G$-bundle, with Lie group $G$, and $H\subseteq
G$, denote by $q:P\to N$ the corresponding principal $H$-bundle and
$\pi:N:=P/H\to M$ the projection. We assume $H$ to be naturally reductive,
i.e. the existence of an orthogonal complement $\mathfrak{m}$ satisfying

$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}\quad\text{and}\quad\mathrm{Ad}_{G}(H)\mathfrak{m}\subseteq\mathfrak{m}.$

A connection $\omega\in\Omega^{1}(P,\mathfrak{g})$ on $P$ (e.g. if
$P=\mathrm{Fr}(M)$), will induce a splitting

$TN=\mathcal{V}\oplus\mathcal{H}$

with $\mathcal{V}:=\ker\pi_{*}=q_{*}(\ker p_{*})$ and
$\mathcal{H}:=q_{*}(\ker\omega).$

Let $\underline{\mathfrak{m}}\to N$ be the vector bundle associated to $q$
with fibre $\mathfrak{m}$, points of which are the $H$-equivalence classes
defined by the infinitesimal action of $w\in\mathfrak{m}$ on $z\in P$:

$z\bullet w:=[(z,w)]_{H}=\mathcal{I}(q_{*}(w_{z}^{*})),$

where $w_{z}^{*}$ is a fundamental left-invariant vector field.

$\textstyle{\hbox to0.0pt{\hss$z\in$
}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{{\underline{\mathfrak{m}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox
to0.0pt{ $:=P\times_{H}\mathfrak{m}\ni z\bullet w$\hss}}$$\textstyle{\hbox
to0.0pt{\hss$y\in$
}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{I}}$$\textstyle{\hbox
to0.0pt{\hss$x\in$ }M}$

This defines a vector bundle isomorphism

$\begin{array}[]{rcccl}\mathcal{I}&:&\mathcal{V}&\tilde{\rightarrow}&\underline{\mathfrak{m}}\\\
&&q_{*}(w_{z}^{*})&\mapsto&z\bullet w\end{array}$ (2.2)

and the $\mathfrak{m}$-component
$\omega_{\mathfrak{m}}\in\Omega^{1}(P,\mathfrak{m})$ of the connection is
$H$-equivariant and $q$-horizontal, so it projects to a _homogeneous
connection form_ $f\in\Omega^{1}(N,\underline{\mathfrak{m}})$ defined by:

$f(q_{*}(Z)):=z\bullet\omega_{\mathfrak{m}}(Z)\quad\text{for}\quad Z\in
T_{z}P.$ (2.3)

On $\pi$-vertical vectors, $f$ coincides with the canonical isomorphism (2.2),
while $\pi$-horizontal vectors are in the kernel:

$f(v_{y})=\mathcal{I}(v_{y}^{\mathcal{V}}),\quad\text{for}\quad v_{y}\in
T_{y}N.$ $\textstyle{\hbox to0.0pt{\hss$v_{z}\in$
}TP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{*}}$$\scriptstyle{p_{*}}$$\scriptstyle{\omega}$$\scriptstyle{\omega_{\mathfrak{m}}}$$\textstyle{\mathfrak{g}=\mathfrak{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\oplus\mathfrak{h}}$$\scriptstyle{z\bullet}$$\textstyle{{\underline{\mathfrak{m}}}}$$\textstyle{TN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\scriptstyle{f}$$\textstyle{\mathcal{V}\hbox
to0.0pt{
$\oplus\;\mathcal{H}$\hss}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{I}}$$\textstyle{TM}$

Let $\mathcal{F}$ be a rank $r$ sub-bundle of $\bigoplus\mathcal{T}^{p,q}(M)$,
with fibre $V=\mathbb{R}^{r}$. When $P=\mathrm{Fr}(M)$, we have a natural
monomorphism of principal bundles
$\rho:P\hookrightarrow\mathrm{Fr}(\mathcal{F})$, which identifies, at each
$x\in M$, the element $z_{x}\in P_{x}$ with a frame of $\mathcal{F}_{x}$,
i.e., with a linear isomorphism onto the typical fibre:

$\rho(z_{x}):\mathcal{F}_{x}\tilde{\to}\;V.$

A section $\xi\in\Gamma(\mathcal{F})$ is a _geometric structure_ , modelled on
a fixed element $\xi_{0}\in V$, if, for any $x\in M$, there is a frame of
$T_{x}M$ identifying $\xi(x)$ and $\xi_{0}$. Suppose now $H\subseteq
G=\mathrm{SO}^{g}(n)$ fixes the model structure $\xi_{0}$:

$H=\operatorname{Stab}(\xi_{0}).$ (2.4)

In view of (2.4), a _universal section_ $\Xi\in\Gamma(N,\pi^{*}(\mathcal{F}))$
is well-defined by

$\displaystyle\Xi(y):=y^{*}\xi_{0}.$ (2.5)

Explicitly, one assigns to $y\in N$ the vector of $\mathcal{F}_{\pi(y)}$ whose
coordinates are given by $\xi_{0}$ in any frame $\rho(z_{\pi(y)})$.

$\textstyle{\pi^{*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}p}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\Xi}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$

Now to each section $\sigma\in\Gamma(M,N)$ one associates a geometric
structure $\xi\in\Gamma(M,\mathcal{F})$ modelled on $\xi_{0}$ by

$\displaystyle\xi_{\sigma}:=\sigma^{*}\Xi=\Xi\circ\sigma,$ (2.6)

and, conversely, to a given geometric structure $\xi\in\Gamma(M,\mathcal{F})$,
stabilised by $H$, one associates, at $x\in M$, an $H$-class of frames of
$T_{x}M$ which, in turn, identifies an element of $\pi^{-1}(x)$, i.e. an
element $\sigma(x)$ in the fibre of $\pi:N\to M$ over $x\in M$.

####  2.1.2 Deformations of $H$-structures

We establish some fundamental facts on the general evolution of an
$H$-structure $\xi$ under $\mathrm{GL}(m,\mathbb{R})$-action. Notably, we
identify the role of the stabiliser of $\xi$ and its implications on the
evolution of the metric. A key result for the ensuing study of solitons is the
action on $\xi$ of the metric itself.

Separating symmetric (and traceless therein) and skew-symmetric parts of
$(1,1)$-tensors, we have an orthogonal splitting

$\displaystyle\operatorname{End}(TM)=\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus
S_{0}\oplus\Omega^{2}.$

Identifying the irreducible components
$\mathfrak{h}\simeq\Omega^{2}_{\mathfrak{h}}$ and
$\mathfrak{m}\simeq\Omega^{2}_{\mathfrak{m}}$ in
$\Omega^{2}\simeq\mathfrak{so}(\dim M)$, we further decompose $\Omega^{2}(M)$
as

$\displaystyle\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus
S_{0}\oplus\Omega^{2}_{\mathfrak{m}}\oplus\Omega^{2}_{\mathfrak{h}}.$ (2.7)

With respect to this splitting, we can write $A=\frac{1}{\dim
M}(\operatorname{tr}A)g+A_{0}+A_{\mathfrak{m}}+A_{\mathfrak{h}}$ where $A_{0}$
is a symmetric traceless $2$-tensor.

In view of (2.4) and (2.6), the orthogonal projection along
$\Omega^{2}_{\mathfrak{h}}$ can be geometrically interpreted as selecting the
non-trivial infinitesimal deformations of $\xi$ as an $H$-structure. To see
that more clearly, write $\xi\in\mathcal{T}^{p,q}$ in local coordinates
$\\{x_{1},\dots,x_{m}\\}$:

$\xi=\sum\xi^{i_{1}\dots i_{p}}_{j_{1}\dots j_{q}}\frac{\partial}{\partial
x_{i_{1}}}\otimes\dots\otimes\frac{\partial}{\partial x_{i_{p}}}\otimes
dx_{j_{1}}\otimes\dots\otimes dx_{j_{q}}.$

A local change of frame $\mathrm{g}\in\mathrm{GL}(m,\mathbb{R})$ acts on
$T_{p}M$ and on $T^{*}_{p}M$, respectively, by

$\begin{array}[]{rrll}\mathrm{g}:&T_{p}M&\to&T_{p}M\\\
&X&\mapsto&\mathrm{g}^{-1}.X\end{array}\quad\text{and}\quad\begin{array}[]{rrll}\mathrm{g}:&T^{*}_{p}M&\to&T^{*}_{p}M\\\
&\alpha&\mapsto&\mathrm{g}.\alpha.\end{array}$

By extension, $\mathrm{g}$ acts on $\mathcal{T}^{p,q}$ by

$\displaystyle\mathrm{g}.\xi=\sum\xi^{i_{1}\dots i_{p}}_{j_{1}\dots
j_{q}}\mathrm{g}^{-1}.\frac{\partial}{\partial
x_{i_{1}}}\otimes\dots\otimes\mathrm{g}^{-1}.\frac{\partial}{\partial
x_{i_{p}}}\otimes\mathrm{g}.dx_{j_{1}}\otimes\dots\otimes\mathrm{g}.dx_{j_{q}}.$
(2.8)

Differentiating the action of $e^{tA}\in\mathrm{GL}(m,\mathbb{R})$ at the
identity, with $A\in\mathfrak{gl}(m,\mathbb{R})$ and
$t\in\,]-\varepsilon,\varepsilon[$, we have:

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}e^{tA}.\xi=\sum\xi^{i_{1}\dots
i_{p}}_{j_{1}\dots j_{q}}\sum_{r,s=1}^{p,q}\Big{\\{}$
$\displaystyle-\frac{\partial}{\partial x_{i_{1}}}\otimes\dots\otimes
A.\frac{\partial}{\partial
x_{i_{r}}}\otimes\dots\otimes\frac{\partial}{\partial x_{i_{p}}}\otimes
dx_{j_{1}}\otimes\dots\otimes dx_{j_{q}}$
$\displaystyle+\frac{\partial}{\partial
x_{i_{1}}}\otimes\dots\otimes\frac{\partial}{\partial x_{i_{p}}}\otimes
dx_{j_{1}}\otimes\dots\otimes A.dx_{j_{s}}\otimes\dots\otimes
dx_{j_{q}}\Big{\\}}.$ (2.9)

######  Definition 2.2.

Let $\xi\in\mathcal{T}^{p,q}$ be a tensor on $M$ and
$A\in\operatorname{End}(TM)$, and define the _infinitesimal deformations_ of
$\xi$ by:

$\begin{array}[]{rcl}\diamond\,\xi:\quad\operatorname{End}(TM)&\to&\mathcal{F}=\mathcal{T}^{p,q}\\\
A&\mapsto&A\diamond\xi:=\left.\frac{d}{dt}\right|_{t=0}e^{tA}.\xi.\end{array}$

The _net degree_ of $\xi$ is defined as the difference $\ell:=q-p$ between its
numbers of covariant and contravariant indices.

######  Proposition 2.3.

Let $H\subset G=\mathrm{SO}(m)$. Let $A\in\mathfrak{gl}(m,\mathbb{R})$ act on
an $H$-structure $\xi\in\mathcal{F}=\mathcal{T}^{p,q}$ by (2.1.2). Then, in
terms of the splitting (2.7), the following assertions hold:

  1. 1.

$\ker(\diamond\,\xi)\supseteq\Omega^{2}_{\mathfrak{h}}$.

  2. 2.

A general $GL(m,\mathbb{R})$-variation of $\xi$ can be written as
$\frac{d}{dt}\xi=A\diamond\xi$, for some $A=B+C$, with $B\in S^{2}(M)$ and
$C\in\Omega^{2}_{\mathfrak{m}}$.

  3. 3.

If $g_{t}$ is a co-evolving metric on $M$, and
$H\subset\mathrm{SO}^{g_{t}}(m)$ along the flow, then $\frac{d}{dt}g_{t}=2B$.
Equivalently, any isometric variation of $\xi$ is characterised by
$A=C\in\Omega^{2}_{\mathfrak{m}}$.

###### Proof.

Item 1. follows from Definition 2.2, and 2. is a trivial consequence of 1. and
(2.7). The first assertion in 3. can be proved following [Kar08, Prop 3.1] for
the general case; the second assertion is then a consequence of 2.. ∎

In particular, if $A$ is the metric tensor, i.e. $A_{ij}=\delta_{ij}$ in
normal coordinates at a given point, then

$A.\frac{\partial}{\partial x_{i_{r}}}=\frac{\partial}{\partial
x_{i_{r}}};\quad A.dx_{j_{s}}=dx_{j_{s}}$

so indeed:

######  Lemma 2.4.

Let $g$ be a Riemannian metric and $\ell$ be the net degree of $\xi$; then

$g\diamond\xi=\ell.\xi.$

######  Lemma 2.5.

The torsion $T=d^{\mathcal{V}}\sigma$ is identified with an element of
$\Omega^{1}(M,\sigma^{*}\mathcal{F}_{\mathfrak{m}})$ by the relation

$\displaystyle\nabla_{Y}\xi$ $\displaystyle=(Y\lrcorner
T)\diamond\xi,\quad\forall\,Y\in\mathscr{X}(M).$ (2.10)

###### Proof.

We know from [LS19, Lemma 1] that $\nabla_{Y}\Xi=\mathrm{f}(Y).\Xi$, where
$\mathrm{f}\in\Omega^{1}(N,\underline{\mathfrak{m}})$ is the homogeneous
connection form associated to the connection
$\omega\in\Omega^{1}(P,\mathfrak{g})$ and the splitting
$TN=\mathcal{V}\oplus\mathcal{H}$:

$\mathrm{f}(q_{*}(Z))=z\bullet\omega_{\mathfrak{m}}(Z)\quad\text{for}\quad
Z\in T_{z}P,z\in P.$

Since $\xi=\Xi\circ\sigma$, we have

$\displaystyle\nabla_{Y}\xi$
$\displaystyle=\nabla_{Y}(\Xi\circ\sigma)=(\nabla_{d\sigma(Y)}\Xi)\circ\sigma$
$\displaystyle=f(d\sigma(Y)).\Xi\circ\sigma=\mathcal{I}(d^{\mathcal{V}}\sigma(Y)).\xi$
$\displaystyle=(Y\lrcorner T)\diamond\xi.\qed$

Given $Y\in\mathscr{X}(M)$, in view of the splitting (2.7) of
$\operatorname{End}(TM)$, we may write

$\nabla
Y=\frac{1}{2}\mathcal{L}_{Y}g+\nabla_{\mathfrak{m}}(Y)+\nabla_{\mathfrak{h}}(Y),$

where $\nabla_{k}(Y):=\pi_{k}(\nabla Y)\in\Omega^{2}_{k}$, in particular
$\nabla_{\mathfrak{h}}Y\in\Omega^{2}_{\mathfrak{h}}\subset\ker(\diamond\,\xi)$.

######  Lemma 2.6.

In terms of the torsion $T=d^{\mathcal{V}}\sigma$, the Lie derivative of a
$H$-structure is

$\displaystyle\mathcal{L}_{Y}\xi$ $\displaystyle=(Y\lrcorner
T+\frac{1}{2}\mathcal{L}_{Y}g+\nabla_{\mathfrak{m}}(Y))\diamond\xi.$ (2.11)

###### Proof.

If $\xi$ is a $(p,q)$-form then, for $X_{i}\in\Gamma(TM)$ and
$\sigma_{j}\in\Gamma(T^{*}M)$,

$\displaystyle(\mathcal{L}_{Y}\xi)(X_{i},\sigma_{j})=$ $\displaystyle\
Y(\xi(X_{i},\sigma_{j}))-\xi(\mathcal{L}_{Y}X_{i},\sigma_{j})-\xi(X_{i},\mathcal{L}_{Y}\sigma_{j})$
$\displaystyle=$ $\displaystyle\
(\nabla_{Y}\xi)(X_{i},\sigma_{j})+\xi(\nabla_{Y}X_{i},\sigma_{j})+\xi(X_{i},\nabla_{Y}\sigma_{j})$
$\displaystyle-\xi(\nabla_{Y}X_{i},\sigma_{j})+\xi(\nabla_{X_{i}}Y,\sigma_{j})-\xi(X_{i},\mathcal{L}_{Y}\sigma_{j})$

but $\mathcal{L}_{Y}\sigma_{j}=\nabla_{Y}\sigma_{j}+\sigma_{j}(\nabla Y),$ so

$\displaystyle(\mathcal{L}_{Y}\xi)(X_{i},\sigma_{j})$
$\displaystyle=(\nabla_{Y}\xi)(X_{i},\sigma_{j})+\xi(\nabla_{X_{i}}Y,\sigma_{j})-\xi(X_{i},\sigma_{j}(\nabla
Y)).$

On the other hand, by (2.10),

$\displaystyle(\nabla_{Y}\xi)(X_{i},\sigma_{j})$ $\displaystyle=((Y\lrcorner
T)\diamond\xi)(X_{i},\sigma_{j})$

while, using normal coordinates,

$\displaystyle\xi(\nabla_{X_{i}}Y,\sigma_{j})=\xi(\frac{1}{2}(\mathcal{L}_{Y}g)(X_{i}),\sigma_{j})+\xi((\nabla_{\mathfrak{m}}Y)(X_{i}),\sigma_{j})+\xi((\nabla_{\mathfrak{h}}Y)(X_{i}),\sigma_{j})$
$\displaystyle=\frac{1}{2}(\mathcal{L}_{Y}g)\Big{(}X_{i},\frac{\partial}{\partial
x_{k}}\Big{)}\xi\Big{(}\frac{\partial}{\partial
x_{k}},\sigma_{j}\Big{)}+g\Big{(}(\nabla_{\mathfrak{m}}Y)(X_{i}),\frac{\partial}{\partial
x_{k}}\Big{)}\xi\Big{(}\frac{\partial}{\partial
x_{k}},\sigma_{j}\Big{)}+g\Big{(}(\nabla_{\mathfrak{h}}Y)(X_{i}),\frac{\partial}{\partial
x_{k}}\Big{)}\xi\Big{(}\frac{\partial}{\partial x_{k}},\sigma_{j}\Big{)}$

and

$\displaystyle\xi(X_{i},\sigma_{j}(\nabla
Y))=\xi(X_{i},\frac{1}{2}(\mathcal{L}_{Y}g)(\sigma_{j}))+\xi(X_{i},(\nabla_{\mathfrak{m}}Y)(\sigma_{j}))+\xi(X_{i},(\nabla_{\mathfrak{h}}Y)(\sigma_{j}))$
$\displaystyle=\frac{1}{2}(\mathcal{L}_{Y}g)(\sigma_{j},dx_{k})\xi(X_{i},dx_{k})+(\nabla_{\mathfrak{m}}Y)(\sigma_{j},dx_{k})\xi(X_{i},dx_{k})+(\nabla_{\mathfrak{h}}Y)(\sigma_{j},dx_{k})\xi(X_{i},dx_{k}).$

So putting these three terms together, and keeping in mind Definition 2.2:

$\displaystyle\mathcal{L}_{Y}\xi$ $\displaystyle=(Y\lrcorner
T)\diamond\xi+\frac{1}{2}(\mathcal{L}_{Y}g)\diamond\xi+\nabla_{\mathfrak{m}}Y\diamond\xi+\nabla_{\mathfrak{h}}Y\diamond\xi.$

Now $H$ is the stabiliser of $\xi$, so the last term vanishes:
$\nabla_{\mathfrak{h}}Y\diamond\xi=0$. ∎

####  2.1.3 Harmonic $H$-solitons and self-similar solutions

The objective of this sub-section is to prove that solitons, i.e. generalised
fixed points of the flow, are the same thing as self-similar solutions, in the
sense that they are preserved by a family of diffeomorphisms parametrised by
time.

