# On the Global Well-Posedness of the Einstein-Yang-Mills System

Petar Griggs<EMAIL_ADDRESS>Department of Mathematics, Harvard
University  Department of Physics, Harvard University  Puskar Mondal
<EMAIL_ADDRESS>Department of Mathematics, Harvard University
Center of Mathematical Sciences and Applications, Harvard University

###### Abstract

In this paper, we present a partial result on the global well-posedness of the
Cauchy problem for the Einstein-Yang-Mills system in the constant mean
extrinsic curvature spatial harmonic and generalized Coulomb gauges as
introduced in [1]. We give a small-data global existence theorem for a family
of $n+1$ dimensional spacetimes with $n\geq 4$, utilizing energy arguments
presented in [2]. We observe that these energy arguments will fail for $n=3$
due to the conformal invariance of the $3+1$ Yang-Mills equations and present
a gauge-covaraiant formulation of the Einstein-Yang-Mills system in $3+1$
dimensions to show that an energy argument cannot be used to prove the global
well-posedness result, regardless of the choice of gauge.

##  1 Introduction

One of the fundamental questions in the study of hyperbolic partial
differential equations is the issue of long time existence. In gauge theories
these issues are made complex due to the fact that the choice of gauge plays
an important role in the existence problem. As an example, the spacetime
harmonic was utilized to prove the local well-posedness of the vacuum
Einstein’s equations by [3]. This gauge was believed to be unstable for large
time. However, [4] later proved the global existence of the Minkowski space in
spacetime harmonic gauge for small data perturbations, suggesting otherwise.
Another familiar example is the use of the Lorentz gauge in the Maxwell
theory. Use of these gauges cast the equations of motion into a hyperbolic
system whose local well-posedness theory follows using standard methods. Long
time existence issues however are substantially more complicated in the case
of non-linear equations and require careful attention to the detail of the
non-linearities present. In several cases choice of gauge plays an important
role, in the sense that while one gauge develops coordinate singularity in
finite time, other gauges may be able to exhaust the entire time interval. In
addition to these gauge choices which are naturally adapted to equations in
fully spacetime covariant form, there are other gauges suitable for studying
the equations on a spacetime $M$ of the product type $\mathbb{R}\times\Sigma$.
One of the most convenient choices is the constant mean curvature spatial
harmonic gauge (CMCSH gauge) introduced by Andersson and Moncrief [5] to study
the vacuum Einstein equations. Application of CMCSH gauge turns Einstein’s
equations into a coupled elliptic-hyperbolic system, whose local well-
posedness was proven in [5]. In the context of Yang-Mills theory, the Coulomb
gauge is a natural analog of the spatial harmonic gauge, and it was used by
Klainerman and Machedon [6] to study the local and global existence problems
of the Yang-Mills equations on Minkowski spacetime.

In addition to the fundamental studies presented in the previous paragraph,
numerous studies in the literature deal with local and global existence
problems of the Einstein and Yang-Mills equations. However, we will only
describe the studies that are relevant to our problem. Firstly, we mention the
work of Lindblad and Rodnianski [4], which used the spacetime harmonic gauge
to obtain a global stability result of Minkowski space. LeFloch and Ma [7]
used the same gauge to establish a small data global existence result of the
Einstein-Klein-Gordon system. In addition to the stability problem of
Minkowski space, spacetime harmonic gauge is also used in the context of the
cosmological stability problem by [8, 9]. Beyond these examples, there are
numerous other studies, e.g. [10, 11, 12], that utilize this particular choice
of gauge. Andersson and Moncrief [2] proved an asymptotic stability result of
the Milne universe utilizing the CMCSH gauge. Later, Andersson and Fajman used
the CMCSH gauge to prove a small data global existence result for the
Einstein-Vlasov system on a Milne spacetime [13]. In addition, [14] obtained a
rough data local well-posedness result for the vacuum Einstein equations, [15]
proved a local well-posedness result of the Einstein-Vlasov system, [16]
studied the stability of the Milne universe in the presence of a perfect
fluid, and [17] studied the asymptotic stability of the Milne universe coupled
to a Klein-Gordon field, among others. There are numerous studies where CMCSH
gauge is used to study the vacuum gravity problem or gravity coupled to matter
fields, for example [18, 19, 20]. A substantial amount of study has also been
done in the Yang-Mills sector. Eardley and Moncrief [21, 22] established the
global existence of Yang-Mills fields on a Minkowski background in temporal
gauge. Later, [23] extended the global existence result to globally hyperbolic
curved spacetimes in the same gauge. As mentioned earlier, using Coulomb
gauge, Klainerman and Machedon [6] established a global existence result in
energy norm. Tao [24] subsequently presented a below-energy-norm local
existence theorem for the Yang-Mills equations on Minkowski space in temporal
gauge. There are countless other studies which are less relevant in the
current context.

Our current article is motivated by the desire to extend the result of [2] to
include a non-trivial Yang-Mills source. We employ a generalized Coulomb (GC)
gauge to cast the Yang-Mills equations into a coupled elliptic-hyperbolic
system, mirroring the gravitational sector which is also reduced to a coupled
elliptic-hyperbolic system by the application of the CMCSH gauge. A local
well-posedness of this coupled elliptic hyperbolic system in CMCSHGC gauge was
established in [1]. However, we find that the energy argument presented in [2]
for pure gravity fails for the Einstein-Yang-Mills system in $3+1$ dimensions.
In particular, the decay of Yang-Mills fields are weak in an appropriate sense
and as such fail to overcome the non-linearities present through a
straightforward energy argument. This is tied to the conformal invariance of
Yang-Mills equations in $3+1$ dimensions: the Milne universe is conformal to a
cylinder spacetime that exhibits no decay, and so it is natural that Yang-
Mills fields would not have an uniform decay on the $3+1$ Milne background.
This problem does not occur in the presence of Maxwell sources, since the
later is linear [25] and uniform decay for the Maxwell field is not required
to close the argument. However, in $n+1$ with $n\geq 4$ dimensions, a modified
energy type argument works and we obtain the desired asymptotic behaviour that
leads to small data global existence result. The result essentially depends on
the spectrum of a Lichnerowicz-type Laplacian (acting on $2-$tensors) as well
as that of a Hodge-type Laplacian (acting on $1-$forms). The additional
dependence on the spectrum of a Hodge-type Laplacian originates from the Yang-
Mills sector.

The failure of the energy method in $3+1$ dimensions necessitates applications
of sophisticated analytical techniques such as light-cone estimates [26, 27].
We believe that the $3+1$ Milne model is stable under coupled Einstein-Yang-
Mills perturbations and we are unable to prove it at present using energy
estimates alone. We note that Minkowski space is stable under coupled
Einstein-Yang-Mills perturbations [28], and the proof required sophisticated
double null energy estimates; such a technique fails in the current context
due to non-trivial topology of the spacetime. We intend to address this
problem in the future.

The structure of the article is as follows. In sections 2 and 3, we present a
brief overview of the background spacetimes we are studying and the
construction of the Yang-Mills theory as in [1]. We also discuss the gauge-
fixing and re-scaling that we must do to study the dynamics over time of the
geometric quantities that describe the spatial structure of the spacetime. We
describe the small-data scenario of the Einstein-Yang-Mills system.

In section 4, we give a series of estimates for the gauge variables, while in
section 5 we construct a energy analogous to that presented in [2] and find
leading-order estimates on the decay of the energies. The decay of the energy
gives us a global existence result for the Cauchy initial value problem of the
Einstein-Yang-Mills problem. Finally, in section 6, we present a gauge
covariant formulation of the Einstein-Yang-Mills in $3+1$ dimensions to
demonstrate that an energy argument of the form used for $n+1$, with $n\geq
4$, dimensions will fail.

##  2 Preliminaries

Here we will briefly review the setup, terminology, and notation that we will
use throughout the rest of this work. Take an integer $n>3$. Let $M$ be a
globally hyperbolic $n+1$ dimensional spacetime manifold. As $M$ is globally
hyperbolic, we may foliate the spacetime by closed, spatial Cauchy
hypersurfaces $\Sigma$, such that $M$ is decomposed as
$(0,\infty)\times\Sigma$. Let $\mathcal{M}_{\Sigma}$ denote the space of
Riemannian metrics on a constant time hypersurface $\Sigma$, and let
$\mathcal{E}_{\alpha}\subset\mathcal{M}_{\Sigma}$ be the subset of negative
Einstein metrics with Einstein constant $-\alpha$, i.e., the set of all
metrics $\gamma$ that satisfy

$\mathrm{Ric}[\gamma]=-\alpha\gamma.$

We will restrict our interest in this paper to metrics in the space
$\mathcal{E}_{\alpha}$, where we may re-scale the metric such that
$\alpha=\frac{n-1}{n^{2}}$. If we take a $\gamma\in\mathcal{E}_{\alpha}$, we
then have a hyperbolic metric $\tilde{\gamma}$ on $M$ given by

$\tilde{\gamma}=-\frac{n^{2}}{t^{4}}\mathrm{d}t\otimes\mathrm{d}t+\frac{1}{t^{2}}{\gamma}_{ij}\>\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$
(1)

for $t\in(0,\infty)$. Observe that $\tilde{\gamma}$ also has vanishing Ricci,
and indeed Riemann, curvature and hence is a solution of the vacuum Einstein’s
equations. This solution will serve as the background geometry in our later
analysis, and we will aim to study whether small perturbations to both the
geometry and matter source decay to yield again a metric of this form.

Throughout the paper, we will use $g$ to denote the dynamical metric that
satisfies Einstein’s equations in the ADM formalism. We will denote by
$\gamma$ a $C^{\infty}$ background metric that satisfies the so-called shadow-
metric condition with respect to $g$; see 2.1.1 for a discussion of the
conditions we impose on $\gamma$. Note that we will always have
$\gamma\in\mathcal{E}_{\alpha}$.

We view tensors as sections of vector bundles, and we will frequently make use
of index notation when working with tensors. We let Greek indices
($\mu,\nu,\dots$) denote tensors on the spacetime manifold $M$, and let Latin
indices from the middle of the alphabet ($i,j,\dots$) denote tensors on the
spatial hypersurface $\Sigma$.

Now, if $u$ and $v$ are sections of a rank-$(m,n)$ tensor bundle over
$\Sigma$, we define an inner product as

$\langle
u,v\rangle=\int_{\Sigma}\gamma_{i_{1}j_{1}}\dots\gamma_{i_{n}j_{n}}\gamma^{k_{1}\ell_{1}}\dots\gamma^{k_{m}\ell_{m}}u^{i_{1}\dots
i_{n}}{}_{k_{1}\dots k_{m}}v^{j_{1}\dots
j_{n}}{}_{\ell_{1}\dots\ell_{m}}\>\mu_{g},$ (2)

where $\mu_{g}$ is the volume form $\sqrt[\ ]{\det
g}\mathrm{d}x^{1}\wedge\dots\wedge\mathrm{d}x^{n}$. We will occasionally abuse
this notation and use $\mu_{g}$ to simply denote $\sqrt[\ ]{\det g}$, though
it will be clear from context when we do or do not include the measure
$\mathrm{d}x^{1}\wedge\dots\wedge\mathrm{d}x^{n}$.

We will denote the covariant derivative with respect to the metrics $g$ and
$\gamma$ as $\nabla$ and $\nabla[\gamma]$, respectively. We will use this type
of notation when referring to other metric-derived quantities, such as the
Riemann curvature $R$ and $R[\gamma]$, but will often still explicitly specify
the Christoffel symbols as $\Gamma[g]$ and $\Gamma[\gamma]$. We will
occasionally use $\nabla[\gamma]^{i}$ to denote $i$-times repeated covariant
derivatives.

For derivatives of the rank-$(m,n)$ tensors $u$ and $v$, we define the inner
product

$\langle\nabla[\gamma]u,\nabla[\gamma]v\rangle=\int_{\Sigma}\gamma_{i_{1}j_{1}}\dots\gamma_{i_{n}j_{n}}\gamma^{k_{1}\ell_{1}}\dots\gamma^{k_{m}\ell_{m}}g^{rs}(\nabla[\gamma]_{r}u)^{i_{1}\dots
i_{n}}{}_{k_{1}\dots k_{m}}(\nabla[\gamma]_{s}v)^{j_{1}\dots
j_{n}}{}_{\ell_{1}\dots\ell_{m}}\>\mu_{g}.$ (3)

The inner product for higher derivatives of $u$ and $v$ is defined
analogously. Now, define the twisted rough Laplacian as the second-order
differential operator

$\Delta_{g}^{\gamma}=-\frac{1}{\sqrt[\
]{\mu_{g}}}\nabla[\gamma]_{i}(g^{ij}\mu_{g}\nabla[\gamma]_{j}).$ (4)

We note that the twisted Laplacian is self-adjoint with respect to the inner
product (3). We define the usual covariant Laplacian as
$\Delta_{g}=-g^{ij}\nabla_{i}\nabla_{j}$, such that $\Delta_{g}$ has non-
negative spectrum. From $\Delta_{g}^{\gamma}$, we can define two additional
differential operators which will be key to the later analysis. First, let
$\mathcal{L}_{g,\gamma}$ denote the second-order differential operator that
acts on two-forms $h$ as

$\mathcal{L}_{g,\gamma}h_{ij}=\Delta^{\gamma}_{g}h_{ij}-2g^{mk}g^{n\ell}R[\gamma]_{imjn}h_{k\ell}.$
(5)

Similarly, let $\mathfrak{L}_{g,\gamma}$ denote the second-order differential
operator that acts on one-forms $\omega$ as

$\mathfrak{L}_{g,\gamma}\omega_{i}=\Delta^{\gamma}_{g}\omega_{i}-g^{j\ell}R[\gamma]^{m}{}_{\ell
ij}\omega_{m}.$ (6)

We note that $\mathfrak{L}_{\gamma,\gamma}=\Delta_{\gamma}-\text{Ric}[\gamma]$
is related to the Hodge Laplacian
$\Delta_{H}\equiv(\mathrm{d}\mathrm{d}^{\star}+\mathrm{d}^{\star}\mathrm{d})$
by

$\mathfrak{L}_{\gamma,\gamma}\omega_{i}=\Delta_{H}\omega_{i}-2\mathrm{Ric}[\gamma]^{m}{}_{i}\omega_{m}$

and, like the Hodge Laplacian, will also have non-negative spectrum as
$\mathrm{Ric}[\gamma]$ is negative-definite. For $g$ near $\gamma$, we will
then also have the spectrum of $\mathfrak{L}_{g,\gamma}$ is non-negative. We
will also assume that $\mathcal{L}_{g,\gamma}$ has non-negative spectrum for
$g$ near $\gamma$, motivated by the fact that there is no known compact,
negative Einstein manifold for which $\mathcal{L}_{\gamma,\gamma}$ has
negative eigenvalues [2]. The assumption of non-negativity of these operators
spectra will be key in the later energy arguments.

For the Yang-Mills theory, we recall the setup from [1]. Let $P$ be a
principle $G$-bundle over $M$ where $G$ is any compact, semi-simple Lie group
of dimension $d_{G}$. Let $\mathfrak{g}$ denote the Lie algebra of $G$ and
take $V$ to be a real $d_{V}$-dimensional representation of $\mathfrak{g}$.
Because $G$ is compact, we may find a positive-definite adjoint-invariant
metric on $\mathfrak{g}$. We take $\\{(\chi_{A})^{a}{}_{b}\\}_{A=1}^{d_{G}}$
to be a basis of $\mathfrak{g}$ in a $d_{V}$-dimensional representation $V$
for which

$-\operatorname{Tr}(\chi_{A}\chi_{B})=(\chi_{A})^{a}{}_{b}(\chi_{B})^{b}{}_{a}=\delta_{AB},$

where $a,b\in\\{1,\dots,d_{V}\\}$ denote indices at the level of the
representation $V$. We will reserve Latin indices from the beginning of the
alphabet ($a,b,\dots$) for the indices of the representation $V$. Occasionally
we will suppress these indices and simply write the Lie algebra-valued
$k$-form $A^{a}{}_{b}$ as $A$, where $A$ is then understood to be a
$d_{V}\times d_{V}$ real matrix. With this choice of metric, we may define a
gauge invariant inner product on $\mathfrak{g}$. With $A$ and $B$ both Lie
algebra-valued $(m,n)$-tensors, we have an inner product given by

$\langle
A,B\rangle=\int_{\Sigma}\gamma_{i_{1}j_{1}}\dots\gamma_{i_{n}j_{n}}\gamma^{k_{1}\ell_{1}}\dots\gamma^{k_{m}\ell_{m}}A^{a}{}_{b}{}^{i_{1}\dots
i_{n}}{}_{k_{1}\dots k_{m}}B^{a}{}_{b}{}^{j_{1}\dots
j_{n}}{}_{\ell_{1}\dots\ell_{m}}\>\mu_{g}.$ (7)

Note that $A$ and $B$ must themselves be gauge covariant objects for the inner
product $\langle A,B\rangle$ to be gauge invariant. We may also extend this
definition to the derivatives $\nabla[\gamma]A$ and $\nabla[\gamma]B$ in a
fashion identical to (3).

Finally, we will have a connection $1$-form on $P$, which for the chosen basis
and representation we will denote as
$\tilde{A}^{a}{}_{b}=\tilde{A}^{a}{}_{b\mu}\mathrm{d}x^{\mu}$. From
$\tilde{A}$, we may define the Yang-Mills field strength as

$\tilde{F}^{a}{}_{b\mu\nu}=\partial_{\mu}\tilde{A}^{a}{}_{b\nu}-\partial_{\nu}\tilde{A}^{a}{}_{b\mu}+\big{[}\tilde{A},\tilde{A}\big{]}^{a}{}_{b\mu\nu},$
(8)

where $[\bullet,\bullet]$ is the Lie bracket of $\mathfrak{g}$ and corresponds
to the usual matrix commutator in the representation $V$. Note that we fix the
Yang-Mills coupling constant as unity since we are working at a classical
level thus do not allow running of the coupling constant. We also note that
$\tilde{A}$ is not gauge covariant, though we may construct a gauge covariant
quantity by subtracting from $\tilde{A}$ another $\mathfrak{g}$-valued
$1$-form. We also emphasize that we will view $\tilde{A}$ as a perturbation of
the background geometry, which is a solution to the vacuum Einstein’s
equations.

Using the various inner products defined above, we may define the
$L^{2}(\Sigma)$ norm of a tensor $v$ by $\lVert v\rVert_{L^{2}}=\langle
v,v\rangle^{\frac{1}{2}}$, where the appropriate inner product is used based
on the particular type of object that $v$ is. The $L^{2}(\Sigma)$ norm will be
especially important, as we will conduct our analysis in the Sobolev spaces
$H^{s}$, where we fix $s>\frac{n}{2}+1$. The choice of such an $s$ ensures
that we have the embedding $H^{s-1}(\Sigma)\hookrightarrow
L^{\infty}(\Sigma)$, where the $L^{\infty}$ norm of a generic Lie algebra-
value rank-$(m,n)$ tensor $v$ is given by

$\lVert
v\rVert_{L^{\infty}}=\sup_{\Sigma}\big{(}\gamma_{i_{1}j_{1}}\dots\gamma_{i_{n}j_{n}}\gamma^{k_{1}\ell_{1}}\dots\gamma^{k_{m}\ell_{m}}v^{a}{}_{b}{}^{i_{1}\dots
i_{n}}{}_{k_{1}\dots k_{m}}v^{b}{}_{a}{}^{j_{1}\dots
j_{n}}{}_{\ell_{1}\dots\ell_{m}}\big{)}^{\frac{1}{2}}$

and is defined analogously for the other types of objects we have discussed.

###  2.1 Gauge Fixing of Evolution Equations

In this section we will discuss how we construct the Einstein-Yang-Mills
system and the gauge choices that we must make to study the evolution of the
geometric variables. This is largely an overview of the construction of the
Einstein-Yang-Mills system presented in [1].

As we work in the ADM formalism, we foliate $M$ by spacelike hypersurfaces
$\Sigma_{t}$ and decompose the vector field $\partial_{t}$ on the manifold $M$
into components orthogonal to and parallel to $\Sigma_{t}$, writing

$\partial_{t}=N\mathbf{n}+X.$ (9)

Here the shift vector field $X=X^{i}\partial_{x_{i}}$ is a section of the
tangent bundle $T\Sigma_{t}$, while the lapse function $N$ controls the normal
distance between two constant $t$ hypersurfaces. From this decomposition, we
may write Einstein’s equations in the form of evolution equations for the
Riemannian metric $g$ and second fundamental form $k$ of $\Sigma_{t}$. We
denote by $\tau=\operatorname{tr}_{g}(k)$ the trace of the second fundamental
form. The quantity $\tau$ is the mean extrinsic curvature of $\Sigma_{t}$ as a
surface embedded in $M$, and will, under suitable assumptions, provide us with
a natural time function. We note that, because the Milne universe model that
we are considering is expanding, we have $\tau<0$.

As we will consider a matter source in Einstein’s equations given by a Yang-
Mills stress-energy tensor, we will also wish to decompose the connection
$1$-form $\tilde{A}$ and field strength $\tilde{F}$ into components parallel
to and orthogonal to the hypersurface $\Sigma_{t}$. We do so by writing

$\displaystyle\tilde{A}^{a}{}_{b}=A^{a}{}_{b}-\tilde{g}(\tilde{A}^{a}{}_{b},\mathbf{n})\mathbf{n},$
(10)
$\displaystyle\tilde{F}^{a}{}_{b}=\mathcal{E}^{a}{}_{b}\otimes\mathbf{n}-\mathbf{n}\otimes\mathcal{E}^{a}{}_{b}+F^{a}{}_{b}.$
(11)

Here $A^{a}{}_{b}$ and $\mathcal{E}^{a}{}_{b}$ are Lie algebra-valued one-
forms, with $\mathcal{E}^{a}{}_{bi}$ the components of the electric field.
Meanwhile
$F^{a}{}_{bij}=\partial_{i}A^{a}{}_{bj}-\partial_{j}A^{a}{}_{bi}+[A,A]^{a}{}_{bij}$
is a Lie algebra-valued two form on $\Sigma_{t}$ related to the magnetic
field. The value $\tilde{g}(\tilde{A}^{a}{}_{b},\partial_{t})$ is denoted as
$A^{a}{}_{b0}$. After performing this splitting, we find a system of evolution
and constraint equations for the quantities $(k,g,\mathcal{E},A)$; see
equations (26)-(31) in [1]. However, to obtain a hyperbolic system of
evolution equations, we need to make an appropriate choice of gauge. In
particular, the system of evolution and constraint equations leave us free to
make choices of $N$, $X$, and a quantity related to $A^{a}{}_{b0}$.