######  Definition 2.7.

Let $\\{\sigma_{t}\\}$ be a solution of the harmonic section flow (HSF), and
let $\\{\xi_{t}:=\sigma_{t}^{*}\Xi\\}$ be the corresponding family of
$H$-structures. We say that it is a _self-similar solution_ if there exist a
function $\rho(t)$, with $\rho(0)=1$, an $H$-structure $\xi:=\xi_{\sigma}$ and
a family of diffeomorphisms $f_{t}:M\rightarrow M$ such that

$\xi_{t}=\rho(t)^{\ell}{f_{t}}^{*}\xi,\quad\forall\ t\in\,[0,{\rm T}[,$

where $\ell$ is the net degree of $\xi$, cf. Definition 2.2.

Denoting by $\\{W_{t}\\}\subset\mathscr{X}(M)$ the infinitesimal generator of
$\\{f_{t}\\}\subset\operatorname{Diff}{M}$, the _stationary vector field_ of a
self-similar solution is defined by

$X_{t}:=(f_{t}^{-1})_{*}W_{t}\in\mathscr{X}(M),\quad\forall\ t\in\,[0,{\rm
T}[.$ (2.12)

######  Lemma 2.8.

Along the harmonic section flow of a self-similar solution, as in Definition
2.7, we have $g=\rho(t)^{2}f_{t}^{*}g$, hence

$\mathcal{L}_{X_{t}}g=-2(\log\rho)^{\prime}g.$ (2.13)

###### Proof.

Asking that $\xi_{t}$ remains an $H$-structure, for
$H\subset\mathrm{SO}^{g_{t}}(n)$, implies the metric must scale under the
dilation prescribed by self-similarity:

$g_{t}=\rho(t)^{2}f_{t}^{*}g.$

Since the (HSF) is actually isometric, $g_{t}\equiv g$ and so:

$\displaystyle 0$
$\displaystyle=\frac{d}{dt}g_{t}=2(\log\rho)^{\prime}g+\mathcal{L}_{X_{t}}g.\qed$

We can now prove that solitons of the harmonic section flow (HSF) are exactly
self-similar solutions. This bears close analogy to Lemmata 2.15 and 2.17 in
[DGK21], to the proof of which the reader is referred for details of the
following computations.

######  Lemma 2.9.

Given a self-similar solution $\\{\xi_{t}\\}_{t\in\,[0,{\rm T}[,}$ of the
harmonic section flow (HSF), the stationary vector field $X_{t}$ and the
torsion $T_{t}$ satisfy the following relation:

$\operatorname{div}{T_{t}}=X_{t}\lrcorner T_{t}+\nabla_{\mathfrak{m}}(X_{t}),$

where
$\nabla_{\mathfrak{m}}:=\pi_{\mathfrak{m}}\circ\nabla:\mathscr{X}(M)\to\Omega^{2}_{\mathfrak{m}}(M)$.

###### Proof.

Using (HSF), (2.13), (2.11), and the fact that
$g_{t}\diamond\xi_{t}=\ell\xi_{t}$ [Lemma 2.4],

$\displaystyle\operatorname{div}T_{t}\diamond{\xi_{t}}=\frac{d}{dt}\xi_{t}$
$\displaystyle=\ell(\log\rho)^{\prime}\xi_{t}+\mathcal{L}_{X_{t}}\xi_{t}$
$\displaystyle=\ell(\log\rho)^{\prime}\xi_{t}+(X_{t}\lrcorner
T+\frac{1}{2}\mathcal{L}_{X_{t}}g_{t}+\nabla_{\mathfrak{m}}({X_{t}}))\diamond\xi_{t}$
$\displaystyle=(X_{t}\lrcorner
T+\nabla_{\mathfrak{m}}({X_{t}}))\diamond\xi_{t},$

where the first and third terms in the second row cancel out, by (2.13). ∎

######  Definition 2.10.

A _harmonic $H$-soliton_ on a Riemannian manifold $(M,g)$ is a triple
$(\hat{\xi},\hat{X},c)$, consisting of an $H$-structure $\hat{\xi}$ with
$H\subset\mathrm{SO}^{g}(n)$, a vector field $\hat{X}\in\mathscr{X}(M)$, and
$c\in\mathbb{R}$, such that

$\displaystyle\left\\{\begin{array}[]{rcl}\mathcal{L}_{\hat{X}}g&=&cg\\\
\operatorname{div}{\hat{T}}&=&\hat{X}\lrcorner\hat{T}+\nabla_{\mathfrak{m}}(\hat{X}).\end{array}\right.$
(2.16)

Let us ponder in general whether a given harmonic $H$-soliton
$(\hat{\xi},\hat{X},c)$ induces a self-similar solution, i.e.,
$\xi_{t}=\rho(t)^{\ell}{f_{t}}^{*}\xi_{0}$, for some given function $\rho$ and
$\xi_{0}=\hat{\xi}$. For a real function $\alpha$ and instant
$\hat{t}\in\mathbb{R}$ to be determined, such that $\alpha(\hat{t})=1$,
consider the $1$-parameter family of diffeomorphisms $\\{f_{t}:M\to M\\}$
defined by

$\displaystyle\left\\{\begin{array}[]{rcl}\frac{d}{dt}f_{t}&=&\alpha(t)\hat{X}\circ
f_{t}\\\ f_{\hat{t}}&=&\operatorname{Id}_{M},\end{array}\right.$

and the vector field $W_{t}:=\alpha(t)\hat{X}$. In particular,
$\frac{d}{dt}f_{t}^{*}=\alpha f_{t}^{*}\circ\mathcal{L}_{\hat{X}}.$ Now
$g_{t}:=\rho(t)^{2}f_{t}^{*}g_{0}$ satisfies:

$\displaystyle 0=\frac{d}{dt}g_{t}$
$\displaystyle=\\{(\log\rho^{2})^{\prime}+\alpha c\\}g_{t}$
$\displaystyle\Leftrightarrow\qquad\alpha(t)$
$\displaystyle=-\frac{2}{c}(\log\rho)^{\prime},\quad\text{for}\quad c\neq 0.$

Imposing $\rho(0)=1=\alpha(\hat{t})$, this is solved, for instance, by

$\displaystyle\rho(t)=|t|^{\frac{1}{2}}$
$\displaystyle\quad\text{and}\quad\alpha(t)=\mp\frac{1}{t},\quad\text{for}\quad
c=\pm 1=:-\hat{t},$ $\displaystyle\rho(t)=1$
$\displaystyle\quad\text{and}\quad\alpha(t)=1,\quad\text{for}\quad
c=0=:\hat{t},$

but also by more general rates (see Remark 2.12 below).

We now wish to determine the stationary family
$\\{X_{t}\\}\subset\mathscr{X}(M)$ predicted in Lemma 2.9:

$\operatorname{div}{T_{t}}=X_{t}\lrcorner T_{t}+\nabla_{\mathfrak{m}}(X_{t}).$

We begin by observing that the vector field $\hat{X}$, which generates the
soliton’s translational symmetry, is itself constant in time:

$\frac{d}{dt}(f_{t})^{-1}_{*}\hat{X}=\mathcal{L}_{W_{t}}\hat{X}=\mathcal{L}_{\alpha\hat{X}}\hat{X}=0,$

so indeed $(f_{t})^{-1}_{*}\hat{X}=\hat{X}$, for all $t$ along the flow.
Therefore, we obtain the desired vector field by setting
$X_{t}:=(f_{t})^{-1}_{*}W_{t}=\alpha(t)(f_{t})^{-1}_{*}\hat{X}=\alpha(t)\hat{X}$,
as in Lemma 2.9. In summary:

######  Proposition 2.11.

Suppose that an $H$-structure $\hat{\xi}$ on $M$, a vector field
$\hat{X}\in\mathscr{X}(M)$ and $c\in\\{-1,0,1\\}$ define an harmonic soliton.
Then there exist:

  * •

a function $\rho\in C^{\infty}(\mathbb{R},\mathbb{R}^{+})$ such that
$\rho(0)=1$,

  * •

a function $\alpha\in C^{\infty}(\operatorname{Dom}(\alpha),\mathbb{R})$, an
instant $\hat{t}\in\operatorname{Dom}(\alpha)$ such that $\alpha(\hat{t})=1$,
and an interval $\hat{I}\subset\operatorname{Dom}(\alpha)$ containing both
$\\{\hat{t}\\}$ and $\\{\hat{t}\cdot\infty\\}$,

such that the $1$-parameter family $\\{f_{t}\\}\subset\operatorname{Diff}{M}$
defined by

$\displaystyle\left\\{\begin{array}[]{rcl}\frac{d}{dt}f_{t}&=&\alpha(t)\hat{X}\circ
f_{t}\\\ f_{\hat{t}}&=&\operatorname{Id}_{M},\quad
t\in\hat{I},\end{array}\right.$

induces a self-similar solution of the (HSF) by
$\xi_{t}:=\rho(t)^{\ell}f_{t}^{*}\hat{\xi}$. Moreover, the family of vector
fields $\\{X_{t}:=\alpha(t)\hat{X}\\}\subset\mathscr{X}(M)$ satisfies the
harmonic soliton condition of Lemma 2.9:

$\operatorname{div}{T_{t}}=X_{t}\lrcorner
T_{t}+\nabla_{\mathfrak{m}}(X_{t}),\quad\forall\ t\in\hat{I}.$

######  Remark 2.12.

The data $(\rho,\alpha,\hat{t})$ in Proposition 2.11 can be explicitly
constructed in the following way:

$c=\pm 1$:

Take any $(\rho,\alpha,\hat{t})$ solving

$\alpha(t)=-\frac{2}{c}(\log\rho)^{\prime}(t)\quad\text{and}\quad\rho(0)=1=\alpha(\hat{t}).$

For instance, given any $p>0$,

$\rho(t)=|t|^{p},\quad\alpha(t)=\mp\frac{2p}{t},\quad\hat{t}=-2pc,\quad\text{and}\quad\hat{I}=\begin{cases}]-\infty,\hat{t}],&c=1;\\\
[\hat{t},+\infty[,&c=-1,\end{cases}$

$c=0$:

take $\rho,\alpha\equiv 1$, $\hat{t}=0$ and $\hat{I}=\mathbb{R}$.

###  2.2 Harmonic sections

In this sub-section, we take a slight detour through Riemannian submersions
$\pi:(N,\eta)\to(M,g)$ with totally geodesic fibres, and their sections
$\sigma:M\to N$. The kernel of $d\pi$ defines a vertical distribution in $TN$
and the metric $\eta$ defines the horizontal sub-bundle as its orthogonal
complement.

If $M$ is compact, the (vertical) energy is then the $L^{2}$-norm of the
vertical component of $d\sigma$:

$E^{\mathcal{V}}(\sigma)=\frac{1}{2}\int_{M}|d^{\mathcal{V}}\sigma|^{2}\,\operatorname{vol_{g}}.$

This defines a variational principle for sections of $\pi$ and its critical
points, coined _harmonic sections_ , are characterised, following the same
formal principles as in §2.1, by the Euler-Lagrange equation:
$\tau^{\mathcal{V}}(\sigma)=0.$ Since $\pi$ is a Riemannian submersion with
totally geodesic fibres, $\tau^{\mathcal{V}}(\sigma)$ is actually the vertical
part of the tension field $\tau(\sigma)$ from harmonic maps theory
Wood1986,Wood1997. The analytic study of the corresponding heat equation

$\displaystyle\frac{d\sigma_{t}}{dt}=\tau^{\mathcal{V}}(\sigma_{t})$ (2.17)

aims at finding sufficient conditions for convergence towards a harmonic
section. This will require the local expression of the vertical second
fundamental form, which will lead to a Weitzenböck formula on the swapping of
derivatives for the vertical tension field and, taking inner-products, an
adapted Bochner formula. Under bounded geometry, we obtain a Bochner-type
inequality along the harmonic flow.

As a consequence of these formulas, we prove wide-ranging Shi-type estimates
on the derivatives of solutions of the harmonic flow. This result naturally
holds for $H$-structures and will be exploited for
$H=\mathrm{Spin}(7)\subset\mathrm{SO}{(8)}$ later on.

####  2.2.1 Vertical second fundamental form

To compute the vertical second fundamental form of a section
$\sigma\in\Gamma(\pi)$, choose a point $p\in M$ and local coordinates
$x_{1},\dots,x_{m}$ on a domain $U$ around $p$ and $y_{1},\dots,y_{n}$ local
coordinates on $N$ such that $y_{i}=x_{i}\circ\pi$ for $i\leq m$ and for all
$x\in U$, $y_{m+1}|_{\pi^{-1}(x)},\dots,y_{n}|_{\pi^{-1}(x)}$ are local
coordinates111This part greatly benefited from discussions with C.M. Wood. at
$\pi^{-1}(x)$.

For $i\leq m$, let $\partial_{i}$ be the coordinate field for $x_{i}$ and
$\tilde{\partial}_{a}$ the coordinate field for $y_{a}$ and for $1\leq i\leq
m$, we have $d\pi(\tilde{\partial}_{i})={\partial}_{i}$, while
$\tilde{\partial}_{r}$ is vertical for $m+1\leq r\leq n$ (but
$\tilde{\partial}_{i}$ is not, a priori, horizontal). We assume summation on
repeated indices and take the index convention that $1\leq i,j,k,l\leq m$ and
$m+1\leq u,v,s,p,r,t\leq n$.

In this coordinate system, the differential of the section $\sigma$ is
$d\sigma\Big{(}\frac{\partial}{\partial
x_{i}}\Big{)}=\frac{\partial\sigma^{u}}{\partial
x_{i}}\tilde{\partial}_{u}+\tilde{\partial}_{i}$, and its vertical
differential is therefore

$\displaystyle d^{\mathcal{V}}\sigma\Big{(}\frac{\partial}{\partial
x_{i}}\Big{)}$ $\displaystyle=\Big{(}\frac{\partial\sigma^{u}}{\partial
x_{i}}+\eta_{is}\eta^{su}\Big{)}\tilde{\partial}_{u}.$

An easy computation shows that

$\displaystyle|d^{\mathcal{V}}\sigma|^{2}$
$\displaystyle=g^{ij}\eta_{uv}\frac{\partial\sigma^{u}}{\partial
x_{i}}\frac{\partial\sigma^{v}}{\partial
x_{j}}+2g^{ij}\eta_{iv}\frac{\partial\sigma^{v}}{\partial
x_{j}}+g^{ij}\eta_{is}\eta_{ju}\eta^{su}.$

The vertical second fundamental form is

$\displaystyle\alpha(X,Y)$
$\displaystyle=(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(X,Y)=\nabla_{X}^{\sigma^{-1}\mathcal{V}}(d^{\mathcal{V}}\sigma(Y))-(d^{\mathcal{V}}\sigma)(\nabla^{M}_{X}Y)$

and, writing $\frac{\partial}{\partial x_{i}}=\partial_{i}$, we compute
henceforth $\alpha(\partial_{i},\partial_{j})$:

$\displaystyle\alpha(\partial_{i},\partial_{j})=\Big{(}\frac{\partial^{2}\sigma^{u}}{\partial
x_{i}\partial x_{j}}+\frac{\partial}{\partial
x_{i}}(\eta_{js}\eta^{su})\Big{)}\tilde{\partial}_{u}+\Big{(}\frac{\partial\sigma^{u}}{\partial
x_{j}}+\eta_{js}\eta^{su}\Big{)}\nabla_{d\sigma(\partial_{i})}^{\mathcal{V}}\tilde{\partial}_{u}-\Gamma^{k}_{ij}\Big{(}\frac{\partial\sigma^{u}}{\partial
x_{k}}+\eta_{ks}\eta^{su}\Big{)}\tilde{\partial}_{u}.$

But
$\tilde{\Gamma}^{k}_{uv}=dy^{k}(\nabla_{\tilde{\partial}_{v}}\tilde{\partial}_{u})=0$,
because the fibres are totally geodesic, so

$\displaystyle\nabla_{d\sigma(\partial_{i})}^{\mathcal{V}}\tilde{\partial}_{u}$
$\displaystyle=\frac{\partial\sigma^{v}}{\partial
x_{i}}\tilde{\Gamma}^{w}_{uv}\tilde{\partial}_{w}+\Big{(}\tilde{\Gamma}^{k}_{iu}\eta_{ks}\eta^{sw}+\tilde{\Gamma}^{w}_{iu}\Big{)}\tilde{\partial}_{w}$

since
$\tilde{\partial}^{\mathcal{V}}_{k}=\eta_{ks}\eta^{sw}\tilde{\partial}_{w}$.
Now, denoting by $\tilde{\partial}^{z}_{i}$ the horizontal part of
$\tilde{\partial}_{i}$,

$\displaystyle\frac{\partial}{\partial x_{i}}(\eta_{js}\eta^{su}\circ\sigma)$
$\displaystyle=(\tilde{\partial}^{z}_{i})(\eta_{js}\eta^{su})+(\eta_{is}\eta^{sv}+\frac{\partial\sigma^{v}}{\partial
x_{i}})\tilde{\partial}_{v}(\eta_{js}\eta^{su}).$ (2.18)

Similarly,

$\displaystyle\tilde{\Gamma}^{r}_{ij}$
$\displaystyle=dy^{r}(\nabla_{\tilde{\partial}_{i}}\tilde{\partial}_{j})=dy^{r}(\nabla_{\tilde{\partial}^{z}_{i}}\tilde{\partial}^{z}_{j})+\tilde{\partial}^{z}_{i}(\eta_{js}\eta^{sr})+\eta_{js}\eta^{sv}\tilde{\Gamma}^{r}_{iv}-\eta_{js}\eta^{sv}\eta_{iw}\eta^{wu}\tilde{\Gamma}^{r}_{uv}+\eta_{is}\eta^{su}\tilde{\Gamma}^{r}_{ju}.$

By O’Neill’s connection formulas for Riemannian submersions [ONe66], and
denoting also by $\mathcal{V}$ (resp. $\mathcal{H}$) the projection onto the
vertical (resp. horizontal) distribution, we have

$\displaystyle
dy^{r}(\mathcal{H}\nabla_{\tilde{\partial}^{z}_{i}}\tilde{\partial}^{z}_{j})$
$\displaystyle=-\Gamma^{k}_{ij}\eta_{ks}\eta^{sr},\quad\mathcal{V}\nabla_{\tilde{\partial}^{z}_{i}}\tilde{\partial}^{z}_{j}=\frac{1}{2}\mathcal{V}[\tilde{\partial}^{z}_{i},\tilde{\partial}^{z}_{j}],$

hence
$dy^{r}(\nabla_{\tilde{\partial}^{z}_{i}}\tilde{\partial}^{z}_{j})=\frac{1}{2}dy^{r}(\mathcal{V}[\tilde{\partial}^{z}_{i},\tilde{\partial}^{z}_{j}])-\Gamma^{k}_{ij}\eta_{ks}\eta^{sr}$.