We will fix the gauge variables $N$ and $X$ that arise from the ADM
decomposition using the constant mean extrinsic curvature spatial harmonic
(CMCSH) gauge. We first set the mean extrinsic curvature $\tau$ to be constant
on each leaf in the foliation of $M$, thus enabling $\tau$ to serve as a time
function. To make this choice of time function, we must know that the
spacetime $M$ admits a constant mean extrinsic curvature hypersurface. As
discussed in [29, 30, 31, 32], not all spacetimes will admit such a
hypersurface. However the background metric $\tilde{g}$ will, and so we will
assume that the perturbed spacetime we work with does as well. In particular,
we will take $t=\tau$. Explicitly computing $\frac{\partial\tau}{\partial t}$
from the definition $\tau=\operatorname{tr}_{g}(k)$ then yields an elliptic
equation for the lapse function $N$; see Eq. (34) of [1]. To fix the shift
vector field $X$, we utilize the spatial harmonic gauge, which is the
requirement that the tension field vanishes, i.e.,

$V^{k}=g^{ij}\big{(}\Gamma[g]^{k}{}_{ij}-\Gamma[\gamma]^{k}{}_{ij}\big{)}=0.$
(12)

Taking the time derivative of $V^{k}$ in the spatial harmonic gauge and
imposing the constraint on $\nabla\widehat{k}$ obtained from the Codazzi
equation, we may obtain an elliptic equation for $X$; see Eq. (37) of [1]. We
note that in the spatial harmonic gauge, the twisted rough Laplacian
$\Delta^{\gamma}_{g}$ will take the simplified form

$\Delta^{\gamma}_{g}=-g^{mn}\nabla[\gamma]_{m}\nabla[\gamma]_{n}+V^{m}\nabla[\gamma]_{m}=-g^{mn}\nabla[\gamma]_{m}\nabla[\gamma]_{n}.$
(13)

Finally, we must fix a Yang-Mills gauge variable. We define
$\varphi\in\mathfrak{g}$ to be
$\varphi^{a}{}_{b}=A^{a}{}_{b0}-A^{a}{}_{bi}X^{i}$ to be this gauge variable.
We do not directly use the temporal coordinate of the connection $1$-form
$A_{0}$ as it would require a higher regularity than $\varphi$ and thus the
later energy arguments would not close. To obtain a constraint on $\varphi$,
we adopt the generalized Coulomb gauge, given by the condition

$g^{ij}\nabla^{\widehat{A}}[\gamma]_{i}(A_{j}-\widehat{A}_{j})\coloneqq
g^{ij}\nabla[\gamma]_{i}(A_{j}-\widehat{A}_{j})+g^{ij}\big{[}\widehat{A}_{i},(A_{j}-\widehat{A}_{j})\big{]}=0.$
(14)

Here $\nabla^{\widehat{A}}[\gamma]$ is the gauge covariant derivative with
respect to the smooth background connection $1$-form $\widehat{A}$. Taking the
time derivative of (14) and imposing the constraint on $\nabla\mathcal{E}$
obtained from Gauss’ law then yields an elliptic equation on $\varphi$. The
full set of evolution and constraint equations is given in Section 3 of [1],
though we do not present them here as we will, in Section 3.1, give a scaled
form of the equations; see (20)–(26).

####  2.1.1 Shadow Metric Condition

If $\gamma_{0}\in\mathcal{E}_{(n-1)/n^{2}}$ is the background metric near
which we perturb to get the dynamical metric $g$, we will not, in general,
have that $g$ also lies in $\mathcal{E}_{(n-1)/n^{2}}$. However, as we wish to
study the stability of $M$, we will need to study whether the dyanmical metric
$g$ converges to an element of $\mathcal{E}_{(n-1)/n^{2}}$. For this, we
present the shadow metric condition, introduced in [2].

To begin, we must introduce the concept of the deformation space of a fixed
metric $\gamma_{0}\in\mathcal{E}_{(n-1)/n^{2}}$. First, define the set
$\mathcal{S}_{\gamma_{0}}$ to those metrics $g\in\mathcal{M}_{\Sigma}$ such
that the identity map $(M,g)\mapsto(M,\gamma_{0})$ is harmonic. This is
equivalent to the tension field of $g$ and $\gamma_{0}$ vanishing, i.e.,

$\mathcal{S}_{\gamma_{0}}=\big{\\{}g\in\mathcal{M}_{\Sigma}\>|\>g^{ij}\big{(}\Gamma[g]^{k}_{ij}-\Gamma[\gamma_{0}]^{k}{}_{ij})=0\big{\\}}.$

Hence $\mathcal{S}_{\gamma_{0}}$ will be the set of metrics that satisfy the
CMCSH gauge condition. We have that $\mathcal{S}_{\gamma_{0}}$ defined in this
way is both a submanifold of the space $\mathcal{M}_{\Sigma}$ close to
$\gamma$ and a slice for the diffeomorphism group, that it is invariant under
isometries of $\gamma_{0}$; c.f. [2] and [33]. With
$\mathcal{S}_{\gamma_{0}}$, we may define the deformation space of
$\gamma_{0}$ as follows: let $\mathcal{V}\subset\mathcal{E}_{(n-1)/n^{2}}$ be
the connected component of $\gamma_{0}$ and $\mathcal{S}_{\gamma_{0}}$ the
slice defined above. The deformation space of $\gamma_{0}$ is then defined to
be the set $\mathcal{N}=\mathcal{V}\cap\mathcal{S}_{\gamma_{0}}$. Throughout
the remainder of this paper we will use $\mathcal{N}$ to refer to the
deformation space of a fixed $\gamma_{0}\in\mathcal{E}_{(n-1)/n^{2}}$ and
$\gamma$ to refer to a general element of $\mathcal{N}$.

The deformation space of $\gamma_{0}$ is thus the set of background metrics
which we expect the dynamical metric $g$ of Einstein’s equations to converge
to; namely they are the negative Einstein metrics that preserve the CMCSH
gauge condition. Note that by Theorem 5.26 of [33] we may find a set of
coordinates such that $\gamma\in\mathcal{E}_{(n-1)/n^{2}}$ is smooth. It is
thus reasonable for us to assume that $\mathcal{N}\subset\mathcal{M}_{\Sigma}$
has a $C^{\infty}$ topology so that we may smoothly evolve metrics within the
space.

By the discussion in Section 2.3 of [2], we have that the formal tangent space
$T_{\gamma}\mathcal{N}$ for some $\gamma\in\mathcal{N}$ is given by

$T_{\gamma}\mathcal{N}=\\{h\in\mathrm{Sym}^{2}(\Sigma)\>|\>h=h^{\text{TT}\parallel}+L_{Y^{\parallel}}\gamma\\},$
(15)

where $\mathrm{Sym}^{2}(\Sigma)$ is the set of symmetric, rank-$2$ covariant
tensors on $\Sigma$, $h^{\text{TT}\parallel}$ is a transverse-traceless tensor
with respect to $\gamma$, and $Y^{\parallel}$ is a vector field that solves

$(DV)_{j}(L_{Y^{\parallel}}\gamma)^{ij}=h^{\text{TT}\parallel
mn}\big{(}\Gamma[\gamma]^{i}{}_{mn}-\Gamma[\gamma_{0}]^{i}{}_{mn}\big{)},$
(16)

with $DV$ the Fréchet derivative of the tension field
$V^{k}=\gamma^{ij}(\Gamma[\gamma]^{k}{}_{ij}-\Gamma[\gamma_{0}]^{k}{}_{ij})$
taken with respect to $\gamma$. It turns out that $\mathcal{N}$ is a finite-
dimensional submanifold of $\mathcal{M}_{\Sigma}$ [33], and so we let
$\\{q^{\alpha}\\}_{\alpha=1}^{\dim\mathcal{N}}$ be a set of local coordinates
of $\mathcal{N}$. If $\gamma\in\mathcal{N}$, then we may construct a basis for
the formal tangent space $T_{\gamma}\mathcal{N}$ as
$\big{\\{}\frac{\partial\gamma}{\partial
q^{\alpha}}\big{\\}}_{\alpha=1}^{\dim\mathcal{N}}$. For an arbitrary
Riemannian metric $g\in\mathcal{M}_{\Sigma}$, we can define the $L^{2}$
projection with respect to the metric $\gamma$ as

$\bigg{\langle}g,\frac{\partial\gamma}{\partial
q^{\alpha}}\bigg{\rangle}_{\gamma}=-\int_{\Sigma}\gamma^{im}\gamma^{jn}g_{ij}\frac{\partial\gamma_{mn}}{\partial
q^{\alpha}}\>\mu_{\gamma}.$ (17)

With this inner product, we can state the shadow metric condition as follows:
if $g\in\mathcal{M}$ and $\gamma\in\mathcal{N}$, then $\gamma$ is a shadow
metric of $g$ if

$\bigg{\langle}g-\gamma,\frac{\partial\gamma}{\partial
q^{\alpha}}\bigg{\rangle}_{\gamma}=0$ (18)

for all $\alpha\in\\{1,\dots,\dim\mathcal{N}\\}$. By Lemma 4.3 of [2], we have
that this shadow metric condition will always be uniquely satisfied by some
$\gamma\in\mathcal{N}$ for $g\in\mathcal{M}_{\Sigma}$ in a sufficiently small
neighborhood of $\gamma_{0}$.

Geometrically we may view the shadow metric condition as being that
$u\coloneqq g-\gamma$ is orthogonal to the deformation space $\mathcal{N}$, or
equivalently that $\gamma$ is the projection of $g$ onto $\mathcal{N}$.
Indeed, the evolution of $g$ will be entirely determined by the horizontal
evolution of $\gamma$ and the orthogonal evolution of $u$. As $\gamma$ will
evolve within the deformation space $\mathcal{N}$, then to see whether $g$
converges to a point in $\mathcal{N}$ it will suffice to study whether the
orthogonal component $u$ vanishes. In light of the view of $\gamma$ as the
projection of $g$, we note that the shadow metric condition (18) will give us
an estimate of the time evolution $\partial_{T}\gamma$ of the form

$\lVert\partial_{T}\gamma\rVert\leq C\lVert\partial_{T}g\rVert_{L^{2}}\leq
C\lVert\partial_{T}g\rVert_{H^{s-1}}$ (19)

if $g$ lies in a suitably small neighborhood of $\gamma_{0}$. Here $C>0$ is a
constant depending only on the background geometry, and the norm of
$\partial_{T}\gamma$ is arbitrary as $\gamma$, and hence $\partial_{T}\gamma$,
lies in a finite-dimensional vector space. We also note that, if
$\mathcal{L}_{\gamma_{0},\gamma_{0}}$ has strictly positive spectrum, then
$\mathcal{N}=\\{\gamma_{0}\\}$ [2]. This will not, however, affect our
analysis as we may still trivially bound $\partial_{T}\gamma=0$ by
$C\lVert\partial_{T}g\rVert_{H^{s-1}}\geq 0$.

Finally, we remark that one may also wish to define a similar condition for
the analogous quantity in the Yang-Mills theory, namely the connection one-
form $A$. However, unlike the deformation space of metrics, the deformation
space of connection one-forms, which we denote $\mathcal{A}$, is not in
general well-understood. In fact, as even the topology of $\mathcal{A}$ is
unknown, we cannot hope to perform an analogous decomposition to the dynamics
of $A$ and study whether $A$ will converge to a non-vacuum solution of the
Einstein’s equations. Since the vanishing of the Yang-Mills curvature form $F$
will yield a solution to the Einstein-Yang-Mills system, we will perturb the
connection one-form $A$ about the fixed flat background connection
$\hat{A}\equiv 0$ throughout the rest of this work.

##  3 Rescaled Einstein’s Equations

###  3.1 Rescaled Elliptic and Constraint Equations

We follow [2] in rescaling the system of evolution and constraint equations in
the CMCSH and generalized Coulomb gauges to obtain a system of equations
describing the behavior of dimensionless quantities. However, unlike the
vacuum Einstein equations studied by Andersson and Moncrief, we will find that
the Yang-Mills matter source blocks the rescaled system from being fully
autonomous. The rescalling will, however, ensure that all factors of $\tau$
only appear alongside the rescaled Yang-Mills variables $\mathcal{E}$ and
$F=\mathrm{d}A+[A,A]$. This is a consequence of the Yang-Mills matter field
not being scale invariant in $n+1$, with $n\geq 4$, dimensions, and physically
can be thought of as the hyperbolic structure of the spacetime dispersing the
matter source and causing the Yang-Mills field strength to vanish as
$\tau\rightarrow 0$. We note, however, that in $3+1$ dimensions, the Yang-
Mills equations are conformally invariant.

As in [2], we will take $\tau=\operatorname{tr}_{g}(k)$ to have dimensions of
$(\text{length})^{-1}$ and the spatial coordinates $x^{i}$ to be
dimensionless. For the quantities derived from the geometry of $\Sigma$,
namely $(k,g,N,X)$, we obtain the scaling

$\displaystyle[g]$ $\displaystyle\sim(\text{length})^{2},$
$\displaystyle\quad\quad[k]$ $\displaystyle\sim(\text{length}),$
$\displaystyle[N]$ $\displaystyle\sim(\text{length})^{2},$ $\displaystyle[X]$
$\displaystyle\sim(\text{length}).$

Because $k$ has an implicit dependence on $\tau$ as its trace, we will want to
instead study the evolution of the traceless part of $k$, which we will denote
$\widehat{k}$. Explicitly, we let $\widehat{k}=k-(\tau/n)g$, such that
$\operatorname{tr}_{g}(\widehat{k})=0$. Observe that we still have
$[\widehat{k}]\sim(\text{length})$.

To determine the scaling of the Yang-Mills variables
$(\mathcal{E},A,\varphi)$, we impose the condition that the gauge covariant
derivative $\nabla^{A}[g]=\nabla[g]+[A,\bullet]$ should remain scale
invariant. This then gives us the scaling

$[A_{i}]\sim(\text{length})^{0},\quad\quad[\mathcal{E}]\sim(\text{length}),\quad\quad[\varphi]\sim(\text{length})^{-1}.$

We also note that $[F]=[A_{i}]\sim(\text{length})^{0}$. Knowing how each of
the quantities scales with length, we can redefine the variables as

$\displaystyle
g\rightarrow\tau^{2}g,\quad\quad\widehat{k}\rightarrow\tau^{-1}\widehat{k},\quad\quad
N\rightarrow\tau^{2}N,\quad\quad X\rightarrow\tau^{-1}X,$ $\displaystyle
A_{i}\rightarrow
A_{i},\quad\quad\mathcal{E}\rightarrow\tau\mathcal{E},\quad\quad\varphi\rightarrow\tau^{-1}\varphi,$

where the new $(\widehat{k},g,\mathcal{E},A,N,X,\varphi)$ are all
dimensionless quantities. We will finally remove the explicit $\tau$
dependence by defining the dimensionless time coordinate $T$ by

$T=-\ln\bigg{(}\frac{\tau}{\tau_{0}}\bigg{)},$

where $\tau_{0}\in(-\infty,0)$ is some arbitrary reference time. This allows
us to write in the rescaled Einstein’s equations
$\tau\partial_{\tau}=-\partial_{T}$ and $\tau=\tau_{0}e^{-T}$. We also note
that, because $\tau\in(-\infty,0)$, we have that $T\in\mathbb{R}$ and in
particular as we evolve forward in time such that $\tau\rightarrow 0$, then
$T\rightarrow\infty$.

Substituting the dimensionless variables
$(\widehat{k},g,\mathcal{E},A,N,X,\varphi,T)$ into the evolution and
constraint equations given in eqns. (46)–(51) of [1], we find the system of
scale-invariant evolution and constraint equations to be as follows. The
scale-invariant evolution equations are given by

$\displaystyle\partial_{T}A^{a}{}_{bi}$
$\displaystyle=-N\mathcal{E}^{a}{}_{bi}-\partial_{i}\varphi^{a}{}_{b}-[A_{i},\varphi]^{a}{}_{b}-X^{j}\nabla[\gamma]_{j}A^{a}{}_{bi}-A^{a}{}_{bj}\nabla[\gamma]_{i}X^{j}$
(20) $\displaystyle\partial_{T}\mathcal{E}^{a}{}_{bi}$
$\displaystyle=-\bigg{(}\frac{(n-2)N}{n}-1\bigg{)}\mathcal{E}^{a}{}_{bi}+(\nabla_{j}N)F^{a}{}_{bi}{}^{j}+N(\nabla_{j}F^{a}{}_{bi}{}^{j}-\nabla_{i}C^{a}{}_{b})$
$\displaystyle\quad-N[F_{i}{}^{j},A_{j}]^{a}{}_{b}-[\mathcal{E}_{i},\varphi]^{a}{}_{b}-X^{k}\nabla_{k}\mathcal{E}^{a}{}_{bi}-\mathcal{E}^{a}{}_{bk}\nabla_{i}X^{k}+2N\widehat{k}_{i}^{j}\mathcal{E}^{a}{}_{bj}$
(21) $\displaystyle\partial_{T}g_{ij}$
$\displaystyle=-\frac{2}{n}(n-N)g_{ij}+2N\widehat{k}_{ij}-L_{X}g_{ij}$ (22)
$\displaystyle\partial_{T}\widehat{k}_{ij}$
$\displaystyle=-\bigg{(}\frac{(n-2)N}{n}+1\bigg{)}\widehat{k}_{ij}-\frac{N-1}{n}g_{ij}+\nabla_{i}\nabla_{j}N-N\big{\\{}R_{ij}-\alpha_{ij}-2\widehat{k}_{ik}\widehat{k}_{j}{}^{k}\big{\\}}$
$\displaystyle\quad-\tau_{0}^{2}e^{-2T}N\bigg{\\{}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}-g^{k\ell}F^{a}{}_{bik}F^{b}{}_{aj\ell}-\frac{1}{n-1}g^{k\ell}\mathcal{E}^{a}{}_{bk}\mathcal{E}^{b}{}_{a\ell}g_{ij}$
$\displaystyle\quad+\frac{1}{2(n-1)}g^{km}g^{\ell
n}F^{a}{}_{bk\ell}F^{b}{}_{amn}g_{ij}\bigg{\\}}-L_{X}\widehat{k}_{ij}.$ (23)

Here $\alpha_{ij}$ and $C^{a}{}_{b}$ has been defined as

$\displaystyle\alpha_{ij}$
$\displaystyle=\frac{1}{2}(\nabla_{i}V_{j}+\nabla_{j}V_{i})$ $\displaystyle
C^{a}{}_{b}$
$\displaystyle=g^{ij}\nabla^{\widehat{A}}[\gamma]_{i}(A^{a}{}_{bj}-\widehat{A}^{a}{}_{bj})$

and must vanish for the evolution equations to be hyperbolic, as is the case
in the chosen gauges; c.f. Section 3 of [1]. We note that the first term in
(21) will, to leading order in the small data context we work in, vanish for
$n=3$. This ultimately will prevent the subsequent energy arguments from
closing. While this is a feature of the gauge and scaling choices made
earlier, in section 6, we give an gauge-covariant argument as to why an energy
estimate will not be sufficient to obtain the desired global existence result
in $n=3$ spatial dimensions.

Meanwhile, the elliptic equations for the gauge variables take the form

$\displaystyle\Delta_{g}N$
$\displaystyle+\bigg{(}|\widehat{k}|^{2}+\frac{1}{n}+\frac{n-2}{n-1}\tau_{0}^{2}e^{-2T}g^{k\ell}\mathcal{E}^{a}{}_{bk}\mathcal{E}^{b}{}_{a\ell}+\frac{\tau_{0}^{2}}{2(n-1)}e^{-2T}g^{k\ell}g^{mn}F^{a}{}_{bkm}F^{b}{}_{a\ell
n}\bigg{)}N=1$ (24) $\displaystyle\Delta_{g}X^{i}$
$\displaystyle-R^{i}{}_{j}X^{j}+L_{X}V^{i}=-2\widehat{k}^{ij}\nabla_{j}N+\bigg{(}1-\frac{2}{n}\bigg{)}\nabla^{i}N-2\tau_{0}^{2}e^{-2T}Ng^{ik}g^{\ell
j}F^{a}{}_{bk\ell}\mathcal{E}^{b}{}_{aj}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\,+\bigg{(}2N\widehat{k}^{jk}+\frac{2N}{n}g^{jk}-2\nabla^{j}X^{k}\bigg{)}\big{(}\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk}\big{)}+g^{jk}\partial_{T}\Gamma[\gamma]^{i}{}_{jk}$
(25)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}{}_{b}+g^{ij}[\widehat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}{}_{b}+g^{ij}\nabla_{i}N\mathcal{E}^{a}{}_{bj}-Ng^{ij}[A_{i},\mathcal{E}_{j}]^{a}{}_{b}$
$\displaystyle\quad+g^{ij}[\widehat{A}_{i},N\mathcal{E}_{j}]^{a}{}_{b}+g^{ij}\big{(}\Gamma[g]^{k}{}_{ij}-\Gamma[\gamma]^{k}{}_{ij}\big{)}N\mathcal{E}^{a}{}_{bk}-X^{k}\nabla[\gamma]_{k}g^{ij}\nabla[\gamma]_{i}\big{(}A^{a}{}_{bj}-\widehat{A}^{a}{}_{bj}\big{)}$
$\displaystyle\quad-X^{k}\nabla[\gamma]_{k}g^{ij}[\widehat{A}_{i},A_{j}-\widehat{A}_{j}]^{a}{}_{b}+g^{ij}\nabla[\gamma]_{i}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bj}+g^{ij}\nabla[\gamma]_{i}A^{a}{}_{bk}\nabla[\gamma]_{j}X^{k}$
$\displaystyle\quad+g^{ij}A^{a}{}_{bk}\nabla[\gamma]_{i}\nabla[\gamma]_{j}X^{k}-g^{ij}R[\gamma]^{\ell}{}_{jik}A^{a}{}_{b\ell}X^{k}+g^{ij}[\widehat{A}_{i},A_{k}]^{a}{}_{b}\nabla[\gamma]_{j}X^{k}$
$\displaystyle\quad+\bigg{(}2N\widehat{k}^{ij}+\frac{2N}{n}g^{ij}-\nabla^{i}X^{j}-\nabla^{j}X^{i}\bigg{)}\nabla[\gamma]^{\widehat{A}}_{i}(A^{a}{}_{bj}-\widehat{A}^{a}{}_{bj})=0$
(26)