For the other terms, first observe that

$\displaystyle\nabla_{\tilde{\partial}_{u}}\tilde{\partial}_{j}$
$\displaystyle=\nabla_{\tilde{\partial}_{u}}\tilde{\partial}^{z}_{j}+\tilde{\partial}_{u}(\eta_{jv}\eta^{vw})\tilde{\partial}_{w}+\eta_{jv}\eta^{vw}\nabla_{\tilde{\partial}_{u}}\tilde{\partial}_{w}$

and applying $dy^{w}$ we get

$\displaystyle\tilde{\Gamma}^{w}_{uj}$
$\displaystyle=dy^{w}(\nabla_{\tilde{\partial}_{u}}\tilde{\partial}_{j})=dy^{w}(\nabla_{\tilde{\partial}_{u}}\tilde{\partial}^{z}_{j})+\tilde{\partial}_{u}(\eta_{jv}\eta^{vw})+\eta_{jv}\eta^{vt}\tilde{\Gamma}^{w}_{tu}.$

Since the fibres are totally geodesic, O’Neill’s tensor $T$ vanishes and
$\mathcal{V}\nabla_{\tilde{\partial}_{u}}\tilde{\partial}^{z}_{j}=0$.
Therefore, as $\tilde{\partial}^{z}_{j}$ is basic, one has
$\nabla_{\tilde{\partial}_{u}}\tilde{\partial}^{z}_{j}=\tilde{\Gamma}^{k}_{ju}\tilde{\partial}^{z}_{k}$
and, as we already know that
$dy^{w}(\tilde{\partial}^{z}_{k})=-\eta_{kv}\eta^{vw}$, we obtain

$\displaystyle dy^{w}(\nabla_{\tilde{\partial}_{u}}\tilde{\partial}^{z}_{j})$
$\displaystyle=-\eta_{kv}\eta^{vw}\tilde{\Gamma}^{k}_{ju}.$

In conclusion, the vertical second fundamental form is

$\displaystyle\alpha(\partial_{i},\partial_{j})=$ $\displaystyle\
\Big{\\{}\frac{\partial^{2}\sigma^{u}}{\partial x_{i}\partial
x_{j}}-\Gamma^{k}_{ij}\frac{\partial\sigma^{u}}{\partial
x_{k}}+\tilde{\Gamma}^{u}_{vw}\frac{\partial\sigma^{v}}{\partial
x_{i}}\frac{\partial\sigma^{w}}{\partial x_{j}}$ (2.19)
$\displaystyle+\frac{\partial\sigma^{v}}{\partial
x_{j}}\Big{(}\tilde{\Gamma}^{u}_{iv}+\eta_{ks}\eta^{su}\tilde{\Gamma}^{k}_{iv}\Big{)}+\eta_{js}\eta^{sv}\eta_{kw}\eta^{wu}\tilde{\Gamma}^{k}_{iv}$
$\displaystyle+\frac{\partial\sigma^{v}}{\partial
x_{i}}\Big{(}\tilde{\Gamma}^{u}_{jv}+\eta_{ks}\eta^{su}\tilde{\Gamma}^{k}_{jv}\Big{)}+\eta_{is}\eta^{sv}\eta_{kw}\eta^{wu}\tilde{\Gamma}^{k}_{jv}$
$\displaystyle+\Big{(}\tilde{\Gamma}^{u}_{ij}-\frac{1}{2}dy^{u}(\mathcal{V}[\tilde{\partial}^{z}_{i},\tilde{\partial}^{z}_{j}])\Big{)}\Big{\\}}\tilde{\partial}_{u}$

with
$-2g_{kj}\eta^{uv}\tilde{\Gamma}^{k}_{iu}=dy^{v}(\mathcal{V}[\tilde{\partial}^{z}_{i},\tilde{\partial}^{z}_{j}]).$

######  Remark 2.13.

The vertical second fundamental form of a submersion, unlike its usual
counterpart, is not symmetric. In some cases, the non-symmetric part of
$\alpha$ can be expressed in terms of curvature, e.g. if $\mathcal{H}$ is a
$\mathrm{GL}(n)$-connection on a vector bundle, then it is equal to
$\frac{1}{2}(\pi^{*}R)$, see [Bes08, 9.53, 9.54 and 9.65] and [ONe66, §5].
When $\pi:N\to M$ is a homogeneous fibre bundle, the skew-symmetric part of
vertical second fundamental form is $\frac{1}{2}\sigma^{*}\Phi$, where $\Phi$
is the curvature two-form on $N$ [Woo03, Thm 3.5].

Since $\pi$ is a Riemannian submersion, we have that
$\tau^{\mathcal{V}}(\sigma)=\operatorname{tr}{\alpha}$, so tracing the above
expression yields the vertical tension field of $\sigma$:

$\displaystyle\tau^{\mathcal{V}}(\sigma)=$ $\displaystyle\
g^{ij}\Big{\\{}\frac{\partial^{2}\sigma^{u}}{\partial x_{i}\partial
x_{j}}+\tilde{\Gamma}^{u}_{rs}\frac{\partial\sigma^{r}}{\partial
x_{i}}\frac{\partial\sigma^{s}}{\partial
x_{j}}-\Gamma^{k}_{ij}\frac{\partial\sigma^{u}}{\partial x_{k}}$
$\displaystyle+2\Big{(}\tilde{\Gamma}^{u}_{ir}\frac{\partial\sigma^{r}}{\partial
x_{j}}+\tilde{\Gamma}^{k}_{ir}\Big{(}\frac{\partial\sigma^{r}}{\partial
x_{j}}+\eta_{js}\eta^{sr}\Big{)}\eta_{kw}\eta^{wu}\Big{)}+\tilde{\Gamma}^{u}_{ij}\Big{\\}}\tilde{\partial}_{u},$

where $1\leq i,j,k\leq m$ and $m+1\leq r,s,u,w\leq n$.

######  Remark 2.14.

To obtain the equation of the harmonic flow, work with the Cartesian product
$M\times\mathbb{R}$ and write $\frac{\partial\sigma}{\partial t}$ as
$d\sigma(\partial_{t})$. Then, since each $\sigma_{t}$ is a section and
$\sigma^{l}$ is constant,

$\displaystyle\frac{\partial\sigma}{\partial t}$
$\displaystyle=d\sigma(\partial_{t})=\frac{\partial\sigma^{u}}{\partial
t}\tilde{\partial}_{u}+\frac{\partial\sigma^{l}}{\partial
t}\tilde{\partial}_{l}=\frac{\partial\sigma^{u}}{\partial
t}\tilde{\partial}_{u}=d^{\mathcal{V}}\sigma(\partial_{t}).$

####  2.2.2 Weitzenböck and Bochner formulas

We assume the coordinates $\\{x_{i}\\}$ are normal at the point $p$, so that
$\partial_{k}(g^{ij})|_{p}=\frac{\partial g^{ij}}{\partial x_{k}}|_{p}=0$, and
evaluate all quantities at $p$. Given a section $\sigma:M\to N$, we denote by
$\sigma^{-1}\mathcal{V}$ the pullback of the vertical distribution to the base
manifold $M$. By standard arguments, we have a Weitzenböck formula for the
vertical tension:

$\displaystyle\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\tau^{\mathcal{V}}(\sigma)=$
$\displaystyle\
\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\Big{(}g^{ij}\alpha(\partial_{i},\partial_{j})\Big{)}$
$\displaystyle=$ $\displaystyle\
g^{ij}\Big{\\{}R^{\sigma^{-1}\mathcal{V}}(\partial_{k},\partial_{i})(d^{\mathcal{V}}\sigma(\partial_{j}))+(d^{\mathcal{V}}\sigma)(R^{M}(\partial_{i},\partial_{k})\partial_{j})$
$\displaystyle+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\nabla_{\partial_{j}}^{\sigma^{-1}\mathcal{V}}(d^{\mathcal{V}}\sigma(\partial_{k}))+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}\alpha(\partial_{k},\partial_{j})-\alpha(\partial_{j},\partial_{k})\Big{)}-(d^{\mathcal{V}}\sigma)(\nabla_{\partial_{i}}\nabla_{\partial_{j}}\partial_{k})\Big{\\}}.$

Recall from (2.19) that

$\displaystyle\alpha(\partial_{k},\partial_{j})-\alpha(\partial_{j},\partial_{k})$
$\displaystyle=\mathcal{V}([\tilde{\partial}^{z}_{j},\tilde{\partial}^{z}_{k}])=2g_{ij}\eta^{uv}\tilde{\Gamma}^{i}_{ku}\tilde{\partial}_{v},$

so that
$\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}\alpha(\partial_{k},\partial_{j})-\alpha(\partial_{j},\partial_{k})\Big{)}$
is a curvature term.

Let us now compute the Laplacian of the energy density:

$\displaystyle-\Delta e(\sigma)$
$\displaystyle=-g^{ij}\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}\frac{|d^{\mathcal{V}}\sigma|^{2}}{2}$
$\displaystyle=-\frac{1}{2}g^{ij}\Bigg{[}g^{kl}(\nabla^{2}_{\partial_{i},\partial_{j}}\eta)(d^{\mathcal{V}}\sigma(\partial_{k}),d^{\mathcal{V}}\sigma(\partial_{l}))+4g^{kl}(\nabla_{\partial_{j}}\eta)(\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{i}}(d^{\mathcal{V}}\sigma(\partial_{k})),d^{\mathcal{V}}\sigma(\partial_{l}))$
$\displaystyle\quad+\frac{\partial^{2}g^{kl}}{\partial x_{i}\partial
x_{j}}\eta(d^{\mathcal{V}}\sigma(\partial_{k}),d^{\mathcal{V}}\sigma(\partial_{l}))$
$\displaystyle\quad+2g^{kl}\eta(\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{i}}(d^{\mathcal{V}}\sigma(\partial_{k})),\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{j}}(d^{\mathcal{V}}\sigma(\partial_{l})))+2g^{kl}\eta(\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{i}}\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{j}}(d^{\mathcal{V}}\sigma(\partial_{k})),d^{\mathcal{V}}\sigma(\partial_{l}))\Bigg{]}$

The covariant derivative of the target metric $\eta$ by the pull-back
connection is

$\displaystyle\nabla_{\partial_{j}}\eta$
$\displaystyle=\frac{\partial\sigma^{u}}{\partial
x_{j}}\nabla_{\tilde{\partial}_{u}}\eta+\nabla_{\tilde{\partial}_{j}}\eta$

and its second covariant derivatives are

$\displaystyle\nabla^{2}_{\partial_{i},\partial_{j}}\eta=$ $\displaystyle\
\frac{\partial\sigma^{u}}{\partial x_{j}}\frac{\partial\sigma^{v}}{\partial
x_{i}}\nabla^{2}_{\tilde{\partial}_{v},\tilde{\partial}_{u}}\eta+\frac{\partial\sigma^{u}}{\partial
x_{j}}\nabla^{2}_{\tilde{\partial}_{i},\tilde{\partial}_{u}}\eta+\frac{\partial\sigma^{u}}{\partial
x_{i}}\nabla^{2}_{\tilde{\partial}_{u},\tilde{\partial}_{j}}\eta+\nabla^{2}_{\tilde{\partial}_{i},\tilde{\partial}_{j}}\eta$
$\displaystyle+\Big{(}\frac{\partial^{2}\sigma^{w}}{\partial x_{i}\partial
x_{j}}+\frac{\partial\sigma^{u}}{\partial
x_{j}}\frac{\partial\sigma^{v}}{\partial
x_{i}}\tilde{\Gamma}^{w}_{uv}+\frac{\partial\sigma^{u}}{\partial
x_{j}}\tilde{\Gamma}^{w}_{ui}+\frac{\partial\sigma^{u}}{\partial
x_{i}}\tilde{\Gamma}^{w}_{uj}+\tilde{\Gamma}^{w}_{ij}\Big{)}\nabla_{\tilde{\partial}_{w}}\eta$
$\displaystyle+\Big{(}\frac{\partial\sigma^{u}}{\partial
x_{j}}\tilde{\Gamma}^{k}_{ui}+\frac{\partial\sigma^{u}}{\partial
x_{i}}\tilde{\Gamma}^{k}_{uj}+\tilde{\Gamma}^{k}_{ij}\Big{)}\nabla_{\tilde{\partial}_{k}}\eta.$

On the other hand

$\displaystyle\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{i}}(d^{\mathcal{V}}\sigma(\partial_{k}))=\nabla^{\sigma^{-1}\mathcal{V}}_{\partial_{i}}\Big{[}\Big{(}\frac{\partial\sigma^{u}}{\partial
x_{k}}+\eta_{ks}\eta^{su}\Big{)}\tilde{\partial}_{u}\Big{]}$
$\displaystyle=\Big{[}\frac{\partial^{2}\sigma^{u}}{\partial x_{i}\partial
x_{k}}+\frac{\partial}{\partial
x_{i}}(\eta_{ks}\eta^{su})+\Big{(}\frac{\partial\sigma^{w}}{\partial
x_{k}}+\eta_{ks}\eta^{sw}\Big{)}\Big{(}\frac{\partial\sigma^{v}}{\partial
x_{i}}\tilde{\Gamma}^{u}_{wv}+\tilde{\Gamma}^{u}_{wi}\Big{)}\Big{]}\tilde{\partial}_{u}+\Big{(}\frac{\partial\sigma^{w}}{\partial
x_{k}}+\eta_{ks}\eta^{sw}\Big{)}\tilde{\Gamma}^{k}_{wi}\tilde{\partial}_{k}.$

Therefore, using the Weitzenböck formula and properties of the Levi-Civita
connection yields a Bochner formula for $\sigma$:

$\displaystyle-\Delta e(\sigma)=$ $\displaystyle\
-\frac{1}{2}g^{ij}g^{kl}\Big{(}\frac{\partial\sigma^{t}}{\partial
x_{k}}+\eta_{ks}\eta^{st}\Big{)}\Big{(}\frac{\partial\sigma^{r}}{\partial
x_{l}}+\eta_{ls}\eta^{sr}\Big{)}\Bigg{[}\frac{\partial\sigma^{u}}{\partial
x_{j}}\frac{\partial\sigma^{v}}{\partial
x_{i}}\nabla^{2}_{\tilde{\partial}_{v},\tilde{\partial}_{u}}\eta+\frac{\partial\sigma^{u}}{\partial
x_{j}}\nabla^{2}_{\tilde{\partial}_{i},\tilde{\partial}_{u}}\eta+\frac{\partial\sigma^{u}}{\partial
x_{i}}\nabla^{2}_{\tilde{\partial}_{u},\tilde{\partial}_{j}}\eta$
$\displaystyle+\nabla^{2}_{\tilde{\partial}_{i},\tilde{\partial}_{j}}\eta\Bigg{]}(\tilde{\partial}_{t},\tilde{\partial}_{r})$
$\displaystyle-\frac{1}{2}g^{ij}\frac{\partial^{2}g^{kl}}{\partial
x_{i}\partial x_{j}}\Big{(}\frac{\partial\sigma^{t}}{\partial
x_{k}}+\eta_{ks}\eta^{st}\Big{)}\Big{(}\frac{\partial\sigma^{r}}{\partial
x_{l}}+\eta_{ls}\eta^{sr}\Big{)}\eta_{tr}-|\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma|^{2}$
$\displaystyle+g^{ij}g^{kl}\eta\Big{(}d^{\mathcal{V}}\sigma(\partial_{l}),R^{\sigma^{-1}\mathcal{V}}(\partial_{k},\partial_{i})(d^{\mathcal{V}}\sigma(\partial_{j}))+(d^{\mathcal{V}}\sigma)(R^{M}(\partial_{i},\partial_{k})\partial_{j})$
$\displaystyle+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}2g_{ij}\eta^{uv}\tilde{\Gamma}^{i}_{ku}\tilde{\partial}_{v}\Big{)}-\Big{(}\frac{\partial\Gamma^{m}_{jk}}{\partial
x_{i}}+\Gamma^{p}_{jk}\Gamma^{m}_{ip}\Big{)}d^{\mathcal{V}}\sigma(\partial_{m})\Big{)}-g^{kl}\eta(\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\tau^{\mathcal{V}}(\sigma),d^{\mathcal{V}}\sigma(\partial_{l})).$

Then, still under no assumption on $\sigma$ being harmonic, we obtain a
Bochner-type inequality:

######  Lemma 2.15.

Let $\sigma:M\to N$ be a section of a Riemannian submersion
$\pi:(N,\eta)\to(M,g)$ between compact manifolds, with totally geodesic
fibres, then there exists a universal constant $C>0$ depending only on $(M,g)$
and $(N,\eta)$ such that

$\displaystyle-\Delta
e(\sigma)\leq-|\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma|^{2}+C\Big{(}e^{2}(\sigma)+e(\sigma)+1\Big{)}-(\nabla\tau^{\mathcal{V}}(\sigma),d^{\mathcal{V}}\sigma).$

####  2.2.3 Shi-type estimates along (HSF)

We obtain global derivative estimates along the harmonic flow, with a
technique used by Shi [Shi89] for the Ricci flow, Lotay–Wei [LW17] for the
Laplacian flow of closed ${\rm G}_{2}$-structures and Dwivedi _et al._ [DGK21]
for the isometric flow of ${\rm G}_{2}$-structures. This will later on allow
us to establish long-time existence conditions of the harmonic
$\mathrm{Spin}(7)$-flow.