Throughout the paper, we will collectively refer to the rescaled, gauge-fixed
equations (20)–(26) as the Einstein-Yang-Mills system. Due to the elliptic
contraints, if $\mathcal{D}_{0}$ denotes the space of diffeomorphism on
$\Sigma$ and $\mathcal{G}$ denotes the space of gauge transformations, then
the dynamics of the Einstein-Yang-Mills system will occur on the space
$\mathcal{M}_{\Sigma}/\mathcal{D}_{0}\times\mathcal{A}/\mathcal{G}$.

Note that indices are raised with the dynamical metric $g$, unless otherwise
specified. Recall that because we have no equivalent shadow metric condition
on $\widehat{A}$, we have fixed $\widehat{A}\equiv 0$. However, because
$A-\widehat{A}$ is a gauge invariant tensor while $A$ itself is not, we for
now explicitly write all $\widehat{A}$ terms in the constraint equation for
$\varphi$.

Throughout the analysis in this paper, we will assume that
$(\widehat{k},g,\mathcal{E},A)\in H^{s-1}\times H^{s}\times H^{s-1}\times
H^{s}$ for $s>\frac{n}{2}+1$. From the elliptic equations, we then immediately
see that the gauge variables $(N,X,\varphi)$ each lie in the Sobolev space
$H^{s+1}$.

###  3.2 Small Data

We work in the context of a background geometry given by a metric of the form
in (1), which is a solution to the vacuum Einstein’s equations. From this
unscaled metric, we then have the background solution
$(\widehat{k},g,\mathcal{E},A,N,X,\varphi)=(0,t^{-2}\gamma,0,0,t^{-2}n,0,0)$.
Introducing the rescaled variables, this corresponds to the background
solution

$(\widehat{k},g,\mathcal{E},A,N,X,\varphi)_{\text{background}}=(0,\gamma,0,0,n,0,0).$
(27)

If we define $u\coloneqq g-\gamma$ and $\omega\coloneqq N-n$, we then will
consider the data

$(\widehat{k},u,\mathcal{E},A,\omega,X,\varphi)\in H^{s-1}\times H^{s}\times
H^{s-1}\times H^{s}\times H^{s+1}\times H^{s+1}\times H^{s+1}$

to be small in the small data context we are working in, for
$s>\frac{n}{2}+1$. We will often specify this by writing that
$(\widehat{k},u,\mathcal{E},A,\omega,X,\varphi)\in B_{\delta}(0)$, where
$B_{\delta}(0)$ is a ball of sufficiently small radius $\delta$ centered at
$0$ in the relevant function spaces.

##  4 Elliptic Estimates

To begin, we will recall a number of important inequalities regarding Sobolev
spaces, which we will make frequent use of in the following elliptic and
energy estimates.

######  Proposition 4.1.

Let $OP^{r}$ denote the space of pseudo-differential operators with symbol in
Hömander’s class $S^{r}_{1,0}$. The following inequalities then hold.

  1. 1.

[34] If $\mathcal{P}\in OP^{r}$ for $r\in\mathbb{R}$ and if $s\in\mathbb{R}$,
then $\mathcal{P}:H^{s}\rightarrow H^{s-r}$. That is, if $f\in H^{s}$, then
the estimate

$\lVert\mathcal{P}f\rVert_{H^{s-r}}\leq C\lVert f\rVert_{H^{s}}$ (28)

holds.

  2. 2.

[34] Let $\mathcal{P}\in OP^{r}$ for $r>0$ and take $s\geq 0$. Then for $f\in
H^{r+s}$ and $g\in H^{r+s-1}$,

$\lVert[\mathcal{P},f]g\rVert_{H^{s}}=\lVert\mathcal{P}(fg)-f\mathcal{P}(g)\rVert_{H^{s}}\leq
C\big{(}\lVert\nabla[\gamma]f\rVert_{L^{\infty}}\lVert
g\rVert_{H^{r+s-1}}+\lVert f\rVert_{H^{r+s}}\lVert
g\rVert_{L^{\infty}}\big{)}$ (29)

  3. 3.

[35] If $s>0$ then $H^{s}\cap L^{\infty}$ is an algebra and for $f,g\in
H^{s}\cap L^{\infty}$,

$\lVert fg\rVert_{H^{s}}\leq C\big{(}\lVert f\rVert_{L^{\infty}}\lVert
g\rVert_{L^{2}}+\lVert f\rVert_{L^{2}}\lVert f\rVert_{L^{\infty}}\big{)}.$
(30)

In particular, with the embeddings $H^{s}\hookrightarrow L^{2}$ for all $s$
and $H^{s}\hookrightarrow W^{0,\infty}=L^{\infty}$ for $s>\frac{n}{2}$, then
if $s>\frac{n}{2}$,

$\lVert fg\rVert_{H^{s}}\leq C\lVert f\rVert_{H^{s}}\lVert g\rVert_{H^{s}}.$
(31)

An important case of Proposition 4.1.1 is $\nabla[\gamma]\in OP^{1}$, for
which we have the estimate $\lVert\nabla[\gamma]f\rVert_{H^{s-1}}\leq C\lVert
f\rVert_{H^{s}}$. Moreover, we will make use of—although not always
explicitly—Minkowski’s and Hölder’s inequalities when necessary.

Now we will obtain estimates for the small gauge variables $\omega$, $X$, and
$\varphi$. This, however, will first require an estimate on the tangential
velocity $\partial_{T}\gamma$, given in the following lemma.

######  Lemma 4.2.

Let $\gamma\in\mathcal{N}$ be a metric satisfying the shadow metric condition
$(g-\gamma)\perp\mathcal{N}$ and for which $\lVert
g-\gamma\rVert_{H^{s}}<\delta$ with a sufficiently small $\delta>0$ and
$s>\frac{n}{2}+1$. Then the vector field $\partial_{T}\gamma$ satisfies the
estimate

$\lVert\partial_{T}\gamma\rVert\leq C\big{\\{}(1+\lVert
u\rVert_{H^{s}})(\lVert\omega\rVert_{H^{s+1}}+\lVert
X\rVert_{H^{s+1}})+\lVert\widehat{k}\rVert_{H^{s-1}}(1+\lVert\omega\rVert_{H^{s+1}})\big{\\}},$
(32)

for a constant $C$ depending only on the background geometry.

###### Proof.

Recall (19); that is, from the shadow metric condition, we obtain an estimate
for the time evolution $\partial_{T}\gamma$ of the form

$\lVert\partial_{T}\gamma\rVert\leq C\lVert\partial_{T}g\rVert_{H^{s-1}}.$

In the evolution equation for $g$, we write the Lie derivative
$\mathcal{L}_{X}g$ in terms of $u=g-\gamma$ and $\gamma$ as

$\displaystyle\mathcal{L}_{X}g_{ij}=\mathcal{L}_{X}u_{ij}+\mathcal{L}_{X}\gamma_{ij}=X^{k}\nabla[\gamma]_{k}u_{ij}+u_{ik}\nabla[\gamma]_{j}X^{k}+u_{jk}\nabla[\gamma]_{i}X^{k}+\nabla[\gamma]_{i}X_{j}+\nabla[\gamma]_{j}X_{i}.$

We then have from the evolution equation for $g$ and Minkowski’s inequality,
that $\partial_{T}\gamma$ satisfies the norm bound

$\lVert\partial_{T}\gamma\rVert\leq C\big{\\{}\lVert
u\omega\rVert_{H^{s-1}}+\lVert\omega\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert\widehat{k}\omega\rVert_{H^{s-1}}+\lVert
X\nabla[\gamma]u\rVert_{H^{s-1}}+\lVert
u\nabla[\gamma]X\rVert_{H^{s-1}}+\lVert\nabla[\gamma]X\rVert_{H^{s-1}}\big{\\}},$

where the positive constant $C=C(n,\lVert\gamma\rVert,\dots)$ depends on the
background geometry only. Applying the inequalities from Proposition 4.1 for
$s-1>\frac{n}{2}$, we obtain

$\displaystyle\lVert\partial_{T}\gamma\rVert$ $\displaystyle\leq
C\big{\\{}\lVert\omega\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s-1}}+\lVert\omega\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}\lVert\omega\rVert_{H^{s-1}}$
$\displaystyle\qquad\qquad+\lVert
X\rVert_{H^{s-1}}\lVert\nabla[\gamma]u\rVert_{H^{s-1}}+\lVert\nabla[\gamma]X\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s-1}}+\lVert\nabla[\gamma]X\rVert_{H^{s-1}}\big{\\}}$
$\displaystyle\leq C\big{\\{}\lVert\omega\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s-1}}+\lVert\omega\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}\lVert\omega\rVert_{H^{s-1}}$
$\displaystyle\qquad\qquad+\lVert X\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s}}+\lVert X\rVert_{H^{s}}\lVert u\rVert_{H^{s-1}}+\lVert
X\rVert_{H^{s}}\big{\\}}$

Using the embedding $H^{s_{1}}\hookrightarrow H^{s_{2}}$ for $s_{1}>s_{2}$, we
obtain the desired estimate. ∎

We now study the perturbed lapse function $\omega=N-n$, where we will use the
proof of a similar result from [36] to show that $\omega$ satisfies the
following second-order estimate.

######  Lemma 4.3.

Fix $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of sufficiently small
radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$ that satisfy the
Einstein-Yang-Mills system. Then the lapse gauge variable $\omega=N-n$
satisfies the estimate

$\lVert\omega\rVert_{H^{s+1}}\leq
C\Big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+e^{-2T}\big{(}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{)}\Big{\\}},$ (33)

for a constant $C$ depending only on the background geometry.

###### Proof.

From Eq. (24), the elliptic equation for the perturbation $\omega$ will be
given by

$\displaystyle\Delta_{g}\omega+\bigg{(}|\widehat{k}|^{2}+\frac{1}{n}+\frac{n-2}{n-1}\tau_{0}^{2}e^{-2T}|\mathcal{E}|^{2}$
$\displaystyle+\frac{\tau_{0}^{2}}{2(n-1)}e^{-2T}|F|^{2}\bigg{)}\omega$
$\displaystyle=-n\bigg{(}|\widehat{k}|^{2}+\frac{n-2}{n-1}\tau_{0}^{2}e^{-2T}|\mathcal{E}|^{2}+\frac{\tau_{0}^{2}}{2(n-1)}e^{-2T}|F|^{2}\bigg{)}.$
(34)

We note that the second term of the left-hand side is strictly positive, and
thus the second-order elliptic operator
$\mathcal{P}=\Delta_{g}+\big{(}|\widehat{k}|^{2}+\frac{1}{n}+\frac{n-2}{n-1}\tau_{0}^{2}e^{-2T}|\mathcal{E}|^{2}+\frac{\tau_{0}^{2}}{2(n-1)}e^{-2T}|F|^{2}\big{)}\operatorname{Id}$
has a definite sign. As shown via contradiction in the proof of Lemma 5 in
[36], such an operator will satisfy an estimate of the form
$\lVert\omega\rVert_{H^{s+1}}\leq C\lVert\mathcal{P}\omega\rVert_{H^{s-1}}$
for any $s\geq 1$. Since we have assumed throughout that $s>\frac{n}{2}+1>1$,
we thus have such an estimate with $\mathcal{P}\omega$. Using the expression
for $\mathcal{P}\omega$ in (4), we then may bound the Sobolev norm of $\omega$
as

$\displaystyle\lVert\omega\rVert_{H^{s+1}}$ $\displaystyle\leq
C\bigg{\lVert}-n\bigg{(}|\widehat{k}|^{2}+\frac{n-2}{n-1}\tau_{0}^{2}e^{-2T}|\mathcal{E}|^{2}+\frac{\tau_{0}^{2}}{2(n-1)}e^{-2T}|F|^{2}\bigg{)}\bigg{\rVert}_{H^{s-1}}$
$\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+e^{-2T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+e^{-2T}\lVert
F\rVert_{H^{s-1}}^{2}\big{\\}}$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+e^{-2T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+e^{-2T}\lVert
A\rVert_{H^{s}}^{2}(1+\lVert A\rVert_{H^{s}})^{2}\big{\\}}$

Because we have taken $(\widehat{k},A,\mathcal{E})\in B_{\delta}(0)$, we may
absorb non-leading order terms into the coefficient $C$ by redefining
$C\rightarrow(1+\delta)C$, i.e.,

$\displaystyle\lVert\omega\rVert_{H^{s+1}}$ $\displaystyle\leq
C\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+Ce^{-2T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+Ce^{-2T}\lVert
A\rVert_{H^{s}}^{2}(1+\delta)^{2}$ $\displaystyle\leq
C\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+Ce^{-2T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+C^{\prime}e^{-2T}\lVert
A\rVert_{H^{s}}^{2}$ $\displaystyle\leq
C^{\prime}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})\big{\\}}.$

∎

We note that, as we will study the Cauchy problem and thus take initial data
at some $T_{0}$, we may bound $e^{-T}\leq e^{-T_{0}}$ which may be absorbed
into the constant $C$. However, if terms of the form $e^{T}$ were to appear,
our subsequent energy estimates would fail as these exponential terms would
prevent us from finding a decreasing bound on the total energy. Since $e^{-T}$
terms may neutralize such terms, we will keep, for now, all terms of the form
$e^{-T}$ with their leading order Sobolev norms.

While the elliptic differential operator for $\omega$ has definite sign, those
appearing in the defining equations for $X$ and $\varphi$ do not. This means
that, in general, the elliptic operators acting on $X$ and $\varphi$ will not
have trivial kernel and thus will exhibit a Gribov ambiguity [37, 38].
Geometrically we can view this degeneracy as a breakdown of geodesic normal
coordinates on the gauge-fixed spaces of Riemannian metrics or connection
$1$-forms, as follows.

Let us consider the shift vector field $X$. After fixing the spatial harmonic
gauge, the dynamics of the metric evolution occur on the space
$\mathcal{M}_{\Sigma}/\mathcal{D}_{0}$, where we recall that $\mathcal{D}_{0}$
is the space of diffeomorphisms on $\Sigma$. However, the elliptic equation
defining $X$ is derived from requiring that the spatial harmonic gauge
condition is preserved over time, i.e., $\partial_{T}V^{k}=0$ for the tension
field $V^{k}$. Since geodesics on the gauge-fixed quotient space
$\mathcal{M}_{\Sigma}/\mathcal{D}_{0}$ must preserve the spatial harmonic
gauge condition, we then have that the uniqueness of geodesics will be
determined by the uniqueness of the $X$ defined by (25). In other words,
looking at geodesic normal coordinates, we will have a neighborhood
$B_{\delta}(\gamma)$ of a metric
$\gamma\in\mathcal{M}_{\Sigma}/\mathcal{D}_{0}$ such that the exponential map
used to construct geodesics is injective, which in turn corresponds to the
injectivity of the differential operator in (25) that acts on $X$. However,
for $g\in(\mathcal{M}_{\Sigma}/\mathcal{D}_{0})\setminus B_{\delta}(\gamma)$,
there will not be a unique geodesic between $\gamma$ and $g$, corresponding to
the aforementioned Gribov ambiguity and breakdown in injectivity of the
differential operator defining $X$.

An identical reasoning ought to hold for $\varphi$ with the space of
connection $1$-forms $A$, though as the topology of this space is unknown we
can only draw parallels. We see from this geometric view that we will want to
restrict our focus to neighborhoods in the spaces of Riemannian metrics and
connection $1$-forms on which the elliptic operators defining $X$ and
$\varphi$ remain injective, which we may freely do as we are working in the
context of small data. The following lemma gives the necessary bound for the
shift vector field $X$. We give the full proof, following the derivation given
in [5] and [36], as we will use a similar method to subsequently estimate
$\varphi$.

######  Lemma 4.4.

Fix $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of sufficiently small
radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$ that satisfy the
Einstein-Yang-Mills system. Then the shift vector field $X$ will satisfy the
bound

$\lVert X\rVert_{H^{s+1}}\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}},$ (35)

for a constant $C$ depending only on the background geometry.

###### Proof.

Recall the elliptic equation for $X$ is given as

$\displaystyle\Delta_{g}X^{i}-R^{i}{}_{j}X^{j}+L_{X}V^{i}$
$\displaystyle=-2\widehat{k}^{ij}\nabla_{j}N+\bigg{(}1-\frac{2}{n}\bigg{)}\nabla^{i}N-2\tau_{0}^{2}e^{-2T}Ng^{ik}g^{\ell
j}F^{a}{}_{bk\ell}\mathcal{E}^{b}{}_{aj}$
$\displaystyle\qquad+\big{(}2N\widehat{k}^{jk}-2\nabla^{j}X^{k}\big{)}\big{(}\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk}\big{)}+g^{jk}\partial_{T}\Gamma[\gamma]^{i}{}_{jk}.$

It is necessary to then demonstrate that
$X^{i}\mapsto\Delta_{g}X^{i}+2\nabla^{j}X^{k}(\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk})-R^{i}{}_{j}X^{j}+L_{X}V^{i}$
is an isomorphism of function spaces $H^{s+1}$ and $H^{s-1}$. To do so, we
will consider the flow of $X$ along the pull-back of the tension field which,
in the CMCSH gauge, will vanish.

Let $\psi_{s}$ be the flow of $X$, and consider the pull-back of the tension
field $V^{\flat}$. As the difference of connections transforms as a section of
a suitable vector bundle, this gives

$\big{(}\psi_{s}^{*}V^{\flat}\big{)}^{i}=\big{(}\psi_{s}^{*}g^{-1}\big{)}^{jk}\big{(}\psi_{s}^{*}(\Gamma[g]-\Gamma[\gamma])\big{)}^{i}{}_{jk}=\big{(}\psi_{s}^{*}g^{-1}\big{)}^{jk}\big{(}\Gamma[\psi_{s}^{*}g]^{i}{}_{jk}-\Gamma[\psi_{s}^{*}\gamma]^{i}{}_{jk}\big{)}.$

We may then compute the Lie derivative of $V^{\flat}$ along $X$ as

$\displaystyle(L_{X}V)^{i}$
$\displaystyle={\frac{\mathrm{d}}{\mathrm{d}s}}(\psi_{s}^{*}V^{\flat})^{i}\big{|}$
$\displaystyle=(L_{X}g^{-1})^{jk}\big{(}\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk}\big{)}+\frac{1}{2}g^{jk}g^{i\ell}\big{(}-\nabla_{\ell}(L_{X}g)_{jk}+\nabla_{j}(L_{X}g)_{\ell
k}+\nabla_{k}(L_{X}g)_{\ell j}\big{)}$
$\displaystyle\qquad-\frac{1}{2}g^{jk}\gamma^{i\ell}\big{(}-\nabla[\gamma]_{\ell}(L_{X}\gamma)_{jk}+\nabla[\gamma]_{j}(L_{X}\gamma)_{\ell
k}+\nabla[\gamma]_{k}(L_{X}\gamma)_{\ell j}\big{)}$

Expanding the Lie derivative on the right-hand side with
$(L_{X}g)_{ij}=\nabla_{i}X_{j}+\nabla_{j}X_{i}$ and
$(L_{X}\gamma)_{ij}=\nabla[\gamma]_{i}X_{j}+\nabla[\gamma]_{j}X_{i}$, we then
find that

$\Delta_{g}X^{i}-R^{i}{}_{j}X^{j}+L_{X}V^{i}+2\nabla^{j}X^{k}\big{(}\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk}\big{)}=\Delta^{\gamma}_{g}X^{i}-g^{k\ell}R[\gamma]^{i}{}_{kj\ell}X^{j},$
(36)

The operator acting on $X$ on the left-hand side is $\mathcal{P}$, and the
operator on the right-hand side now has a definite sign. So to show that
$\mathcal{P}$ is an isomorphism of function spaces is to show that the mapping
$\mathcal{P}_{g,\gamma}:X^{i}\mapsto\Delta_{\gamma}^{g}X^{i}-g^{k\ell}R[\gamma]^{i}{}_{kj\ell}X^{j}$
is. As the Cauchy hypersurface is compact and $\mathcal{P}_{g,\gamma}$ is
second-order elliptic, then it will suffice to show that
$\mathcal{P}_{g,\gamma}$ is injective in order to show it is the desired
isomorphism.