######  Proposition 2.16.

Let $\\{\sigma_{t}\\}_{t\in[0,t_{1}[}$ be a solution of the harmonic section
flow (HSF), and assume that there exist $K\geq 1$ and constants $B_{j}$,
$0\leq j\in\mathbb{Z}$, and $D\geq 0$, such that

$|T|\leq K,\quad|\nabla g|,\ |\nabla\eta|\leq
DK,\quad\text{and}\quad|\nabla^{j}R^{M}|,\ |\nabla^{j}R^{N}|\leq
B_{j}K^{j+2},\quad\forall\ j\geq 0.$

Then, for all $m\in\mathbb{N}$, there exist constants $C_{m}=C_{m}(g,\eta)$
such that

$|\nabla^{m}T|\leq C_{m}Kt^{-m/2},\quad\forall\ t\in[0,\frac{1}{K^{4}}].$

###### Proof.

Locally the Weitzenböck formula for $H$-structures becomes

$\displaystyle\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\tau^{\mathcal{V}}(\sigma)=$
$\displaystyle\
g^{ij}\Big{\\{}R^{\sigma^{-1}\mathcal{V}}(\partial_{k},\partial_{i})(d^{\mathcal{V}}\sigma(\partial_{j}))+(d^{\mathcal{V}}\sigma)(R^{M}(\partial_{i},\partial_{k})\partial_{j})+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\nabla_{\partial_{j}}^{\sigma^{-1}\mathcal{V}}(d^{\mathcal{V}}\sigma(\partial_{k}))$
$\displaystyle+2\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}g_{lj}\eta^{uv}\tilde{\Gamma}^{l}_{ku}\tilde{\partial}_{v}\Big{)}-(d^{\mathcal{V}}\sigma)(\nabla_{\partial_{i}}\nabla_{\partial_{j}}\partial_{k})\Big{\\}}+\frac{\partial
g^{ij}}{\partial
x_{k}}\alpha(\partial_{i},\partial_{j})-g^{ij}(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{k},\nabla_{\partial_{i}}\partial_{j}).$

Along the harmonic flow, we have

$\displaystyle\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\frac{\partial\sigma}{\partial
t}=$ $\displaystyle\
g^{ij}\Big{\\{}R^{\sigma^{-1}\mathcal{V}}(\partial_{k},\partial_{i})(d^{\mathcal{V}}\sigma(\partial_{j}))+(d^{\mathcal{V}}\sigma)(R^{M}(\partial_{i},\partial_{k})\partial_{j})+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\nabla_{\partial_{j}}^{\sigma^{-1}\mathcal{V}}(d^{\mathcal{V}}\sigma(\partial_{k}))$
$\displaystyle+2\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}g_{lj}\eta^{uv}\tilde{\Gamma}^{l}_{ku}\tilde{\partial}_{v}\Big{)}-(d^{\mathcal{V}}\sigma)(\nabla_{\partial_{i}}\nabla_{\partial_{j}}\partial_{k})\Big{\\}}+\frac{\partial
g^{ij}}{\partial
x_{k}}\alpha(\partial_{i},\partial_{j})-g^{ij}(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{k},\nabla_{\partial_{i}}\partial_{j}).$

Since

$\displaystyle\nabla_{\partial_{k}}^{\sigma^{-1}\mathcal{V}}\frac{\partial\sigma}{\partial
t}$
$\displaystyle=\Big{[}(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{k},\partial_{t})-(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{t},\partial_{k})\Big{]}+\frac{\partial}{\partial
t}(d^{\mathcal{V}}\sigma(\partial_{k}))$

and

$\displaystyle\Big{[}(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{k},\partial_{t})-(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{t},\partial_{k})\Big{]}=\frac{\partial\sigma^{u}}{\partial
t}\nabla^{N}_{\tilde{\partial}_{k}}\tilde{\partial}_{u}-\frac{\partial}{\partial
t}\Big{(}\eta_{ks}\eta^{su}\Big{)}\tilde{\partial}_{u}-\eta_{ks}\eta^{su}\frac{\partial\sigma^{v}}{\partial
t}\nabla^{N}_{\tilde{\partial}_{v}}\tilde{\partial}_{u},$

the flow equation for $d^{\mathcal{V}}\sigma$ is

$\displaystyle\frac{\partial}{\partial
t}(d^{\mathcal{V}}\sigma(\partial_{k}))=-\frac{\partial\sigma^{u}}{\partial
t}\nabla^{N}_{\tilde{\partial}_{k}}\tilde{\partial}_{u}+\frac{\partial}{\partial
t}\Big{(}\eta_{ks}\eta^{su}\Big{)}\tilde{\partial}_{u}+\eta_{ks}\eta^{su}\frac{\partial\sigma^{v}}{\partial
t}\nabla^{N}_{\tilde{\partial}_{v}}\tilde{\partial}_{u}$
$\displaystyle+g^{ij}\Big{\\{}R^{\sigma^{-1}\mathcal{V}}(\partial_{k},\partial_{i})(d^{\mathcal{V}}\sigma(\partial_{j}))+(d^{\mathcal{V}}\sigma)(R^{M}(\partial_{i},\partial_{k})\partial_{j})+\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\nabla_{\partial_{j}}^{\sigma^{-1}\mathcal{V}}(d^{\mathcal{V}}\sigma(\partial_{k}))$
$\displaystyle+2\nabla_{\partial_{i}}^{\sigma^{-1}\mathcal{V}}\Big{(}g_{lj}\eta^{uv}\tilde{\Gamma}^{l}_{ku}\tilde{\partial}_{v}\Big{)}-(d^{\mathcal{V}}\sigma)(\nabla_{\partial_{i}}\nabla_{\partial_{j}}\partial_{k})\Big{\\}}+\frac{\partial
g^{ij}}{\partial
x_{k}}\alpha(\partial_{i},\partial_{j})-g^{ij}(\nabla^{\mathcal{V}}d^{\mathcal{V}}\sigma)(\partial_{k},\nabla_{\partial_{i}}\partial_{j}).$

We will only indicate the main steps of the induction, using the Ricci
identity (1.8). Write $T$ for $d^{\mathcal{V}}\sigma$ and use a schematic
expression for the flow of $T$. In particular, note that the curvature term
$R^{\sigma^{-1}\mathcal{V}}$ of the pull-back connection will contribute
instances of $T$.

$\displaystyle\frac{\partial}{\partial t}T$ $\displaystyle=\nabla
T*\nabla\eta+T^{3}*R^{N}+T*R^{M}+\Delta T+\nabla T*\nabla g+\nabla
g*\nabla\eta+T*R^{N}$ (2.20)

so the schematic expression for the flow of $\nabla T$ is

$\displaystyle\frac{\partial}{\partial t}\nabla T=$
$\displaystyle\nabla^{2}T*\nabla\eta+\nabla T*T*R^{N}+\nabla
T*T^{2}*R^{N}+T^{4}*\nabla R^{N}+\nabla T*R^{M}+T*\nabla R^{M}$ (2.21)
$\displaystyle+\Delta(\nabla T)+\nabla^{2}T*\nabla g+R^{M}*\nabla\eta+\nabla
g*T*R^{N}+\nabla T*R^{N}+T^{2}*\nabla R^{N}$ (2.22)

since $\nabla(R^{N}\circ\sigma)=T*\nabla R^{N}$, $\nabla(\Delta
T)=\Delta(\nabla T)+\nabla T*R^{N}+T*\nabla R^{M}$ and
$\nabla(\frac{\partial}{\partial t}T)=\frac{\partial}{\partial t}(\nabla
T)+R^{N}*T$.

Therefore the evolution equation for $|T|^{2}$ is

$\displaystyle\frac{\partial}{\partial t}|T|^{2}=$ $\displaystyle\
2\langle\nabla T*\nabla\eta,T\rangle+2\langle T^{3}*R^{N},T\rangle+2\langle
T*R^{M},T\rangle+2\langle\Delta T,T\rangle+2\langle\nabla T*\nabla g,T\rangle$
$\displaystyle+2\langle\nabla g*\nabla\eta,T\rangle+2\langle
T*R^{N},T\rangle,$

so

$\displaystyle\frac{\partial}{\partial t}|T|^{2}\leq$ $\displaystyle\
\Delta|T|^{2}-2|\nabla T|^{2}+C|\nabla\eta||T||\nabla
T|+C|T|^{4}|R^{N}|+C|T|^{2}|R^{M}|+C|\nabla g||T||\nabla T|$
$\displaystyle+C|\nabla g||\nabla\eta||T|+C|R^{N}||T|^{2}$

and the evolution equation for $\nabla T$ implies

$\displaystyle\frac{\partial}{\partial t}|\nabla T|^{2}\leq$
$\displaystyle\Delta|\nabla T|^{2}-2|\nabla^{2}T|^{2}+C|\nabla\eta||\nabla
T||\nabla^{2}T|+C|R^{N}||T||\nabla T|^{2}+C|R^{N}||T|^{2}|\nabla T|^{2}$
$\displaystyle+C|\nabla R^{N}||T|^{4}|\nabla T|+C|R^{M}||\nabla
T|^{2}+C|\nabla R^{M}||T||\nabla T|$ $\displaystyle+C|\nabla g||\nabla
T||\nabla^{2}T|+C|R^{M}||\nabla\eta||\nabla T|+C|\nabla g||R^{N}||T||\nabla
T|+C|R^{N}||\nabla T|^{2}$ $\displaystyle+C|\nabla R^{N}||T|^{2}|\nabla T|.$

The hypotheses imply that

$\displaystyle\frac{\partial}{\partial t}|T|^{2}$
$\displaystyle\leq\Delta|T|^{2}-2|\nabla T|^{2}+CK|T||\nabla
T|+CK^{2}|T|^{4}+CK^{2}|T|^{2}+CK^{2}|T|$

and

$\displaystyle\frac{\partial}{\partial t}|\nabla T|^{2}\leq$
$\displaystyle\Delta|\nabla T|^{2}-2|\nabla^{2}T|^{2}+CK|\nabla
T||\nabla^{2}T|+CK^{2}|T||\nabla T|^{2}+CK^{2}|T|^{2}|\nabla T|^{2}$
$\displaystyle+CK^{3}|T|^{4}|\nabla T|+CK^{2}|\nabla T|^{2}+CK^{3}|T||\nabla
T|+CK^{3}|\nabla T|+CK^{3}|T|^{2}|\nabla T|.$

Base step: Let $f=t|\nabla T|^{2}+\beta|T|^{2}$, for a constant $\beta$, then

$\displaystyle\frac{\partial f}{\partial t}=$ $\displaystyle|\nabla
T|^{2}+t\frac{\partial|\nabla T|^{2}}{\partial
t}+\beta\frac{\partial|T|^{2}}{\partial t}$ $\displaystyle\leq$
$\displaystyle|\nabla T|^{2}+t\Big{(}\Delta|\nabla
T|^{2}-2|\nabla^{2}T|^{2}+CK|\nabla T||\nabla^{2}T|+CK^{2}|T||\nabla
T|^{2}+CK^{2}|T|^{2}|\nabla T|^{2}$ $\displaystyle+CK^{3}|T|^{4}|\nabla
T|+CK^{2}|\nabla T|^{2}+CK^{3}|T||\nabla T|+CK^{3}|\nabla
T|+CK^{3}|T|^{2}|\nabla T|\Big{)}$
$\displaystyle+\beta\Big{(}\Delta|T|^{2}-2|\nabla T|^{2}+CK^{2}|\nabla
T|+CK^{6}\Big{)}.$

By Young’s inequality and $|T|\leq K$,

$\displaystyle\frac{\partial f}{\partial t}\leq$ $\displaystyle|\nabla
T|^{2}+\Delta f-(2-\varepsilon)t|\nabla^{2}T|^{2}+t\Big{(}CK^{4}|\nabla
T|^{2}+CK^{10}\Big{)}$ $\displaystyle+\beta\Big{(}-(2-\varepsilon)|\nabla
T|^{2}+CK^{4}+CK^{6}\Big{)}$ $\displaystyle\leq$ $\displaystyle\Delta
f+(C-\beta(2-\varepsilon))|\nabla T|^{2}+C\beta K^{6},$

so, for $\beta$ large enough,

$\frac{\partial f}{\partial t}\leq\Delta f+C\beta K^{6}.$

Since $f(x,0)\leq\beta K^{2}$, by the maximum principle and the assumption
$0\leq t\leq 1/K^{4}$,

$\displaystyle\sup f(x,t)$ $\displaystyle\leq\beta K^{2}+C\beta tK^{6}\leq
cK^{2}.$

Therefore $t|\nabla T|^{2}\leq CK^{2}$, that is $|\nabla T|\leq CKt^{-1/2}$,
and this proves the base step.

Induction step: From (2.20), we obtain the schematic expression for the flow
of $\nabla^{m}T$ based on

$\begin{aligned} a)&\nabla^{m}\frac{\partial}{\partial
t}T=\frac{\partial}{\partial t}\nabla^{m}T+\nabla^{m-1}(T*R^{N})\\\
b)&\nabla^{m}(\Delta
T)=\Delta(\nabla^{m}T)+\sum_{i=0}^{m}\nabla^{m-i}T*\nabla^{i}R^{M}\\\
c)&\nabla^{m}(T*R^{N})=\sum_{a+b=m}\nabla^{a}T*\nabla^{b}(R^{N}\circ\sigma)\\\
d)&\nabla^{m}(T^{3}*R^{N})=\sum_{i=0}^{m}\nabla^{i}(T^{3})*\nabla^{m-i}(R^{N}\circ\sigma)\\\
h)&\nabla^{m}(\nabla
T*\nabla\eta)=\nabla^{m+1}T*\nabla\eta+\sum_{i=0}^{m-1}\nabla^{i+1}T*\nabla^{m-i}(\nabla\eta\circ\sigma)\end{aligned}\begin{aligned}
e)&\nabla^{m}(T*R^{M})=\sum_{a+b=m}\nabla^{a}T*\nabla^{b}R^{M}\\\
f)&\nabla^{m}(\nabla T*\nabla g)=\sum_{a+b=m}\nabla^{a+1}T*\nabla^{b+1}g\\\
g)&\nabla^{m}(\nabla
g*\nabla\eta)=\sum_{a+b=m}\nabla^{a+1}g*\nabla^{b}(\nabla\eta\circ\sigma)\end{aligned}$

and the Faà-di Bruno formula [Faà55]:

$\nabla^{b}(R^{N}\circ\sigma)=\sum_{\sum_{j=1}^{n}jb_{j}=b}\Big{(}\nabla^{b_{1}+\dots+b_{j}}R^{N}\Big{)}*\prod_{j=1}^{n}\Big{(}\nabla^{j-1}T\Big{)}^{b_{j}}$

where $\prod$ is a product of $*$’s. Then

$\displaystyle\frac{\partial}{\partial t}(|\nabla^{m}T|^{2})=$ $\displaystyle
2\langle\nabla^{m}(\nabla
T*\nabla\eta),\nabla^{m}T\rangle+2\langle\nabla^{m}(T^{3}*R^{N}),\nabla^{m}T\rangle+2\langle\nabla^{m}(T*R^{M}),\nabla^{m}T\rangle$
$\displaystyle+2\langle\nabla^{m}(\Delta
T),\nabla^{m}T\rangle+2\langle\nabla^{m}(\nabla T*\nabla
g),\nabla^{m}T\rangle+2\langle\nabla^{m}(\nabla
g*\nabla\eta),\nabla^{m}T\rangle$
$\displaystyle+2\langle\nabla^{m}(T*R^{N}),\nabla^{m}T\rangle-2\langle\nabla^{m-1}(T*R^{N}),\nabla^{m}T\rangle.$

A combination of the chain rule, the above formulas and the hypotheses yields

$\displaystyle|\nabla^{b}(R^{N}\circ\sigma)|$ $\displaystyle\leq
CK^{2}t^{-b/2};$ $\displaystyle 2\langle\nabla^{m}(\Delta
T),\nabla^{m}T\rangle$
$\displaystyle\leq\Delta|\nabla^{m}T|^{2}-2|\nabla^{m+1}T|^{2}+CK^{2}|\nabla^{m}T|^{2}+CK^{3}t^{-m/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(T*R^{M}),\nabla^{m}T\rangle$
$\displaystyle\leq CK^{2}|\nabla^{m}T|^{2}+CK^{3}t^{-m/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(T*R^{N}),\nabla^{m}T\rangle$
$\displaystyle\leq CK^{2}|\nabla^{m}T|^{2}+CK^{3}t^{-m/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(\nabla T*\nabla g),\nabla^{m}T\rangle$
$\displaystyle\leq
CK|\nabla^{m+1}T||\nabla^{m}T|+CK^{2}|\nabla^{m}T|^{2}+CK^{2}t^{-(m+1)/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(T^{3}*R^{N}),\nabla^{m}T\rangle$
$\displaystyle\leq CK^{4}|\nabla^{m}T|^{2}+CK^{5}t^{-m/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(\nabla T*\nabla\eta),\nabla^{m}T\rangle$
$\displaystyle\leq
CK|\nabla^{m}T||\nabla^{m+1}T|+CK^{2}|\nabla^{m}T|^{2}+CK^{3}t^{-m/2}|\nabla^{m}T|;$
$\displaystyle 2\langle\nabla^{m}(\nabla g*\nabla\eta),\nabla^{m}T\rangle$
$\displaystyle\leq
CK^{3}t^{-m/2}|\nabla^{m}T|+CK^{3}t^{-(m-1)/2}|\nabla^{m}T|;$ $\displaystyle
2\langle\nabla^{m-1}(T*R^{N}),\nabla^{m}T\rangle$ $\displaystyle\leq
CKt^{-m/2}|\nabla^{m}T|.$

Therefore

$\displaystyle\frac{\partial}{\partial t}(|\nabla^{m}T|^{2})$
$\displaystyle\leq\Delta|\nabla^{m}T|^{2}-2|\nabla^{m+1}T|^{2}+CK|\nabla^{m}T||\nabla^{m+1}T|+CK^{2}(1+K^{2})|\nabla^{m}T|^{2}$
$\displaystyle+CK^{2}(K+K^{3})t^{-m/2}|\nabla^{m}T|+CK^{2}t^{-(m+1)/2}|\nabla^{m}T|+CK^{3}t^{-(m-1)/2}|\nabla^{m}T|.$

Now, by Young’s inequality, for $\varepsilon$ small enough, and using $K\geq
1$ and $t\leq 1$, we have

$\displaystyle\frac{\partial}{\partial t}(|\nabla^{m}T|^{2})\leq$
$\displaystyle\Delta|\nabla^{m}T|^{2}-|\nabla^{m+1}T|^{2}+CK^{2}|\nabla^{m}T|^{2}+CK^{2}t^{-1/2}|\nabla^{m}T|^{2}+CK^{5}t^{-m/2}|\nabla^{m}T|$
$\displaystyle+CK^{2}t^{-(m+1)/2}|\nabla^{m}T|$

This remains true replacing $k$ by $m-k$ so that

$\displaystyle\frac{\partial}{\partial t}(|\nabla^{m-k}T|^{2})\leq$
$\displaystyle\Delta|\nabla^{m-k}T|^{2}-|\nabla^{m-k+1}T|^{2}+CK^{4}t^{-(m-k)}+CK^{4}t^{-1/2}t^{-(m-k)}$
$\displaystyle+CK^{6}t^{-(m-k)/2}t^{-(m-k)/2}+CK^{3}t^{-(m-k+1)/2}t^{-(m-k)/2}$
$\displaystyle\leq$
$\displaystyle\Delta|\nabla^{m-k}T|^{2}-|\nabla^{m-k+1}T|^{2}+CK^{4}t^{-(m-k)}+CK^{4}t^{-1/2}t^{-(m-k)}.$

Let
$f_{m}=t^{m}|\nabla^{m}T|^{2}+\beta_{m}\sum_{k}\alpha^{m}_{k}t^{m-k}|\nabla^{m-k}T|^{2}$.

Combining the previous estimates and Young’s inequality again,

$\displaystyle\frac{\partial f_{m}}{\partial t}\leq$ $\displaystyle\Delta
f_{m}+\Big{(}CK^{2}t^{m}+CK^{6}t^{m}+CK^{2}t^{(2m-1)/2}+(C+m)t^{m-1}-\beta_{m}\alpha^{m}_{1}t^{m-1}\Big{)}|\nabla^{m}T|^{2}$
$\displaystyle+\beta_{m}\sum_{k=1}^{m-1}((m-k)\alpha^{m}_{k}-\alpha^{m}_{k+1})t^{m-k-1}|\nabla^{m-k}T|^{2}$
$\displaystyle+C\beta_{m}\sum_{k=1}^{m}\alpha^{m}_{k}(K^{4}+K^{4}t^{-1/2})+CK^{4}.$

But $(m-k)\alpha^{m}_{k}-\alpha^{m}_{k+1}=0$, for $1\leq k\leq m-1$, and for
$\beta_{m}$ large enough, we have

$\displaystyle\frac{\partial f_{m}}{\partial t}$ $\displaystyle\leq\Delta
f_{m}+CK^{4}t^{-1/2}+CK^{4}.$

Since
$f_{m}(0)=\beta_{m}\alpha^{m}_{m}|T|^{2}\leq\beta_{m}\alpha^{m}_{m}K^{2}$, we
invoke the maximum principle to conclude that

$\displaystyle\sup f_{m}$
$\displaystyle\leq\beta_{m}\alpha^{m}_{m}K^{2}+CK^{4}t^{1/2}+CK^{4}t$
$\displaystyle\leq CK^{2}.$

Hence $t^{m}|\nabla^{m}T|^{2}\leq CK^{2}$, that is the $m$-induction step
$|\nabla^{m}T|\leq CKt^{-m/2}$, provided $0\leq t\leq 1/K^{4}$. ∎

######  Remark 2.17.

As described in §2.2.1, $H$-structures are in one-to-one correspondence with
sections of a Riemannian homogeneous fibre bundle, so Proposition 2.16 applies
in particular to $H$-structures.

## 3\. Preliminaries on $\mathrm{Spin}(7)$-structures

###  3.1 Definitions and basic properties

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

A $\mathrm{Spin}(7)$-structure on $M^{8}$ is a reduction of the structure
group of the frame bundle $\mathrm{Fr}(M)$ to the Lie group
$\mathrm{Spin}(7)\subset\mathrm{SO}(8)$. From the point of view of
differential geometry, a $\mathrm{Spin}(7)$-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 $\mathrm{Spin}(7)$-structure on $M$
is a subbundle $\mathcal{A}$ 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.

A $\mathrm{Spin}(7)$-structure $\Phi$ determines a Riemannian metric and an
orientation on $M$ in a nonlinear way. 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

$\displaystyle(g_{\Phi}(v,v))^{2}=-\frac{7^{3}}{6^{\frac{7}{3}}}\frac{(\textup{det}\
B_{ij}(v))^{\frac{1}{3}}}{A(v)^{3}}.$ (3.1)

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

######  Definition 3.1.

Let $\nabla$ be the Levi-Civita connection of the metric $g_{\Phi}$. The pair
$(M^{8},\Phi)$ is a _$\mathrm{Spin}(7)$ -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
$\mathrm{Spin}(7)$-manifold has Riemannian holonomy contained in the subgroup
$\mathrm{Spin}(7)\subset\mathrm{SO}(8)$. Such a parallel
$\mathrm{Spin}(7)$-structure is also called _torsion-free_.

###  3.2 Decomposition of the space of forms

The existence of a $\mathrm{Spin}(7)$-structure $\Phi$ induces a decomposition
of the space of differential forms on $M$ into irreducible $\mathrm{Spin}(7)$
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:

$\displaystyle\Omega^{2}_{7}$
$\displaystyle=\\{\beta\in\Omega^{2}\mid\star(\Phi\wedge\beta)=-3\beta\\},$
(3.2) $\displaystyle\Omega^{2}_{21}$
$\displaystyle=\\{\beta\in\Omega^{2}\mid\star(\Phi\wedge\beta)=\beta\\},$
(3.3)

and

$\displaystyle\Omega^{3}_{8}$ $\displaystyle=\\{X\lrcorner\Phi\mid
X\in\Gamma(TM)\\},$ (3.4) $\displaystyle\Omega^{3}_{48}$
$\displaystyle=\\{\gamma\in\Omega^{3}\mid\gamma\wedge\Phi=0\\}.$ (3.5)

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

$\displaystyle\beta_{ij}\in\Omega^{2}_{7}\iff\beta_{ab}\Phi_{abij}$
$\displaystyle=-6\beta_{ij},$ (3.6)
$\displaystyle\beta_{ij}\in\Omega^{2}_{21}\iff\beta_{ab}\Phi_{abij}$
$\displaystyle=2\beta_{ij}.$ (3.7)

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

$\displaystyle\gamma_{ijk}\in\Omega^{3}_{8}$
$\displaystyle\iff\gamma_{ijk}=X_{l}\Phi_{ijkl}\ \ \ \ \textup{for\ some}\
X\in\Gamma(TM),$ (3.8) $\displaystyle\gamma_{ijk}\in\Omega^{3}_{48}$
$\displaystyle\iff\gamma_{ijk}\Phi_{ijkl}=0.$ (3.9)

If $\pi_{7}$ and $\pi_{21}$ are the projection operators on $\Omega^{2}$, it
follows from (3.6) and (3.7) that

$\displaystyle\pi_{7}(\beta)_{ij}$
$\displaystyle=\frac{1}{4}\beta_{ij}-\frac{1}{8}\beta_{ab}\Phi_{abij},$ (3.10)
$\displaystyle\pi_{21}(\beta)_{ij}$
$\displaystyle=\frac{3}{4}\beta_{ij}+\frac{1}{8}\beta_{ab}\Phi_{abij}.$ (3.11)

Finally, for $\beta_{ij}\in\Omega^{2}_{21}$,

$\displaystyle\beta_{ab}\Phi_{bpqr}$
$\displaystyle=\beta_{pi}\Phi_{iqra}+\beta_{qi}\Phi_{irpa}+\beta_{ri}\Phi_{ipqa},$
(3.12)

which can be used to show that $\Omega^{2}_{21}\equiv\mathfrak{so}(7)$ is the
Lie algebra of $\mathrm{Spin}(7)$, 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
described in Definition 2.2, as in [DGK21] for the $\mathrm{G}_{2}$ case.
Given $A\in\Gamma(T^{*}M\otimes TM)$, define

$\displaystyle
A\diamond\Phi=\frac{1}{24}(A_{ip}\Phi_{pjkl}+A_{jp}\Phi_{ipkl}+A_{kp}\Phi_{ijpl}+A_{lp}\Phi_{ijkp})dx^{i}\wedge
dx^{j}\wedge dx^{k}\wedge dx^{l},$ (3.13)

and hence

$\displaystyle(A\diamond\Phi)_{ijkl}=A_{ip}\Phi_{pjkl}+A_{jp}\Phi_{ipkl}+A_{kp}\Phi_{ijpl}+A_{lp}\Phi_{ijkp}.$
(3.14)

Recall that $\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus S_{0}\oplus\Omega^{2}$,
and $\Omega^{2}$ further splits orthogonally into (3.2) and (3.3), so

$\displaystyle\Gamma(T^{*}M\otimes TM)=\Omega^{0}\oplus
S_{0}\oplus\Omega^{2}_{7}\oplus\Omega^{2}_{21}.$ (3.15)

With respect to this splitting, we can write
$A=\frac{1}{8}(\operatorname{tr}A)g+A_{0}+A_{7}+A_{21}$ where $A_{0}$ is a
symmetric traceless $2$-tensor. The diamond contraction (3.14) defines a
linear map $A\mapsto A\diamond\Phi$, from $\Omega^{0}\oplus
S_{0}\oplus\Omega^{2}_{7}\oplus\Omega^{2}_{21}$ to $\Omega^{4}(M)$. The
following proposition is proved in [Kar08, Prop. 2.3].

######  Proposition 3.2.

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.

We also record the following properties of the $\diamond$ operator.

######  Proposition 3.3.

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

  1. 1.

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

$\displaystyle\star(A\diamond\Phi)$ $\displaystyle=(\bar{A}\diamond\Phi)\ \
\textup{where}\ \ \bar{A}=\frac{1}{4}(\operatorname{tr}A)g_{ij}-A_{ji}.$
(3.16)

  2. 2.

If $A,B\in\Gamma(T^{*}M\otimes TM)$ and $A_{0},\ B_{0}$ denote the traceless
symmetric parts of $A$ and $B$ respectively and $A_{7},\ B_{7}$ denote their
$\Omega^{2}_{7}$ component. Then

$\displaystyle g(A\diamond\Phi,B\diamond\Phi)$
$\displaystyle=\frac{7}{2}(\operatorname{tr}A)(\operatorname{tr}B)+4\operatorname{tr}(A_{0}B_{0})-16\operatorname{tr}(A_{7}B_{7})$
(3.17)

where $A_{0}B_{0}$ and $A_{7}B_{7}$ are matrix multiplications.

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

######  Definition 3.4.

On $(M,\Phi)$, define a $\mathrm{Spin}(7)$-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

$\displaystyle(\Lambda_{\Phi}(\sigma))_{ijkl}=(\sigma\cdot\Phi)_{ijkl}+(\sigma\cdot\Phi)_{iklj}+(\sigma\cdot\Phi)_{iljk}+(\sigma\cdot\Phi)_{jkil}+(\sigma\cdot\Phi)_{jlki}+(\sigma\cdot\Phi)_{klij}.$
(3.18)

######  Proposition 3.5.

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

$\displaystyle\begin{aligned}
\Omega^{4}_{1}&=\\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=-24\sigma\\},\\\
\Omega^{4}_{27}&=\\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=4\sigma\\},\end{aligned}$
$\displaystyle\begin{aligned}
\Omega^{4}_{7}&=\\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=-12\sigma\\},\\\
\Omega^{4}_{35}&=\\{\sigma\in\Omega^{4}\mid\Lambda_{\Phi}(\sigma)=0\\}.\end{aligned}$
(3.19)

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

$\displaystyle\Omega^{4}_{+}=\\{\sigma\in\Omega^{4}\mid\star\sigma=\sigma\\}=\Omega^{4}_{1}\oplus\Omega^{4}_{7}\oplus\Omega^{4}_{27},\
\ \ \ \
\Omega^{4}_{-}=\\{\sigma\in\Omega^{4}\mid\star\sigma=-\sigma\\}=\Omega^{4}_{35}.$
(3.20)

Before we discuss the torsion of a $\mathrm{Spin}(7)$-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

$\displaystyle\Phi_{ijkl}\Phi_{abcl}$
$\displaystyle=g_{ia}g_{jb}g_{kc}+g_{ib}g_{jc}g_{ka}+g_{ic}g_{ja}g_{kb}$
$\displaystyle\quad-g_{ia}g_{jc}g_{kb}-g_{ib}g_{ja}g_{kc}-g_{ic}g_{jb}g_{ka}$
$\displaystyle\quad-g_{ia}\Phi_{jkbc}-g_{ib}\Phi_{jkca}-g_{ic}\Phi_{jkab}$
$\displaystyle\quad-g_{ja}\Phi_{kibc}-g_{jb}\Phi_{kica}-g_{jc}\Phi_{kiab}$
$\displaystyle\quad-g_{ka}\Phi_{ijbc}-g_{kb}\Phi_{ijca}-g_{kc}\Phi_{ijab}$
(3.21) $\displaystyle\Phi_{ijkl}\Phi_{abkl}$
$\displaystyle=6g_{ia}g_{jb}-6g_{ib}g_{ja}-4\Phi_{ijab}$ (3.22)
$\displaystyle\Phi_{ijkl}\Phi_{ajkl}$ $\displaystyle=42g_{ia}$ (3.23)
$\displaystyle\Phi_{ijkl}\Phi_{ijkl}$ $\displaystyle=336.$ (3.24)

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

$\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{abkl}$
$\displaystyle=-\Phi_{ijkl}(\nabla_{m}\Phi_{abkl})-4\nabla_{m}\Phi_{ijab}$
(3.25) $\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{ajkl}$
$\displaystyle=-\Phi_{ijkl}(\nabla_{m}\Phi_{ajkl})$ (3.26)
$\displaystyle(\nabla_{m}\Phi_{ijkl})\Phi_{ijkl}$ $\displaystyle=0.$ (3.27)

We now describe the _torsion_ of a $\mathrm{Spin}(7)$-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 3.6.

The _torsion tensor_ of a $\mathrm{Spin}(7)$-structure $\Phi$ is the element
of $\Omega^{1}_{8}\otimes\Omega^{2}_{7}$ defined by expressing $\nabla\Phi$ in
the light of Proposition 3.2:

$\displaystyle\nabla_{m}\Phi_{ijkl}=(T_{m}\diamond\Phi)_{ijkl}=T_{m;ip}\Phi_{pjkl}+T_{m;jp}\Phi_{ipkl}+T_{m;kp}\Phi_{ijpl}+T_{m;lp}\Phi_{ijkp}$
(3.28)

where $T_{m;ab}\in\Omega^{2}_{7}$, for each fixed $m$.

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

$\displaystyle T_{m;ab}=\frac{1}{96}(\nabla_{m}\Phi_{ajkl})\Phi_{bjkl}$ (3.29)

######  Theorem 3.7.

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

Finally, the torsion satisfies a ‘Bianchi-type identity’. This was first
proved in [Kar08, Theorem 4.2], using the diffeomorphism invariance of the
torsion tensor. We give a different proof below, using the Ricci identity
(1.8).

######  Theorem 3.8.

The torsion tensor $T$ satisfies the following ‘Bianchi-type identity’

$\displaystyle\nabla_{i}T_{j;ab}-\nabla_{j}T_{i;ab}=2T_{i;am}T_{j;mb}-2T_{j;am}T_{i;mb}+\frac{1}{4}R_{jiab}-\frac{1}{8}R_{jimn}\Phi_{mnab}.$
(3.30)

###### Proof.

As remarked in [Kar08], relation (3.30) can be proved by using the Ricci
identity (1.8) and the contraction identities (3.21)-(3.24). We provide the
details for completeness. From (3.29), we have

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

which gives

$\displaystyle\nabla_{n}T_{m;ab}-\nabla_{m}T_{n;ab}$
$\displaystyle=\frac{1}{96}(\nabla_{n}\nabla_{m}-\nabla_{m}\nabla_{n})\Phi_{ajkl}\Phi_{bjkl}+\frac{1}{96}(\nabla_{m}\Phi_{ajkl}\nabla_{n}\Phi_{bjkl}-\nabla_{n}\Phi_{ajkl}\nabla_{m}\Phi_{bjkl})$
$\displaystyle=\frac{1}{96}(-R_{nmas}\Phi_{sjkl}-R_{nmjs}\Phi_{askl}-R_{nmks}\Phi_{ajsl}-R_{nmls}\Phi_{ajks})\Phi_{bjkl}$
$\displaystyle\quad+\frac{1}{96}(T_{m;ap}\Phi_{pjkl}+T_{m;jp}\Phi_{apkl}+T_{m;kp}\Phi_{ajpl}+T_{m;lp}\Phi_{ajkp})$
$\displaystyle\qquad\qquad(T_{n;bs}\Phi_{sjkl}+T_{n;js}\Phi_{bskl}+T_{n;ks}\Phi_{bjsl}+T_{n;ls}\Phi_{bjks})$
$\displaystyle\quad+\frac{1}{96}(T_{n;ap}\Phi_{pjkl}+T_{n;jp}\Phi_{apkl}+T_{n;kp}\Phi_{ajpl}+T_{n;lp}\Phi_{ajkp})$
$\displaystyle\qquad\qquad(T_{m;bs}\Phi_{sjkl}+T_{m;js}\Phi_{bskl}+T_{m;ks}\Phi_{bjsl}+T_{m;ls}\Phi_{bjks})$
(3.31)

where we used the Ricci identity (1.8) on the first line of the second
equality and (3.28) to write the remaining terms of the second equality. Now
using the contraction identities (3.21)-(3.24), an easy albeit tedious
computation gives (3.30). ∎

Using the Riemannian Bianchi identity, we see that

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

hence

$\displaystyle R_{ijkl}\Phi_{ajkl}=0.$ (3.32)

Using this and contracting (3.30) on $j$ and $b$ gives the expression for the
Ricci curvature of a metric induced by a $\mathrm{Spin}(7)$-structure:

$\displaystyle
R_{ij}=4\nabla_{i}T_{a;ja}-4\nabla_{a}T_{i;ja}-8T_{i;jb}T_{a;ba}+8T_{a;jb}T_{i;ba}.$
(3.33)

This also proves that the metric of a torsion-free
$\mathrm{Spin}(7)$-structure is Ricci-flat, a result originally due to Bonan.
Taking the trace of (3.33) gives the scalar curvature $R$:

$\displaystyle
R=4\nabla_{i}T_{a;ia}-4\nabla_{a}T_{i;ia}+8|T|^{2}+8T_{a;jb}T_{j;ba}.$ (3.34)

###  3.3 Proof of Theorem A

This section addresses the proof of the Bryant-type formula for isometric
$\mathrm{Spin}(7)$-structures, which are in bijection with unit spinors
[Mar20, §2]. We derive a $\mathrm{Spin}(7)$ counterpart of the well-known
formula for $\mathrm{G}_{2}$-structures [Bry06, (3.6)].

Fix a $\mathrm{Spin}(7)$-structure $\Phi_{0}$ and identify its corresponding
unit spinor with the identity ${\mathbf{1}}_{+}$. As the twistor space is an
$\mathbb{R}\mathrm{P}^{7}$-bundle, any other isometric
$\mathrm{Spin}(7)$-structure can be identified with a unit spinor $\psi$ and
decomposed into

$\psi=f{\mathbf{1}}_{+}\oplus X,\quad\text{with}\quad
X\in\Gamma(\mathbf{7})\quad\text{and}\quad|f|^{2}+|X|^{2}=1,$ (3.35)

in which $\mathbf{7}$ is the orthogonal complement to the span of
${\mathbf{1}}_{+}$ (later on identified with $\operatorname{Im}\mathbb{O}$).
Then its square is

$\psi\otimes\psi=f^{2}{\mathbf{1}}_{+}\otimes{\mathbf{1}}_{+}+f({\mathbf{1}}_{+}\otimes
X+X\otimes{\mathbf{1}}_{+})+X\otimes
X\in\mathrm{Cl}_{8}\simeq\Lambda\mathbb{R}^{8},$

and we are only interested in its $\Lambda^{4}\mathbb{R}^{8}$-component
$[\psi\otimes\psi]_{4}$.