Suppose, for the sake of contradiction, that $\mathcal{P}_{g,\gamma}$ is not
injective. We thus have a non-zero $Z\in\ker\mathcal{P}_{g,\gamma}$. We may
the integrate $\gamma_{ij}Z^{i}\mathcal{P}_{g,\gamma}Z^{j}\equiv 0$ over the
Cauchy hypersurface $\Sigma$, yielding

$\displaystyle 0$
$\displaystyle=\int_{\Sigma}\big{(}-\gamma_{ij}g^{k\ell}Z^{j}\nabla[\gamma]_{k}\nabla[\gamma]_{\ell}Z^{i}-\gamma_{im}g^{k\ell}Z^{m}R[\gamma]^{i}{}_{kj\ell}Z^{j}\big{)}\>\mu_{\gamma}$
$\displaystyle=\int_{\Sigma}\big{(}\gamma_{ij}g^{k\ell}\nabla[\gamma]_{k}Z^{j}\nabla[\gamma]_{\ell}Z^{i}-\gamma_{im}g^{k\ell}Z^{m}R[\gamma]^{i}{}_{kj\ell}Z^{j}\big{)}\>\mu_{\gamma}$
$\displaystyle=\int_{\Sigma}\big{(}\gamma_{ij}\gamma^{k\ell}\nabla[\gamma]_{k}Z^{j}\nabla[\gamma]_{\ell}Z^{i}+\gamma_{ij}(g^{k\ell}-\gamma^{k\ell})\nabla[\gamma]_{k}Z^{j}\nabla[\gamma]_{\ell}Z^{i}$
$\displaystyle\qquad\qquad-\gamma_{im}\gamma^{k\ell}Z^{m}R[\gamma]^{i}{}_{kj\ell}Z^{j}-\gamma_{im}(g^{k\ell}-\gamma^{k\ell})Z^{m}R[\gamma]^{i}{}_{kj\ell}Z^{j}\big{)}\>\mu_{\gamma}.$

With $\lVert u\rVert_{H^{s}}$ taken to be sufficiently small, we have that the
second and fourth terms in the integrand are domainated by the first and third
terms. Furthermore, because
$\gamma^{k\ell}R[\gamma]^{i}{}_{kj\ell}=R[\gamma]^{i}{}_{j}$ is negative-
definite, we have that the first and third terms of the integrand are
positive. Hence the integrand as a whole may be taken to be non-negative,
implying that $Z$ must be $0$. This yields a contradiction, and so we must
have that $\ker\mathcal{P}_{g,\gamma}=\\{0\\}$. Hence $\mathcal{P}_{g,\gamma}$
is injective, and because $\mathcal{P}=\mathcal{P}_{g,\gamma}$ by (36), we
have that $\mathcal{P}$ is injective and thus an isomorphism of suitable
function spaces. By elliptic regularity, we have an estimate of the form
$\lVert X\rVert_{H^{s+1}}\leq\lVert\mathcal{P}X\rVert_{H^{s-1}}$. Using the
result of Lemma 4.2, it then follows that $X$ will be controlled by

$\displaystyle\lVert X\rVert_{H^{s+1}}\big{\\{}1{}$
$\displaystyle{}-C(\lVert\gamma-\gamma_{0}\rVert_{H^{s}}+\lVert
u\rVert_{H^{s}})(1+\lVert u\rVert_{H^{s}})\big{\\}}\leq
C(1+\lVert\widehat{k}\rVert_{H^{s-1}})\lVert\omega\rVert_{H^{s+1}}$
$\displaystyle+C(1+\lVert\omega\rVert_{H^{s+1}})\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s}}+\tau_{0}^{2}e^{-2T}\lVert
F\rVert_{H^{s-1}}\lVert\mathcal{E}\rVert_{H^{s-1}}\big{\\}}$
$\displaystyle+C(\lVert\gamma-\gamma_{0}\rVert_{H^{s}}+\lVert
u\rVert_{H^{s}})\big{\\{}(1+\lVert u\rVert_{H^{s}})(\lVert
u\rVert_{H^{s}}+\lVert\omega\rVert_{H^{s+1}})+(1+\lVert\omega\rVert_{H^{s+1}})\lVert\widehat{k}\rVert_{H^{s-1}}\big{\\}}.$

Note that $\lVert\gamma-\gamma_{0}\rVert_{H^{s}}\leq C$ for constant $C$
depending only on the background geometry. Hence with sufficiently small
$\lVert u\rVert_{H^{s}}$, and using the estimate for $\omega$ from Lemma 4.3,
the shift vector field $X$ is controlled to leading order by

$\displaystyle\lVert X\rVert_{H^{s+1}}$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}.$

∎

We now give a bound on the norm of the Yang-Mills gauge variable $\varphi$. As
previously mentioned, the proof will be similar in structure to that of Lemma
4.4: we will consider the pull-back of the generalized Coulomb gauge condition
along the flow of $\varphi$ and compute its derivative at $s=0$ to find a form
of the elliptic operator of interest that can then be shown to be an
isomorphism of function spaces.

######  Lemma 4.5.

Fix $\widehat{A}=0$ and $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of
sufficiently small radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$
that satisfy the Einstein-Yang-Mills system. Then the Yang-Mills gauge
variable $\varphi$ will satisfy the estimate if $\widehat{A}=0$:

$\displaystyle\lVert\varphi\rVert_{H^{s+1}}$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}+\lVert
A\rVert_{H^{s}})(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})$ $\displaystyle\qquad\quad+e^{-4T}\lVert
A\rVert_{H^{s}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}},$ (37)

for a constant $C$ depending only on the background geometry.

###### Proof.

To begin, we will show that the mapping $\mathcal{P}:\varphi\mapsto
g^{ij}\nabla[\gamma]_{i}^{A}\nabla[\gamma]_{j}^{A}\varphi+g^{ij}[\widehat{A}_{i}-A_{i},\nabla_{j}^{A}\varphi]$
is an isomorphism of suitable function spaces. Let $\psi_{s}=e^{s\varphi}$ be
the one-parameter subgroup of the Lie group $G$ corresponding to $\varphi$ and
take
$\mathcal{Q}=g^{ij}\nabla[\gamma]_{i}^{\widehat{A}}(A_{j}-\widehat{A}_{j})$.

As the difference of two connections will transform under a gauge
transformation as a suitable section of the associated $V$-bundle, we have
that under a gauge transformation $\mathcal{G}$, $\mathcal{Q}$ will transform
as $\mathcal{Q}\mapsto\mathcal{GQG}^{-1}$. Taking $\mathcal{G}$ to be
$\psi_{s}$ and differentiating with respect to $s$,

An explicit computation gives

$[\varphi,\mathcal{Q}]=-\mathcal{P}\varphi+g^{ij}\Big{\\{}\nabla[\gamma]_{i}^{\widehat{A}}\nabla[\gamma]_{j}^{\widehat{A}}\varphi+\big{[}A_{i}-\widehat{A}_{i},[\widehat{A}_{j},\varphi]\big{]}\Big{\\}},$
(38)

and so in the generalized Coulomb gauge, where $\mathcal{Q}=0$,

$\mathcal{P}\varphi^{a}{}_{b}=g^{ij}\nabla[\gamma]_{i}^{\widehat{A}}\nabla[\gamma]_{j}^{\widehat{A}}\varphi^{a}{}_{b}+g^{ij}\big{[}A_{i}-\widehat{A}_{i},[\widehat{A}_{j},\varphi]\big{]}^{a}{}_{b}.$
(39)

To see that $\mathcal{P}$ is an isomorphism, suppose that there is a non-zero
$\zeta\in\ker\mathcal{P}$. Integrating the gauge-invariant inner product
$\zeta^{b}{}_{a}\mathcal{P}\zeta^{a}{}_{b}\equiv 0$ over $\Sigma$,

$\displaystyle 0$
$\displaystyle=\int_{\Sigma}\Big{(}g^{ij}\zeta^{b}{}_{a}\nabla[\gamma]_{i}^{\widehat{A}}\nabla[\gamma]_{j}^{\widehat{A}}\zeta^{a}{}_{b}+g^{ij}\zeta^{b}{}_{a}\big{[}A_{i}-\widehat{A}_{i},[\widehat{A}_{j},\zeta]\big{]}^{a}{}_{b}\Big{)}\>\mu_{\gamma}$
$\displaystyle=\int_{\Sigma}\Big{(}-\gamma^{ij}\nabla[\gamma]_{i}^{\widehat{A}}\zeta^{b}{}_{a}\nabla[\gamma]_{j}^{\widehat{A}}\zeta^{a}{}_{b}-\big{(}g^{ij}-\gamma^{ij}\big{)}\nabla[\gamma]_{i}^{\widehat{A}}\zeta^{b}{}_{a}\nabla[\gamma]_{j}^{\widehat{A}}\zeta^{a}{}_{b}$
$\displaystyle\qquad\qquad+\gamma^{ij}\zeta^{b}{}_{a}\big{[}A_{i}-\widehat{A}_{i},[\widehat{A}_{j},\zeta]\big{]}^{a}{}_{b}+\big{(}g^{ij}-\gamma^{ij}\big{)}\zeta^{b}{}_{a}\big{[}A_{i}-\widehat{A}_{i},[\widehat{A}_{j},\zeta]\big{]}^{a}{}_{b}\Big{)}\>\mu_{\gamma}$

Now, with $\widehat{A}=0$, the third and fourth terms will drop out.
Meanwhile, with $\lVert u\rVert_{H^{s}}$ small, the second term can be taken
to be smaller in magnitude than the first. Since the first term is negative-
definite, we have that the integrand as a whole will be non-positive. As such,
we must have $\zeta=0$ for the integral to vanish.

We conclude that there can be no non-zero $\zeta\in\ker\mathcal{P}$, and so
since $\mathcal{P}$ has trivial kernel it must be injective. As $\Sigma$ is
compact and $\mathcal{P}$ is second-order elliptic, the injectivity of
$\mathcal{P}$ gives that it is indeed an isomorphism of $H^{s+1}$ and
$H^{s-1}$. So, having established that $\mathcal{P}:\varphi\mapsto
g^{ij}\nabla[\gamma]_{i}^{A}\nabla[\gamma]_{j}^{A}\varphi+g^{ij}[\widehat{A}_{i}-A_{i},\nabla_{j}^{A}\varphi]$
is an isomorphism between function spaces, it then follows by elliptic
regularity and the inequalities outlined in Proposition 4.1 that

$\displaystyle\lVert\varphi\rVert_{H^{s+1}}$ $\displaystyle\leq
C\big{\\{}\lVert\mathcal{E}\rVert_{H^{s-1}}\lVert\omega\rVert_{H^{s+1}}+\lVert
A\rVert_{H^{s}}(1+\lVert\omega\rVert_{H^{s+1}})(\lVert\mathcal{E}\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+(1+\lVert
u\rVert_{H^{s}})\lVert A\rVert_{H^{s}}\lVert X\rVert_{H^{s+1}}\big{\\}}.$ (40)

Using the estimates for the lapse function perturbation $\omega$ and shift
field $X$ from Lemmas 4.3 and 4.4, the desired estimate for $\varphi$ is then
obtained. ∎

Finally, because the deformation space $\mathcal{N}$ is not in general
trivial, recall that we may decompose a tangent vector field to the shadow
metric as $\partial_{T}\gamma=h^{\text{TT}\parallel}+L_{Y^{\parallel}}\gamma$,
with $Y^{\parallel}$ in general non-zero. It will turn out that we will
require an estimate of the sum $X+Y^{\parallel}$ to complete the energy
estimates in the gravitational sector, due to a term of the form
$\mathcal{L}_{X+Y^{\parallel}}\gamma$. We present a bound on the Sobolev norm
for $X+Y^{\parallel}$ below, using the result of [2] that a particular second-
order differential operator is indeed an isomorphism of function spaces.