It is well-known that $[{\mathbf{1}}_{+}\otimes{\mathbf{1}}_{+}]_{4}=\Phi_{0}$
[LM89, p. 355]. In order to compute the cross-term $[{\mathbf{1}}_{+}\otimes
X+X\otimes{\mathbf{1}}_{+}]_{4}$, we take its inner-product with a general
quadruple $e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l}\in\Lambda^{4}\mathbb{R}^{8}$, following eg. the proof of [DH93, Prop
2.1]):

$\displaystyle 16\langle{\mathbf{1}}_{+}\otimes
X+X\otimes{\mathbf{1}}_{+},e_{i}\wedge e_{j}\wedge e_{k}\wedge e_{l}\rangle$
$\displaystyle=16\langle{\mathbf{1}}_{+},(e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l})X\rangle+16\langle X,(e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l}){\mathbf{1}}_{+}\rangle.$

With the identifications
$\mathrm{Cl}_{8}\simeq\mathbb{R}(16)\simeq\operatorname{End}(\mathbb{P})\simeq\mathbb{P}\otimes\mathbb{P}^{*}$
and $\mathbb{P}=S^{+}\oplus S^{-}$, so that
$\mathbb{P}\simeq\mathbb{O}\oplus\mathbb{O}$, with
$S_{+}\simeq\mathbb{O}\times\\{0\\}$ and $S_{-}\simeq\\{0\\}\times\mathbb{O}$,
a general spinor $s$ satisfies:

$(e_{i}\wedge e_{j}\wedge e_{k}\wedge e_{l})s=(e_{i}(e_{j}(e_{k}(e_{l}s))))$

where the right-hand side is a composition of left-multiplications in
$\mathbb{O}$.

Take $e_{i},e_{j},e_{k},e_{l}$ to be distinct unit orthonormal vectors ordered
such that $7\geq l>k>j>i\geq 0$. By standard properties of the octonions
[SW17, (5.6), (5.21), (5.31)], we have

$\displaystyle\langle{\mathbf{1}}_{+}\otimes
X+X\otimes{\mathbf{1}}_{+},e_{i}\wedge e_{j}\wedge e_{k}\wedge e_{l}\rangle$
$\displaystyle=\langle{\mathbf{1}}_{+},(e_{i}(e_{j}(e_{k}(e_{l}X))))\rangle+\langle
X,(e_{i}(e_{j}(e_{k}(e_{l}{\mathbf{1}}_{+}))))\rangle.$

Identifying ${\mathbf{1}}_{+}$ with the identity of $\mathbb{O}$ and defining
the $\mathrm{Spin}(7)$-structure $\Phi_{0}$ by [SW17, Thm 5.20] we obtain

$\displaystyle\langle{\mathbf{1}}_{+}\otimes
X+X\otimes{\mathbf{1}}_{+},e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l}\rangle=2\Phi_{0}(e_{i}X,e_{j},e_{k},e_{l}).$

To compute the term $[X\otimes X]_{4}$, we again take its inner-product with
$e_{i}\wedge e_{j}\wedge e_{k}\wedge e_{l}\in\Lambda^{4}\mathbb{R}^{8}$:

$\langle X\otimes X,e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l}\rangle=\varepsilon(X,e_{l}(e_{k}(e_{j}(e_{i}X)))),$

where $\epsilon$ is an inner-product on $\mathbb{P}$. If
$\Phi_{0}(e_{i},e_{j},e_{k},e_{l})\neq 0$, then

$e_{l}(e_{k}(e_{j}(e_{i}X)))=\begin{cases*}\phantom{-}\Phi_{0}(e_{i},e_{j},e_{k},e_{l})X,\text{
if }X=e_{i},e_{j},e_{k},e_{l}\\\ -\Phi_{0}(e_{i},e_{j},e_{k},e_{l})X\ \
\text{otherwise}.\end{cases*}$

Decomposing $X=X_{ijkl}+X_{*}$, with
$X_{ijkl}\in\operatorname{span}(e_{i},e_{j},e_{k},e_{l})$, we obtain

$\displaystyle\varepsilon(X,e_{l}(e_{k}(e_{j}(e_{i}X))))$
$\displaystyle=\Phi_{0}(e_{i},e_{j},e_{k},e_{l})(|X_{ijkl}|^{2}-|X_{*}|^{2})$
$\displaystyle=\Phi_{0}(e_{i},e_{j},e_{k},e_{l})(2|X_{ijkl}|^{2}-|X|^{2}).$

On the other hand, we have

$\displaystyle 4(X\wedge(X\lrcorner\Phi_{0}))(e_{i}\wedge e_{j}\wedge
e_{k}\wedge e_{l})$
$\displaystyle=|X_{ijkl}|^{2}\Phi_{0}(e_{i},e_{j},e_{k},e_{l}),$

and, at least for such quadruples, the following relation holds:

$\langle X\otimes X,e_{i}\wedge e_{j}\wedge e_{k}\wedge
e_{l}\rangle=(8(X\wedge(X\lrcorner\Phi_{0}))-|X|^{2}\Phi_{0})(e_{i},e_{j},e_{k},e_{l}).$

This formula remains valid for all the other quadruples, so long as $X$ is
pure imaginary.

Summing up the three computations, we have:

$[X\otimes
X]_{4}=(f^{2}-|X|^{2})\Phi_{0}+2f\Theta_{0,X}+8(X\wedge(X\lrcorner\Phi_{0})),$

where $\Theta_{0,X}(a,b,c,d)=\Phi_{0}(a\cdot X,b,c,d)$ is a four-form.

## 4\. Harmonic $\mathrm{Spin}(7)$-flow

###  4.1 Derivation of the flow

In this section we define the harmonic flow of $\mathrm{Spin}(7)$-structures
and prove that it is a negative gradient flow.

######  Definition 4.1 (Isometric $\mathrm{Spin}(7)$-structures.).

Two $\mathrm{Spin}(7)$-structures $\Phi_{1}$ and $\Phi_{2}$ on $M$ are called
_isometric_ if they induce the same Riemannian metric, that is, if
$g_{\Phi_{1}}=g_{\Phi_{2}}$. We will denote by $\llbracket\Phi\rrbracket$ the
space of $\mathrm{Spin}(7)$-structures that are isometric to a given
$\mathrm{Spin}(7)$-structure $\Phi$.

######  Definition 4.2.

Let $\Phi_{0}$ be a fixed initial $\mathrm{Spin}(7)$-structure on $M$. The
_energy functional_ $E$ on the set $\llbracket\Phi_{0}\rrbracket$ is

$\displaystyle
E(\Phi)=\frac{1}{2}\int_{M}|T_{\Phi}|^{2}\operatorname{vol_{g}}_{g_{\Phi}}$
(4.1)

where $T_{\Phi}$ is the torsion of $\Phi$.

At this point it is worth noticing that, whereas the energy functional (4.1)
could be considered over _all_ $\mathrm{Spin}(7)$-structures, in the present
setting it is being considered only over isometric
$\mathrm{Spin}(7)$-structures. The natural question here, inspired by harmonic
map theory, is whether, given a $\mathrm{Spin}(7)$-structure $\Phi_{0}$, there
is a “best” $\mathrm{Spin}(7)$-structure in $\llbracket\Phi_{0}\rrbracket$. An
obvious way to study this question is to consider the negative gradient flow
of $E$.

####  4.1.1 Direct approach: variational calculus

The most general flow of $\mathrm{Spin}(7)$-structures [Kar08] is given by

$\displaystyle\frac{\partial\Phi}{\partial t}=A\diamond\Phi=(h+X)\diamond\Phi$
(4.2)

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
$\mathrm{Spin}(7)$-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 is given by

$\displaystyle\frac{\partial g}{\partial t}=2h_{ij}.$ (4.3)

We start by considering the variation of the torsion $T$ with respect to
variations of the $\mathrm{Spin}(7)$-structures that preserve the metric.

######  Proposition 4.3.

Let $(\Phi_{t})_{t\in(-\varepsilon,\varepsilon)}$ be a smooth family of
$\mathrm{Spin}(7)$-structures in the class $\llbracket\Phi_{0}\rrbracket$. By
(4.2) and (4.3), we can write $\left.\frac{\partial}{\partial
t}\right|_{t=0}\Phi_{t}=X\diamond\Phi$ for some $X\in\Omega^{2}_{7}$. If
$T_{t}$ is the torsion of $\Phi_{t}$ then

$\displaystyle\left.\frac{\partial}{\partial
t}\right|_{t=0}(T_{t})_{m;is}=X_{ip}T_{m;ps}-X_{sp}T_{m;pi}+\pi_{7}(\nabla_{m}X_{is}).$
(4.4)

###### Proof.

As the $\mathrm{Spin}(7)$-structures are varying in an isometry class,
$g_{t}=g_{0}$ for all $t\in(-\varepsilon,\varepsilon)$ and hence the covariant
derivative $\nabla$ is independent of $t$ and so is the volume form. Since
$\left.\frac{\partial}{\partial t}\right|_{t=0}\Phi_{t}=X\diamond\Phi$, by
[Kar08, Lemma 3.3], $\left.\frac{\partial}{\partial
t}\right|_{t=0}\nabla\Phi_{t}=X\diamond\nabla\Phi+\nabla X\diamond\Phi$.
Explicitly, we have

$\displaystyle\left.\frac{\partial}{\partial t}\right|_{t=0}(\Phi_{t})_{ijkl}$
$\displaystyle=X_{ip}\Phi_{pjkl}+X_{jp}\Phi_{ipkl}+X_{kp}\Phi_{ijpl}+X_{lp}\Phi_{ijkp},$
(4.5) $\displaystyle\left.\frac{\partial}{\partial
t}\right|_{t=0}(\nabla_{m}(\Phi_{t})_{ijkl})$
$\displaystyle=X_{ip}\nabla_{m}\Phi_{pjkl}+X_{jp}\nabla_{m}\Phi_{ipkl}+X_{kp}\nabla_{m}\Phi_{ijpl}+X_{lp}\nabla_{m}\Phi_{ijkp}$
$\displaystyle\quad+(\nabla_{m}X_{ip})\Phi_{pjkl}+(\nabla_{m}X_{jp})\Phi_{ipkl}+(\nabla_{m}X_{kp})\Phi_{ijpl}+(\nabla_{m}X_{lp})\Phi_{ijkp}.$
(4.6)

Using the above equations and (3.29), we compute

$\displaystyle 96\left.\frac{\partial}{\partial t}\right|_{t=0}(T_{t})_{m;is}$
$\displaystyle=\left.\frac{\partial}{\partial
t}\right|_{t=0}((\nabla_{m}\Phi_{ijkl})\Phi_{sjkl})$
$\displaystyle=\Phi_{sjkl}\left.\frac{\partial}{\partial
t}\right|_{t=0}(\nabla_{m}(\Phi_{t})_{ijkl})+(\nabla_{m}\Phi_{ijkl})\left.\frac{\partial}{\partial
t}\right|_{t=0}(\Phi_{t})_{ijkl}$
$\displaystyle=\Big{[}X_{ip}\nabla_{m}\Phi_{pjkl}+X_{jp}\nabla_{m}\Phi_{ipkl}+X_{kp}\nabla_{m}\Phi_{ijpl}+X_{lp}\nabla_{m}\Phi_{ijkp}$
$\displaystyle\quad+(\nabla_{m}X_{ip})\Phi_{pjkl}+(\nabla_{m}X_{jp})\Phi_{ipkl}+(\nabla_{m}X_{kp})\Phi_{ijpl}+(\nabla_{m}X_{lp})\Phi_{ijkp}\Big{]}\Phi_{sjkl}$
$\displaystyle\quad+\nabla_{m}\Phi_{ijkl}\Big{[}X_{ip}\Phi_{pjkl}+X_{jp}\Phi_{ipkl}+X_{kp}\Phi_{ijpl}+X_{lp}\Phi_{ijkp}\Big{]}.$

Using the contraction identities (3.21)-(3.24), the above becomes

$\displaystyle 96\left.\frac{\partial}{\partial t}\right|_{t=0}(T_{t})_{m;is}$
$\displaystyle=96X_{ip}T_{m;ps}-96X_{sp}T_{m;pi}+42\nabla_{m}X_{is}+\nabla_{m}X_{jp}(6g_{is}g_{pj}-6g_{ij}g_{ps}-4\Phi_{ipsj})$
$\displaystyle\quad+\nabla_{m}X_{kp}(6g_{is}g_{pk}-6g_{ik}g_{ps}-4\Phi_{ipsk})+\nabla_{m}X_{lp}(6g_{is}g_{pl}-6g_{il}g_{ps}-4\Phi_{ipsl})$
$\displaystyle=96X_{ip}T_{m;ps}-96X_{sp}T_{m;pi}+42\nabla_{m}X_{is}-18\nabla_{m}X_{is}-12\nabla_{m}X_{jp}\Phi_{isjp}$
$\displaystyle=96X_{ip}T_{m;ps}-96X_{sp}T_{m;pi}+24\nabla_{m}X_{is}-12\nabla_{m}X_{jp}\Phi_{isjp}$

which, using (3.10) for the last two terms, is precisely (4.4). ∎

Let $E$ be the energy functional from (4.1), restricted to the isometry class
$\llbracket\Phi\rrbracket$ of $\mathrm{Spin}(7)$-structures. The following
result motivates our formulation of the harmonic $\mathrm{Spin}(7)$-flow, cf.
Definition 1.1.

######  Lemma 4.4.

The gradient of $16E:\llbracket\Phi\rrbracket\rightarrow\mathbb{R}_{\geq 0}$
at the point $\Phi$ is $-\operatorname{div}T\diamond\Phi$, where $T$ is the
torsion of $\Phi$. In other words, if
$(\Phi_{t})_{t\in(-\varepsilon,\varepsilon)}$ is a smooth family of
$\mathrm{Spin}(7)$-structures in $\llbracket\Phi_{0}\rrbracket$ and
$\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}=\xi$, then

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}16E(\Phi_{t})=-\int_{M}\langle\operatorname{div}T\diamond\Phi,\xi\rangle\operatorname{vol_{g}}.$

###### Proof.

We use Proposition 4.3 to compute

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}E(\Phi_{t})$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}\frac{1}{2}\int_{M}(T_{t})_{m;is}(T_{t})_{m;is}\operatorname{vol_{g}}$
$\displaystyle=\int_{M}T_{m;is}(X_{ip}T_{m;ps}-X_{sp}T_{m;pi}+\pi_{7}(\nabla_{m}X_{is}))\operatorname{vol_{g}}$
$\displaystyle=\int_{M}T_{m;is}(X_{ip}T_{m;ps}-X_{sp}T_{m;pi}+\frac{1}{4}\nabla_{m}X_{is}-\frac{1}{8}\nabla_{m}X_{jp}\Phi_{isjp})\operatorname{vol_{g}}.$

The first two terms vanish, as $T_{m;is}T_{m;ps}$ and $T_{m;is}T_{m;pi}$ are
symmetric in $i,\ p$ and $s,\ p$, respectively, and $X\in\Omega^{2}_{7}$. We
integrate by parts on the remaining two terms:

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}E(\Phi_{t})$
$\displaystyle=\int_{M}-\frac{1}{4}\nabla_{m}T_{m;is}X_{is}+\frac{1}{8}\nabla_{m}T_{m;is}X_{jp}\Phi_{isjp}\operatorname{vol_{g}}$
$\displaystyle=\int_{M}-\frac{1}{4}\nabla_{m}T_{m;is}X_{is}-\frac{3}{4}\nabla_{m}T_{m;is}X_{is}\operatorname{vol_{g}}$
$\displaystyle=-\int_{M}\langle\operatorname{div}T,X\rangle\operatorname{vol_{g}}.$

Using Proposition 3.3–$2.$, we see that $A,B\in\Omega^{2}_{7}$ satisfy
$\langle A\diamond\Phi,B\diamond\Phi\rangle=16\langle A,B\rangle$, so the
above equation becomes

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}16E(\Phi_{t})=-16\int_{M}\langle\operatorname{div}T,X\rangle\operatorname{vol_{g}}=\int_{M}\langle\operatorname{div}T\diamond\Phi,X\diamond\Phi\rangle\operatorname{vol_{g}}.$
(4.7)

The result now follows from the splitting
$\Omega^{4}=\Omega^{4}_{1}\oplus\Omega^{4}_{7}\oplus\Omega^{4}_{27}\oplus\Omega^{4}_{35}$,
with $\Omega^{4}_{7}=\\{Y\diamond\Phi\mid Y\in\Omega^{2}_{7}\\}$ and
$T_{m;is}\in\Omega^{2}_{7}$, for each $m$. ∎

Comparing (4.2) and (HF), we see that $h\equiv 0$ for the harmonic flow, and
hence it is indeed a flow of isometric $\mathrm{Spin}(7)$-structures.

####  4.1.2 Alternative approach: representation theory

The harmonicity condition for $\mathrm{Spin}(7)$-structures can be inferred as
an instance of the general framework in §2.1, stemming from the representation
theory of $\mathrm{Spin}(7)$. The key ingredient is the connection form $f$,
which identifies the vertical of the tangent bundle of the twistor space with
$\mathfrak{m}$, the (naturally reductive) complement of
$\mathfrak{so}(7)=\mathfrak{spin}(7))\subset\mathfrak{so}(8)$. Sections of
this space correspond to $\mathrm{Spin}(7)$-structures and restricting
ourselves to the vertical part means we only look at variations through
$\mathrm{Spin}(7)$-structures.

If $\tilde{\Phi}$ is the universal $\mathrm{Spin}(7)$-structure [cf. (2.5)],
then the connection form is characterised by

$\nabla_{A}\tilde{\Phi}=f(A).\tilde{\Phi}.$

Identifying $\mathfrak{so}(8)\simeq\Omega^{2}$, the reductive complement
$\mathfrak{m}$ is then identified with $\Omega^{2}_{7}$ and
$f(A).\tilde{\Phi}$ should be understood as the the natural
$\mathrm{GL}(8)$-action encoded in the diamond operator (3.13). We invert the
$\diamond$ operator with a triple contraction $\lrcorner_{3}$:

$(\beta\diamond\Phi)\lrcorner_{3}\Phi=96\beta.$

The connection form is then given by
$96f(A)=\nabla_{A}\tilde{\Phi}\lrcorner_{3}\tilde{\Phi}$ and, pulling back by
a section $\sigma$ of the twistor space, we obtain:

$f(d\sigma(X))=\tfrac{1}{96}(\nabla_{X}\Phi)\lrcorner_{3}\Phi.$

Everything now follows from this expression, which is precisely the torsion
$T(X)$ of (3.29). Then the (vertical) energy density is
$|d^{\mathcal{V}}\sigma|^{2}=|T|^{2}$, so the energy is precisely the
$L^{2}$-norm of the torsion. The vertical tension field is

$I(\tau^{\mathcal{V}}(\sigma))=\sum_{1}^{8}\nabla_{e_{i}}(T(e_{i}))-T(\nabla_{e_{i}}e_{i})=\operatorname{div}T$

and the flow $\frac{d\sigma_{t}}{dt}=\tau^{\mathcal{V}}(\sigma_{t})$ is
equivalent to $I(\frac{d\sigma_{t}}{dt})=I(\tau^{\mathcal{V}}(\sigma_{t}))$
where $I$ plays the role of an extended connection $f$ on $M\times\mathbb{R}$.
We know that $I(\tau^{\mathcal{V}}(\sigma_{t}))=\operatorname{div}T_{t}$ and,
generalising to $M\times\mathbb{R}$ (or at least an interval) all the previous
objects, we have

$I\left(\frac{d\sigma_{t}}{dt}\right)=\tfrac{1}{96}\frac{d{\Phi}_{t}}{dt}\lrcorner_{3}{\Phi}_{t}.$

On the other hand, since
$\operatorname{div}T_{t}=\tfrac{1}{96}(\operatorname{div}T_{t}\diamond{\Phi}_{t})\lrcorner_{3}{\Phi}_{t}\in\Omega^{2}_{7}$,
and $\lrcorner_{3}{\Phi}_{t}$ is an isomorphism on $\Omega^{2}_{7}$ (its
kernel being $\Omega^{2}_{21}$), we recover indeed

$\frac{d{\Phi}_{t}}{dt}=\operatorname{div}T_{t}\diamond{\Phi}_{t}.$

######  Remark 4.5.

This flow is the vertical part of the harmonic map flow, so short time
existence of (HF) follows from general properties of parabolic equations, cf.
[LS19].

###  4.2 Immediate consequences from the harmonic $H$-flow

Let us collect from the outset a number of analytic properties of the harmonic
$\mathrm{Spin}(7)$-flow which can be effortlessly deduced from the general
theory in §2.

####  4.2.1 Harmonic $\mathrm{Spin}(7)$-solitons and self-similar solutions

The specific notion of self-similar solution to (HF), derived from Definition
2.7, is the following:

######  Definition 4.6.

Let $\\{\Phi_{t}\\}_{t\in(a,b)}$ be a solution of the harmonic flow (HF) where
$0\in(a,b)$. We say that it is a _self-similar solution_ if there exist a
function $\rho(t)$ with $\rho(0)=1$, a $\mathrm{Spin}(7)$-structure $\Phi_{0}$
and a family of diffeomorphisms $f_{t}:M\rightarrow M$ such that

$\displaystyle\Phi_{t}=\rho(t)^{4}{f_{t}}^{*}\Phi_{0}$

for all $t\in(a,b)$.

For $Y\in\Gamma(TM)$, in view of the splitting (3.15) of $TM^{*}\otimes TM$
and Proposition 3.2, we may write

$\nabla Y=\frac{1}{2}\mathcal{L}_{Y}g+\nabla_{7}(Y)+\nabla_{21}(Y),$

where $\nabla_{k}(Y):=\pi_{k}(\nabla Y)\in\Omega^{2}_{k}$. In particular
$\nabla_{21}Y\in\Omega^{2}_{21}=\ker(\diamond\,\Phi)$. We now use (2.11) to
get

$\mathcal{L}_{Y}\Phi=(Y\lrcorner
T+\frac{1}{2}\mathcal{L}_{Y}g+\nabla_{7}(Y))\diamond\Phi.$ (4.8)

The stationary condition for harmonic $\mathrm{Spin}(7)$-solitons is obtained
from Lemma 2.9:

######  Lemma 4.7.

Given a self-similar solution $\\{\Phi_{t}\\}_{t\in(a,b)}$ of the harmonic
flow (HF), there is a family of vector fields
$\\{X_{t}\\}\subset\mathscr{X}(M)$ such that

$\operatorname{div}{T_{t}}=X_{t}\lrcorner T_{t}+\nabla_{7}(X_{t}),$

where $\nabla_{7}:=\pi_{7}\circ\nabla:\mathscr{X}(M)\to\Omega^{2}_{7}(M)$.

######  Definition 4.8.

A _harmonic $\mathrm{Spin}(7)$-soliton_ on a Riemannian manifold $(M^{8},g)$
is a triple $(\hat{\Phi},\hat{X},c)$, in which $\hat{\Phi}$ is a
$\mathrm{Spin}(7)$-structure with induced metric $g$,
$\hat{X}\in\mathscr{X}(M)$ and $c\in\mathbb{R}$ are such that

$\displaystyle\begin{split}\mathcal{L}_{\hat{X}}g&=cg,\\\
\operatorname{div}\hat{T}&=\hat{X}\lrcorner\hat{T}+\nabla_{7}(\hat{X}).\end{split}$
(4.9)

We obtain the family of stationary vector fields
$\\{X_{t}\\}\subset\mathscr{X}(M)$ predicted in Lemma 4.7, by setting [cf.
(2.12)]

$X_{t}:=(f_{t})^{-1}_{*}W_{t}=\alpha(t)(f_{t})^{-1}_{*}\hat{X}=\alpha(t)\hat{X}.$

Finally, the equivalence between harmonic $\mathrm{Spin}(7)$-solitons and
self-similar solutions to (HF) is summarised in Theorem B, as an instance of
Proposition 2.11.

####  4.2.2 Doubling-time estimates

As an immediate consequence of the general theory of harmonic section flows
[LS19, Thm.2], we obtain a doubling-time estimate for (HF) which says that if
the torsion is bounded at the initial time then it will remain bounded until
some subsequent time.

######  Corollary 4.9 (Doubling-Time Estimate).

Let $\Phi_{t}$ be a solution of the harmonic flow on a compact $8$-manifold
$M$ for $t\in[0,\tau]$. Then there exists $\delta>0$ such that

$\mathcal{T}(t)\leq 2\mathcal{T}(0)$ (4.10)

for all $0\leq t\leq\delta$, where

$\mathcal{T}(t)=\sup_{M}|T(x,t)|.$

####  4.2.3 Shi-type estimates for the harmonic $\mathrm{Spin}(7)$-flow

For $H=\mathrm{Spin}(7)$, the homogeneous fibre bundle $N=P/H\to M^{8}$ of
§2.1 is compact, with fibres isometric to
$\mathrm{SO}(8)/\mathrm{Spin}(7)\simeq\mathbb{RP}^{7}$. In this case,
O’Neill’s curvature formulas [ONe66] enable us to reduce the hypotheses of
Proposition 2.16 to geometric bounds over the base manifold $M$ only.

######  Corollary 4.10.

Let $\Phi_{t}$ be a solution of the harmonic flow (HF) and assume that there
exist $K\geq 1$ and constants $B_{j}$, $0\leq j\in\mathbb{Z}$, such that

$\displaystyle|T|\leq K,\ \ |\nabla^{j}\mathrm{R}|\leq B_{j}K^{j+2},\
\text{for all}\ j\geq 0.$

Then, for all $m\in\mathbb{N}$, there exist constants $C_{m}=C_{m}(g)$ such
that for all $t\in[0,\frac{1}{K^{4}}]$, we have

$\displaystyle|\nabla^{m}T|\leq C_{m}Kt^{-m/2}.$

###  4.3 New analytic preliminaries for the harmonic $\mathrm{Spin}(7)$-flow

We now derive some properties which are context-specific, and cannot
heretofore be obtained directly as instances of the general theory for
$H$-flows.

####  4.3.1 Parabolic rescaling

We discuss the ‘parabolic rescaling’ for (HF). 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 $c$. We will use this notion
frequently in the paper.

######  Lemma 4.11.

Let $c>0$ be a constant. If $\\{\Phi(t)\\}$ is a solution of the harmonic
$\mathrm{Spin}(7)$-flow (HF) with $\Phi(0)=\Phi$, then
$\widetilde{\Phi}(\widetilde{t})=c^{4}\Phi(c^{2}t)$ is still a solution of
(HF) with $\widetilde{\Phi}(0)=c^{4}\Phi$. In particular,

$\widetilde{T}=c^{2}T\quad\text{and}\quad\operatorname{div}_{\widetilde{g}}\widetilde{T}=\operatorname{div}_{g}T.$

###### Proof.

Define a new $\mathrm{Spin}(7)$-structure $\widetilde{\Phi}=c^{4}\Phi$. It
follows from (3.1) and the expressions of $B_{ij}$, $A$ in §3 that
$\widetilde{g}=c^{2}g$, as well as $\widetilde{g}^{-1}=c^{-2}g^{-1}$ and
$\widetilde{\operatorname{vol_{g}}}=c^{8}\operatorname{vol_{g}}$. Recall that
we are suppressing the writing of $g^{-1}$ terms because we are using an
orthonormal frame. Hence (3.29) yields $\widetilde{T}=c^{2}T$. Thus, as a
$2$-form,
$\operatorname{div}_{\widetilde{g}}\widetilde{T}=\operatorname{div}_{g}T$, and
therefore
$(\operatorname{div}_{\widetilde{g}}\widetilde{T})\diamond\widetilde{\Phi}=c^{-2}c^{4}(\operatorname{div}_{g}T\diamond\Phi)=c^{2}(\operatorname{div}_{g}T)\diamond\Phi$.
The main assertion now follows from (HF). ∎

Since $\widetilde{\nabla}=\nabla$, we note, for later use that if
$\widetilde{\Phi}=c^{4}\Phi$ then

$\displaystyle|\widetilde{\nabla}^{j}\widetilde{\mathrm{R}}|_{\widetilde{g}}=c^{-(2+j)}|\nabla^{j}\mathrm{R}|_{g},\
\ \ \ \ \ \ \ |\widetilde{\nabla}^{j}\widetilde{T}|_{g}=c^{-(1+j)}|\nabla
T|_{g}\cdot$ (4.11)

####  4.3.2 Torsion evolution

We determine the evolution equations of the torsion and its norm under the
harmonic $\mathrm{Spin}(7)$-flow, following a process similar to the
$\mathrm{G}_{2}$ case [DGK21, Lemma 3.1].

Let $\\{\partial_{t},e_{1},\dots,e_{7}\\}$ be an orthonormal (geodesic) frame
and use $t$ and $i$ as shorthands for $\partial_{t}$ and $e_{i}$. First we use
the formula

$(R(i,j)T)(a,b,c)=-T(R(i,j)a,b,c)-T(a,R(i,j)b,c)-T(a,b,R(i,j)c)$

to derive a formula for the Laplacian of the torsion of a
$\mathrm{Spin}(7)$-structure.

######  Lemma 4.12.

Let $\Delta=+\operatorname{tr}{\nabla_{i}\nabla_{i}}$ be the Laplacian
operator, then

$\displaystyle(\Delta T)_{m;ab}=$
$\displaystyle\nabla_{m}\nabla_{i}T_{i;ab}-T_{q;ab}R_{imiq}-T_{i;qb}R_{imaq}-T_{i;aq}R_{imbq}+2\nabla_{i}T_{i;ap}T_{m;bp}+2T_{i;ap}\nabla_{i}T_{m;bp}$
$\displaystyle-2\nabla_{i}T_{m;ap}T_{i;bp}-2T_{m;ap}\nabla_{i}T_{i;pb}+\tfrac{1}{4}\nabla_{i}R_{miab}-\tfrac{1}{8}\nabla_{i}R_{mipq}\Phi_{pqab}-\tfrac{1}{8}R_{mipq}\nabla_{i}\Phi_{pqab}.$

###### Proof.

Use the $\mathrm{Spin}(7)$ Bianchi-type identity (3.30) to compute:

$\displaystyle\nabla_{i}\nabla_{i}T_{m;ab}=$ $\displaystyle\
\nabla_{i}(\nabla_{i}T_{m;ab})$ $\displaystyle=$ $\displaystyle\
\nabla_{i}(\nabla_{m}T_{i;ab}+2T_{i;ap}T_{m;pb}-2T_{m;ap}T_{i;pb}+\tfrac{1}{4}R_{miab}-\tfrac{1}{8}R_{mipq}\Phi_{pqab})$
$\displaystyle=$ $\displaystyle\
\nabla_{i}\nabla_{m}T_{i;ab}+2\nabla_{i}T_{i;ap}T_{m;pb}+2T_{i;ap}\nabla_{i}T_{m;pb}-2\nabla_{i}T_{m;ap}T_{i;pb}-2T_{m;ap}\nabla_{i}T_{i;pb}$
$\displaystyle+\tfrac{1}{4}\nabla_{i}R_{miab}-\tfrac{1}{8}\nabla_{i}R_{mipq}\Phi_{pqab}-\tfrac{1}{8}R_{mipq}\nabla_{i}\Phi_{pqab}$

and then use

$\displaystyle\nabla_{i}\nabla_{m}T_{i;ab}=\nabla_{m}\nabla_{i}T_{i;ab}-T_{q;ab}R_{imiq}-T_{i;qb}R_{imaq}-T_{i;aq}R_{imbq}.\qed$

######  Proposition 4.13.

Let $\\{\Phi_{t}\\}$ be a solution of the harmonic $\mathrm{Spin}(7)$-flow
(HF); then its torsion evolves according to the equation

$\displaystyle 4\frac{\partial}{\partial t}T_{m;is}=$ $\displaystyle\ 4(\Delta
T)_{m;is}$
$\displaystyle+\nabla_{a}T_{m;bc}\Big{(}4T_{a;bp}\Phi_{pcis}+T_{a;ip}\Phi_{bcps}+T_{a;sp}\Phi_{bcip}\Big{)}+T_{m;bc}\nabla_{a}T_{a;bp}\Phi_{pcis}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}$
$\displaystyle+T_{m;bc}T_{a;bp}\Big{(}T_{a;pq}\Phi_{qcis}+T_{a;cq}\Phi_{pqis}+2T_{a;iq}\Phi_{pcqs}+2T_{a;sq}\Phi_{pciq}\Big{)}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}T_{a;ip}\Big{(}T_{a;pq}\Phi_{bcqs}+2T_{a;sq}\Phi_{bcpq}\Big{)}+\tfrac{1}{2}T_{m;bc}T_{a;sp}T_{a;pq}\Phi_{bciq}$
$\displaystyle+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}-T_{a;qc}R_{ambq}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}.$

###### Proof.

The general expression for the flow is

$\frac{\partial}{\partial
t}T_{m;is}=X_{ip}T_{m;ps}-X_{sp}T_{m;pi}+\pi_{7}(\nabla_{m}X_{is}),$

where
$(\pi_{7}\beta)_{ij}=\tfrac{1}{4}\beta_{ij}-\tfrac{1}{8}\beta_{ab}\Phi_{abij}$.
Take $X_{is}=(\operatorname{div}T)_{is}=\nabla_{a}T_{a;is}$ for the harmonic
$\mathrm{Spin}(7)$-flow, so

$\displaystyle\frac{\partial}{\partial
t}T_{m;is}=\nabla_{r}T_{r;ip}T_{m;ps}-\nabla_{r}T_{r;sp}T_{m;pi}+\pi_{7}(\nabla_{m}(\nabla_{r}T_{r;is})).$
(4.12)

But

$\displaystyle\nabla_{m}\nabla_{r}T_{r;is}=$ $\displaystyle\ (\Delta
T)_{m;is}+T_{q;is}R_{amaq}+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2\nabla_{a}T_{a;ip}T_{m;sp}-2T_{a;ip}\nabla_{a}T_{m;sp}$
$\displaystyle+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{m;ip}\nabla_{a}T_{a;ps}-\tfrac{1}{4}\nabla_{a}R_{mais}+\tfrac{1}{8}\nabla_{a}R_{mapq}\Phi_{pqis}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$

and note that $T_{q;is}R_{amaq}$ and
$-\tfrac{1}{4}(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})=-\pi_{7}(\nabla_{a}R_{mais})$
are in $\Omega^{2}_{7}$, so

$\displaystyle 4\pi_{7}(\nabla_{m}\nabla_{r}T_{r;is})=$ $\displaystyle\
4\pi_{7}((\Delta
T)_{m;is})+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
(4.13)
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2\nabla_{a}T_{a;ip}T_{m;sp}-2T_{a;ip}\nabla_{a}T_{m;sp}$
$\displaystyle+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{m;ip}\nabla_{a}T_{a;ps}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-\tfrac{1}{2}T_{a;qc}R_{ambq}\Phi_{bcis}-\tfrac{1}{2}T_{a;bq}R_{amcq}\Phi_{bcis}+\nabla_{a}T_{a;bp}T_{m;cp}\Phi_{bcis}$
$\displaystyle+T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\nabla_{a}T_{m;bp}T_{a;cp}\Phi_{bcis}-T_{m;bp}\nabla_{a}T_{a;pc}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}.$

Since $T\in\Omega^{2}_{7}$, we have

$\displaystyle T_{q;bc}\Phi_{bcis}$ $\displaystyle=-6T_{q;is};$
$\displaystyle(96)^{2}T_{m;rp}T_{n;sp}$
$\displaystyle=6\nabla_{m}\Phi_{rjkl}\nabla_{n}\Phi_{sjkl}-9\nabla_{m}\Phi_{rjkl}\nabla_{n}\Phi_{sjbc}\Phi_{klbc};$
$\displaystyle 96T_{a;qc}\Phi_{bcis}$
$\displaystyle=-6\nabla_{a}\Phi_{qbis}+3\nabla_{a}\Phi_{qbkl}\Phi_{klis}+3\nabla_{a}\Phi_{qikl}\Phi_{klsb}+3\nabla_{a}\Phi_{qskl}\Phi_{klbi}.$

The first two terms on line 4 of (4.13) combine to give

$-\tfrac{1}{2}T_{a;qc}R_{ambq}\Phi_{bcis}-\tfrac{1}{2}T_{a;bq}R_{amcq}\Phi_{bcis}=-T_{a;qc}R_{ambq}\Phi_{bcis},$

the third term of line 4 and the third term on line 5 of (4.13) give

$\nabla_{a}T_{a;bp}T_{m;cp}\Phi_{bcis}-T_{m;bp}\nabla_{a}T_{a;pc}\Phi_{bcis}=0.$

The first two terms on line 5 of (4.13) combine to give

$T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\nabla_{a}T_{m;bp}T_{a;cp}\Phi_{bcis}=2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis},$

so

$\displaystyle 4\pi_{7}(\nabla_{m}\nabla_{r}T_{r;is})=$ $\displaystyle\
4\pi_{7}((\Delta
T)_{m;is})+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2\nabla_{a}T_{a;ip}T_{m;sp}-2T_{a;ip}\nabla_{a}T_{m;sp}$
$\displaystyle+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{m;ip}\nabla_{a}T_{a;ps}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}$

and

$\displaystyle 4\frac{\partial}{\partial t}T_{m;is}=$ $\displaystyle\
4\nabla_{a}T_{a;ip}T_{m;ps}-4\nabla_{a}T_{a;sp}T_{m;pi}+4\pi_{7}(\nabla_{m}\nabla_{r}T_{r;is})$
$\displaystyle=$ $\displaystyle\
4\nabla_{a}T_{a;ip}T_{m;ps}-4\nabla_{a}T_{a;sp}T_{m;pi}+4\pi_{7}((\Delta
T)_{m;is})+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2\nabla_{a}T_{a;ip}T_{m;sp}-2T_{a;ip}\nabla_{a}T_{m;sp}$
$\displaystyle+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{m;ip}\nabla_{a}T_{a;ps}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}$
$\displaystyle=$ $\displaystyle\
6\nabla_{a}T_{a;ip}T_{m;ps}-2\nabla_{a}T_{a;sp}T_{m;pi}+4\pi_{7}((\Delta
T)_{m;is})+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}.$

We compute $4\pi_{7}((\Delta T)_{m;is})$:

$\displaystyle 4\pi_{7}((\Delta T)_{m;is})$ $\displaystyle=(\Delta
T)_{m;is}-\tfrac{1}{2}(\Delta T)_{m;bc}\Phi_{bcis}$
$\displaystyle=(\nabla_{a}\nabla_{a}T)_{m;is}-\tfrac{1}{2}(\nabla_{a}\nabla_{a}T)_{m;bc}\Phi_{bcis}.$

Differentiating $T_{m;bc}\Phi_{bcis}=-6T_{m;is}$ twice, we have

$(\nabla_{a}\nabla_{a}T)_{m;bc}\Phi_{bcis}+2\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}+T_{m;bc}(\nabla_{a}\nabla_{a}\Phi)_{bcis}=-6(\nabla_{a}\nabla_{a}T)_{m;is},$

so

$\displaystyle 4\pi_{7}((\Delta T)_{m;is})$ $\displaystyle=(\Delta
T)_{m;is}-\tfrac{1}{2}(\nabla_{a}\nabla_{a}T)_{m;bc}\Phi_{bcis}$
$\displaystyle=(\Delta
T)_{m;is}-\tfrac{1}{2}\Big{(}-6(\nabla_{a}\nabla_{a}T)_{m;is}-2\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}-T_{m;bc}(\nabla_{a}\nabla_{a}\Phi)_{bcis}\Big{)}$
$\displaystyle=4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}+\tfrac{1}{2}T_{m;bc}(\nabla_{a}\nabla_{a}\Phi)_{bcis},$

and the third term expands as follows:

$\displaystyle(\nabla_{a}\nabla_{a}\Phi)_{bcis}=$ $\displaystyle\
\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}+\nabla_{a}T_{a;ip}\Phi_{bcps}+\nabla_{a}T_{a;sp}\Phi_{bcip}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}.$

So

$\displaystyle 4\pi_{7}((\Delta T)_{m;is})=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}+\nabla_{a}T_{a;ip}\Phi_{bcps}+\nabla_{a}T_{a;sp}\Phi_{bcip}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;ip}\Phi_{bcps}+\nabla_{a}T_{a;sp}\Phi_{bcip}\Big{)}$
$\displaystyle=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle-3T_{m;ps}\nabla_{a}T_{a;ip}-3T_{m;ip}\nabla_{a}T_{a;sp}.$

In conclusion,

$\displaystyle 4\frac{\partial}{\partial t}T_{m;is}=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle-3T_{m;ps}\nabla_{a}T_{a;ip}-3T_{m;ip}\nabla_{a}T_{a;sp}$
$\displaystyle+6\nabla_{a}T_{a;ip}T_{m;ps}-2\nabla_{a}T_{a;sp}T_{m;pi}+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis},$

so

$\displaystyle 4\frac{\partial}{\partial t}T_{m;is}=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}\nabla_{a}T_{a;bp}\Phi_{pcis}+\nabla_{a}T_{a;cp}\Phi_{bpis}$
$\displaystyle+T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;cp}\nabla_{a}\Phi_{bpis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}$
$\displaystyle=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}2\nabla_{a}T_{a;bp}\Phi_{pcis}+2T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}.$

Inserting

$\displaystyle\nabla_{m}\Phi_{ijkl}=T_{m;ip}\Phi_{pjkl}+T_{m;jp}\Phi_{ipkl}+T_{m;kp}\Phi_{ijpl}+T_{m;lp}\Phi_{ijkp}$

into first order terms, we get

$\displaystyle 4\frac{\partial}{\partial t}T_{m;is}=$ $\displaystyle\ 4(\Delta
T)_{m;is}+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}2\nabla_{a}T_{a;bp}\Phi_{pcis}+2T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}$
$\displaystyle-
T_{a;qc}R_{ambq}\Phi_{bcis}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}$
$\displaystyle=$ $\displaystyle\ 4(\Delta T)_{m;is}$
$\displaystyle+\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}+T_{m;bc}\nabla_{a}T_{a;bp}\Phi_{pcis}+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}$
$\displaystyle-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}\Big{(}2T_{a;bp}\nabla_{a}\Phi_{pcis}+T_{a;ip}\nabla_{a}\Phi_{bcps}+T_{a;sp}\nabla_{a}\Phi_{bcip}\Big{)}$
$\displaystyle+4T_{q;is}R_{amaq}-(\nabla_{a}R_{mais}-\tfrac{1}{2}\nabla_{a}R_{mapq}\Phi_{pqis})$
$\displaystyle+T_{a;qs}R_{amiq}+T_{a;iq}R_{amsq}+\tfrac{1}{8}R_{mapq}\nabla_{a}\Phi_{pqis}-T_{a;qc}R_{ambq}\Phi_{bcis}-\tfrac{1}{16}R_{mapq}\nabla_{a}\Phi_{pqbc}\Phi_{bcis}.$

Moreover,

$\displaystyle i)\quad$
$\displaystyle\nabla_{a}T_{m;bc}\nabla_{a}\Phi_{bcis}+T_{m;bc}\nabla_{a}T_{a;bp}\Phi_{pcis}+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}$
$\displaystyle-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}$
$\displaystyle=$ $\displaystyle\
\nabla_{a}T_{m;bc}\Big{(}T_{a;bp}\Phi_{pcis}+T_{a;cp}\Phi_{bpis}+T_{a;ip}\Phi_{bcps}+T_{a;sp}\Phi_{bcip}\Big{)}+T_{m;bc}\nabla_{a}T_{a;bp}\Phi_{pcis}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp}+2T_{a;bp}\nabla_{a}T_{m;cp}\Phi_{bcis}$
$\displaystyle=$ $\displaystyle\
\nabla_{a}T_{m;bc}\Big{(}4T_{a;bp}\Phi_{pcis}+T_{a;ip}\Phi_{bcps}+T_{a;sp}\Phi_{bcip}\Big{)}+T_{m;bc}\nabla_{a}T_{a;bp}\Phi_{pcis}$
$\displaystyle+3\nabla_{a}T_{a;ip}T_{m;ps}+\nabla_{a}T_{a;sp}T_{m;pi}-2T_{a;ip}\nabla_{a}T_{m;sp}+2\nabla_{a}T_{m;ip}T_{a;sp};$
$\displaystyle ii)\quad$ $\displaystyle
4T_{m;bc}T_{a;bp}\nabla_{a}\Phi_{pcis}+\tfrac{1}{2}T_{m;bc}T_{a;ip}\nabla_{a}\Phi_{bcps}+\tfrac{1}{2}T_{m;bc}T_{a;sp}\nabla_{a}\Phi_{bcip}$
$\displaystyle=$ $\displaystyle\
T_{m;bc}T_{a;bp}\Big{(}T_{a;pq}\Phi_{qcis}+T_{a;cq}\Phi_{pqis}+T_{a;iq}\Phi_{pcqs}+T_{a;sq}\Phi_{pciq}\Big{)}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}T_{a;ip}\Big{(}T_{a;bq}\Phi_{qcps}+T_{a;cq}\Phi_{bqps}+T_{a;pq}\Phi_{bcqs}+T_{a;sq}\Phi_{bcpq}\Big{)}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}T_{a;sp}\Big{(}T_{a;bq}\Phi_{qcip}+T_{a;cq}\Phi_{bqip}+T_{a;iq}\Phi_{bcqp}+T_{a;pq}\Phi_{bciq}\Big{)}$
$\displaystyle=$ $\displaystyle\
T_{m;bc}T_{a;bp}\Big{(}T_{a;pq}\Phi_{qcis}+T_{a;cq}\Phi_{pqis}+2T_{a;iq}\Phi_{pcqs}+2T_{a;sq}\Phi_{pciq}\Big{)}$
$\displaystyle+\tfrac{1}{2}T_{m;bc}T_{a;ip}\Big{(}T_{a;pq}\Phi_{bcqs}+2T_{a;sq}\Phi_{bcpq}\Big{)}+\tfrac{1}{2}T_{m;bc}T_{a;sp}T_{a;pq}\Phi_{bciq}.$

Gathering terms yields the Proposition. ∎

######  Proposition 4.14.

If $\\{\Phi_{t}\\}$ is a solution of the harmonic $\mathrm{Spin}(7)$-flow (HF)
and $T$ is its torsion, then the evolution equation for $|T|^{2}$ is

$\displaystyle 2\frac{\partial}{\partial t}|T|^{2}$
$\displaystyle=2\Delta|T|^{2}-4|\nabla
T|^{2}+16T_{a;bp}T_{m;bc}T_{a;pq}T_{m;qc}+16T_{a;bp}T_{m;bc}T_{a;cq}T_{m;pq}$
$\displaystyle\quad+16T_{a;qs}T_{m;is}R_{amiq}+4T_{q;is}T_{m;is}R_{amaq}-4T_{m;is}\nabla_{a}R_{mais}.$

######  Remark 4.15.

Even though we obtained the doubling-time estimate in Corollary 4.9 and the
Shi-type estimates in Corollary 4.10 for (HF) as a consequence of the general
theory for (HSF), we could also derive those estimates from Proposition 4.14
and the maximum principle, as one does for the $\mathrm{G}_{2}$ case in
[DGK21, Prop. 3.2 & Thm. 3.3]. Hopefully the general theory will be able to
spare people the effort of similar lengthy computations in the future.

We also have the following local derivative estimates for the torsion tensor
along (HF). The proof is similar to [DGK21, Theorem 3.7] so we omit the
details. We first define the parabolic cylinder of radius $r$ centred at
$(x_{0},t_{0})$ as

$\displaystyle P_{r}(x_{0},t_{0})=\\{(x,t)\in M\times\mathbb{R}\mid
d(x,x_{0})\leq r,\ t_{0}-r^{2}\leq t\leq t_{0}\\}.$

######  Corollary 4.16 (Local derivative estimates).

Let $\Phi(t)$ be a solution to (HF) on $M^{8}$. Let $x_{0}\in M$ and
$t_{0}\geq 0$ such that $\Phi(t)$ is defined at least up to $t_{0}$. There
exist constants $s>0$ and $C_{m}$ for all $m\geq 1$ such that if $|T|\leq K$
and $|\nabla^{j}\mathrm{R}|\leq B_{j}K^{2+j}$ for all $j\geq 0$ in some
parabolic cylinder $P_{r}(x_{0},t_{0})$ with $r\leq s$ and
$K\geq\frac{1}{r^{2}}$, then we have

$\displaystyle|\nabla^{m}T|\leq C_{m}K^{m+1}$ (4.14)

in the smaller parabolic cylinder $P_{\frac{r}{2^{m}}}(x_{0},t_{0})$.

###  4.4 Compactness

We want to prove a Hamilton-type compactness theorem for the solutions of
(HF). Apart from aiding in the understanding of singularity formation, it will
also be useful in the applications of the monotonicity formula.

####  4.4.1 Cheeger-Gromov compactness for $\mathrm{Spin}(7)$-structures

We first prove a very general Cheeger-Gromov type compactness theorem for
$\mathrm{Spin}(7)$-structures which, while believed by experts, has not
appeared in the literature yet.

######  Definition 4.17.

Let $\\{M_{i}\\}$ be a sequence of $8$-manifolds and let $p_{i}\in M_{i}$, for
each $i$. Let $\Phi_{i}$ be a $\mathrm{Spin}(7)$-structure on $M_{i}$ such
that the associated Riemannian metric $g_{i}$ on $M_{i}$ are complete, for
each $i$. Let $M$ be an $8$-manifold with $p\in M$ and $\Phi$ a
$\mathrm{Spin}(7)$-structure on $M$. We say that

$\displaystyle(M_{i},\Phi_{i},p_{i})\rightarrow(M,\Phi,p)\ \ \ \ \textup{as}\
i\rightarrow\infty$

if there exists a sequence of compact subsets $\Omega_{i}\subset M$ exhausting
$M$ with $p_{i}\in\textup{int}(\Omega_{i})$ for each $i$ and a sequence of
diffeomorphisms $F_{i}:\Omega_{i}\rightarrow F_{i}(\Omega_{i})\subset M_{i}$
with $F_{i}(p)=p_{i}$ such that on every compact subset $K$ of $M$ and for
every $\varepsilon>0$, there exists $i_{0}=i_{0}(\varepsilon)$ such that for
$i\geq i_{0}$,

$\displaystyle\underset{x\in K}{\textup{sup}}\
|\nabla^{k}(F_{i}^{*}\Phi_{i}-\Phi)|_{g_{0}}<\varepsilon,\ \ \ \ \ \ \ \
\textup{for}\ k\geq 0,$

uniformly on $K$. Here the covariant derivative is with respect to any fixed
metric $g_{0}$ on $M$.

The proof of the following compactness theorem is similar in spirit to the
compactness theorems for complete pointed Riemannian manifolds [Ham95, Theorem
2.3], and for $\mathrm{G}_{2}$-structures by Lotay–Wei [LW17, Theorem 7.1].

######  Theorem 4.18.

Let $(M_{i}^{8},\Phi_{i},g_{i})$ be a sequence of smooth manifolds with
$\mathrm{Spin}(7)$-structures and with the induced Riemannian metrics, and
suppose that, for each $i$, we have $p_{i}\in M_{i}$ and $g_{i}$ is complete.
Suppose further, that

$\displaystyle\underset{i}{\textup{sup}}\ \underset{x\in M_{i}}{\textup{sup}}\
\left(|\nabla^{k}_{g_{i}}\mathrm{R}_{g_{i}}(x)|^{2}_{g_{i}}+|\nabla^{k+1}_{g_{i}}T_{i}(x)|^{2}_{g_{i}}\right)^{\frac{1}{2}}<\infty,\quad\forall\
k\geq 0,$ (4.15)

where $T_{i}$, $\mathrm{R}_{g_{i}}$ are the torsion of $\Phi_{i}$ and the
curvature of $g_{i}$, respectively, and the injectivity radius of
$(M_{i},g_{i})$ at $p_{i}$ satisfies

$\displaystyle\textup{inf}\ inj(M_{i},g_{i},p_{i})>0,\quad\forall\ i.$

Then there exist a manifold $(M^{8},\Phi)$ with a $\mathrm{Spin}(7)$-structure
and a point $p\in M$ such that, subsequentially,

$\displaystyle(M_{i},\Phi_{i},p_{i})\rightarrow(M,\Phi,p),\quad as\
i\rightarrow\infty.$

###### Proof.

Throughout the proof, we will continue to use the index $i$, even after taking
a subsequence. By (4.15), we have a uniform bound on the Riemann curvature
tensor, and the injectivity radius is positive, so the Cheeger–Gromov
compactness theorem for complete pointed Riemannian manifolds [Ham95, Theorem
2.3] yields a complete Riemannian manifold $(M^{8},g)$ and a point $p\in M$
such that, after passing to a subsequence,

$\displaystyle(M_{i},g_{i},p_{i})\rightarrow(M,g,p),\quad\textup{as}\
i\rightarrow\infty.$ (4.16)

Explicitly, there exist nested compact sets $\Omega_{i}\subset M$ exhausting
$M$, with $p\in\textup{int}(\Omega_{i})$ for each $i$, and diffeomorphisms
$F_{i}:\Omega_{i}\rightarrow F_{i}(\Omega_{i})\subset M_{i}$ with
$F_{i}(p)=p_{i}$, such that, on any compact subset $K\subset M$ and for every
$\varepsilon>0$, there exists $i_{0}=i_{0}(\varepsilon)$ for which the
following hold uniformly on $K$:

$\displaystyle\underset{x\in K}{\textup{sup}}\
|\nabla^{k}(F_{i}^{*}g_{i}-g)|_{g_{0}}<\varepsilon,\quad\text{for}\quad k\geq
0\quad\text{and}\quad i\geq i_{0},$

Fixing $i$ sufficiently large, let us work on the compact subset
$\Omega_{i}\subset M$. Since $\\{\Omega_{i}\\}$ is nested, for each $j\geq 0$
we have $\Omega_{i}\subset\Omega_{i+j}$ and a diffeomorphism
$F_{i+j}:\Omega_{i+j}\rightarrow F_{i+j}(\Omega_{i+j})\subset M_{i+j}$. Define
the restricted diffeomorphism

$\displaystyle F_{i,j}=\left.F_{i+j}\right|_{\Omega_{i}}:\Omega_{i}\rightarrow
F_{i+j}(\Omega_{i})\subset M_{i+j},\quad\forall\ j\geq 0.$

The convergence in (4.16) implies that the sequence
$\\{g_{i,j}=F_{i,j}^{*}g_{i+j}\\}_{j=0}^{\infty}$ of Riemmanian metrics on
$\Omega_{i}$ converges to $g_{i,\infty}=g$ on $\Omega_{i}$, as
$j\rightarrow\infty$.

Let $\nabla,\ \nabla_{i,j}$ be the Levi-Civita connections of $g$ and
$g_{i,j}$ on $\Omega_{i}$, respectively. Let $h=g-g_{i,j}$ and
$A=\nabla-\nabla_{i,j}$ be the difference of the metrics and their connections
on $\Omega_{i}$, respectively. Then locally,

$\displaystyle
A^{c}_{ab}=\frac{1}{2}(g_{i,j})^{cd}(\nabla_{a}h_{bd}+\nabla_{b}h_{ad}-\nabla_{d}h_{ab}).$

Since $g_{i,j}\rightarrow g$ smoothly on $\Omega_{i}$, as
$j\rightarrow\infty$, $g_{i,j}$ and $g$ are equivalent for large enough $j$,
and hence we can take the norms with respect to either of these metrics.
Moreover, $|\nabla^{k}h|_{g}\rightarrow 0$ as $j\rightarrow\infty$, for all
$k\geq 0$. Hence, $A$ is uniformly bounded with respect to $g$ for all large
$j$. Since $\nabla g=0$,

$\displaystyle\nabla^{k}A^{c}_{ab}$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{k}\nabla^{(k+1-l)}(g_{i,j})^{cd}(\nabla^{l}\nabla_{a}h_{bd}+\nabla^{l}\nabla_{b}h_{ad}-\nabla^{l}\nabla_{d}h_{ab})$
$\displaystyle=-\frac{1}{2}\sum_{l=1}^{k}\nabla^{(k+1-l)}(g^{cd}-(g_{i,j})^{cd})(\nabla^{l}\nabla_{a}h_{bd}+\nabla^{l}\nabla_{b}h_{ad}-\nabla^{l}\nabla_{d}h_{ab}).$

Thus there exist constants $c_{k}$ for $k\geq 0$ such that
$|\nabla^{k}A|_{g}\leq c_{k}$ for all $j\geq 0$.

Using the diffeomorphisms $F_{i,j}$, we can define
$\mathrm{Spin}(7)$-structures $\Phi_{i,j}=F^{*}_{i,j}\Phi_{i+j}$ on
$\Omega_{i}$ by pulling back the $\mathrm{Spin}(7)$-structure $\Phi_{i+j}$ on
$M_{i+j}$. We next estimate $|\nabla^{k}\Phi_{i,j}|_{g}$. Since $g$ and
$g_{i,j}$ are equivalent for large $j$, $|\Phi_{i,j}|_{g}\leq
c_{0}|\Phi_{i,j}|_{g_{i,j}}\leq 8c_{0}$ for some constant $c_{0}$. Next,
observe trivially that

$\displaystyle\nabla\Phi_{i,j}=\nabla_{i,j}\Phi_{i,j}+(\nabla-\nabla_{i,j})\Phi_{i,j},$

and since $A$ is uniformly bounded, there is a constant $c_{1}$ such that

$\displaystyle|\nabla\Phi_{i,j}|_{g}\leq