######  Lemma 4.6.

Fix $\widehat{A}=0$ and $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of
sufficiently small radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$
that satisfy the Einstein-Yang-Mills system. Let $Y^{\parallel}$ be a vector
field satisfying (16). The sum $X+Y^{\parallel}$ will then has the norm bound

$\displaystyle\lVert X+Y^{\parallel}\rVert_{H^{s+1}}$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}},$ (41)

for a constant $C$ depending only on the background geometry.

###### Proof.

Define the second-order elliptic operator
$\mathcal{P}_{g,\gamma}:X^{i}\mapsto\Delta_{\gamma}^{g}X^{i}-g^{k\ell}R[\gamma]^{i}{}_{kj\ell}X^{j}$.
Similarly, let $\mathcal{P}_{\gamma,\gamma}$ be given by

$\mathcal{P}_{\gamma,\gamma}:X^{i}\mapsto-\gamma^{k\ell}\nabla[\gamma]_{k}\nabla[\gamma]_{\ell}X^{i}-\gamma^{k\ell}R[\gamma]^{i}{}_{kj\ell}X^{j}.$

As discussed in Section 2.2 and Lemma 6.1 of [2], we have that
$\mathcal{P}_{\gamma,\gamma}$ is an isomorphism of suitable function spaces,
and when applied to $X+Y^{\parallel}$ gives

$\displaystyle\mathcal{P}_{\gamma,\gamma}\big{(}X^{i}+{Y^{i}}^{\parallel}\big{)}$
$\displaystyle=-2\widehat{k}^{ij}\nabla_{j}N+\bigg{(}1-\frac{2}{n}\bigg{)}\nabla^{i}N-2\tau_{0}^{2}e^{-2T}Ng^{ik}g^{\ell
j}F^{a}{}_{bk\ell}\mathcal{E}^{b}{}_{aj}+2N\widehat{k}^{jk}\big{(}\Gamma[g]^{i}{}_{jk}-\Gamma[\gamma]^{i}{}_{jk}\big{)}$
$\displaystyle\quad+(g^{mn}-\gamma^{mn})\partial_{T}\Gamma[\gamma]^{i}{}_{mn}+(g^{mn}-\gamma^{mn})\nabla[\gamma]_{m}\nabla[\gamma]_{n}X^{i}+(g^{mn}-\gamma^{mn})R[\gamma]^{i}{}_{mjn}X^{j}.$

Using elliptic regularity and the product estimates for $s>\frac{n}{2}+1$, we
have that $X+Y^{\parallel}$ is bounded by

$\displaystyle\lVert X+Y^{\parallel}\rVert_{H^{s+1}}$ $\displaystyle\leq
C\lVert\mathcal{P}_{\gamma,\gamma}(X+Y^{\parallel})\rVert_{H^{s-1}}$
$\displaystyle\leq
C\big{\\{}\lVert\omega\rVert_{H^{s}}(1+\lVert\widehat{k}\rVert_{H^{s-1}})+C\lVert
u\rVert_{H^{s}}\big{\\{}\lVert\partial_{T}\gamma\rVert+\lVert
X\rVert_{H^{s+1}}+\lVert X\rVert_{H^{s-1}}\big{\\}}$
$\displaystyle\quad+(1+\lVert\omega\rVert_{H^{s-1}})(e^{-2T}\lVert
F\rVert_{H^{s-1}}\lVert\mathcal{E}\rVert_{H^{s-1}}+\lVert\widehat{k}\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s}})\big{\\}}$ $\displaystyle\leq
C\big{\\{}\lVert\omega\rVert_{H^{s+1}}(1+\lVert\widehat{k}\rVert_{H^{s-1}})+C\lVert
u\rVert_{H^{s}}(1+\lVert
u\rVert_{H^{s}})\big{\\{}\lVert\omega\rVert_{H^{s+1}}+\lVert
X\rVert_{H^{s+1}})\big{\\}}$
$\displaystyle\quad+(1+\lVert\omega\rVert_{H^{s+1}})(e^{-2T}\lVert
A\rVert_{H^{s}}\lVert\mathcal{E}\rVert_{H^{s-1}}(1+\lVert
A\rVert_{H^{s}})+\lVert\widehat{k}\rVert_{H^{s-1}}\lVert
u\rVert_{H^{s}})\big{\\}}$

Applying Lemmas 4.3 and 4.4 then yields the desired estimate for
$X+Y^{\parallel}$. ∎

##  5 Energy Estimates

With the elliptic estimates from the previous section, we may now present an
energy estimate argument to show that the perturbed data will decay. However
we first need to examine the operators $\mathcal{L}_{\gamma_{0},\gamma_{0}}$
and $\mathfrak{L}_{\gamma_{0},\gamma_{0}}$, as they will play a key role in
our analysis. Recall that $\mathcal{L}_{\gamma_{0},\gamma_{0}}$ is assumed to
have non-negative spectrum and $\mathfrak{L}_{\gamma_{0},\gamma_{0}}$ will
always have non-negative spectra. Let $\lambda_{\mathcal{L}}$ and
$\lambda_{\mathfrak{L}}$ denote the smallest non-zero eigenvalues of these two
operators, respectively. We will then define

$\displaystyle\lambda_{E}$ $\displaystyle=\lambda_{\mathcal{L}}-\varepsilon,$
(42) $\displaystyle\lambda_{Y}$
$\displaystyle=\lambda_{\mathfrak{L}}-\varepsilon,$ (43)

for some $\varepsilon>0$ such that $\lambda_{E}$ and $\lambda_{Y}$ are both
strictly positive. As the spectra of $\mathcal{L}_{\gamma,\gamma}$ and
$\mathfrak{L}_{\gamma,\gamma}$ will evolve as $g$ evolves, and hence the
shadow metric $\gamma$, evolves and the smallest non-zero eigenvalue plays a
critical role in obtaining the decay of our energies, we must choose
$\lambda_{E}$ and $\lambda_{Y}$ to be smaller than $\lambda_{\mathcal{L}}$ and
$\lambda_{\mathfrak{L}}$, respectively, such that the decay we obtain is
uniform. Due to the small data scenario we are working in, if we choose a
sufficiently large $\varepsilon$ and sufficiently small neighborhood
$B_{\delta}(0)$ to work in, then we can guarantee that no
$\mathcal{L}_{g,\gamma}$ or $\mathfrak{L}_{g,\gamma}$ has eigenvalues below
$\lambda_{E}$ and $\lambda_{Y}$, respectively.

We now may define the first-order energy
$E^{(1)}=E^{(1)}_{\mathrm{Ein}}+E^{(1)}_{\mathrm{YM}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}$,
where

$\displaystyle E^{(1)}_{\mathrm{Ein}}$
$\displaystyle=\langle\widehat{k},\widehat{k}\rangle+\frac{1}{4}\langle
u,\mathcal{L}_{g,\gamma}u\rangle,$ (44) $\displaystyle E_{\mathrm{YM}}^{(1)}$
$\displaystyle=\langle\mathcal{E},\mathcal{E}\rangle+\langle
A-\widehat{A},\mathfrak{L}_{g,\gamma}(A-\widehat{A})\rangle,$ (45)
$\displaystyle\Gamma^{(1)}_{\mathrm{Ein}}$
$\displaystyle=\frac{1}{n}\langle\widehat{k},u\rangle,$ (46)
$\displaystyle\Gamma^{(1)}_{\mathrm{YM}}$
$\displaystyle=\frac{1}{n}\langle\mathcal{E},A-\hat{A}\rangle,$ (47)

and $c_{E},c_{Y}>0$ are constants given by

$\displaystyle c_{E}$
$\displaystyle=\min\bigg{\\{}\frac{2n^{2}\lambda_{E}}{(n-1)},\;\frac{n-1}{2}\bigg{\\}}$
(48) $\displaystyle c_{Y}$ $\displaystyle=\min\bigg{\\{}\frac{\sqrt[\
]{\lambda_{Y}}}{2},\;\frac{n-3}{2}\bigg{\\}}.$ (49)

We will see later that the correction terms $\Gamma^{(1)}_{\mathrm{Ein}}$ and
$\Gamma^{(1)}_{\mathrm{YM}}$ will be vital in obtaining a decay of the energy.
We will also see that these corrections do not impact the positive-
definiteness of $E^{(1)}$ due to the choice of $c_{E}$ and $c_{Y}$, which will
be bounded from below by the non-negative Sobolev norms of
$(\widehat{k},u,\mathcal{E},A)$. As such, the decay of the energy will imply,
as desired, that the Sobolev norms of $(\widehat{k},u,\mathcal{E},A)$ will
vanish.

From the first-order energy, we may construct higher-order energies. To do so,
we make repeated application of the operators $\mathcal{L}_{g,\gamma}^{i}$ and
$\mathfrak{L}_{g,\gamma}^{i}$ to the small values
$(\widehat{k},u,\mathcal{E},A)$ in the first-order energy. That is, the
higher-order energy terms will be given by

$\displaystyle E^{(i)}$
$\displaystyle=E^{(i)}_{\mathrm{Ein}}+E^{(i)}_{\mathrm{YM}}+c_{E}\Gamma^{(i)}_{\mathrm{Ein}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}},$
$\displaystyle E^{(i)}_{\mathrm{Ein}}$
$\displaystyle=\big{\langle}\widehat{k},\mathcal{L}_{g,\gamma}^{i-1}\widehat{k}\big{\rangle}+\frac{1}{4}\big{\langle}u,\mathcal{L}_{g,\gamma}^{i}u\big{\rangle},$
$\displaystyle E^{(i)}_{\mathrm{YM}}$
$\displaystyle=\big{\langle}\mathcal{E},\mathfrak{L}_{g,\gamma}^{i-1}\mathcal{E}\big{\rangle}+\big{\langle}(A-\widehat{A}),\mathfrak{L}_{g,\gamma}^{i}(A-\widehat{A})\big{\rangle},$
$\displaystyle\Gamma^{(i)}_{\mathrm{Ein}}$
$\displaystyle=\frac{1}{n}\big{\langle}\widehat{k},\mathcal{L}_{g,\gamma}^{i-1}u\big{\rangle},$
$\displaystyle\Gamma^{(i)}_{\mathrm{YM}}$
$\displaystyle=\frac{1}{n}\big{\langle}\mathcal{E},\mathfrak{L}_{g,\gamma}^{i-1}(A-\widehat{A})\big{\rangle}.$

With $s>\frac{n}{2}+1$, the total energy will then be given by

$E_{s}=\sum_{i=1}^{s}E^{(i)}.$ (50)

However we will not need to study the total energy in its entirety. Rather,
due to the embeddings $H^{s_{1}}\hookrightarrow H^{s_{2}}$ for $s_{1}>s_{2}$
and $H^{s}\hookrightarrow L^{2}$ for all $s$, we have that the $L^{2}$ norms
of the derivatives
$(\nabla[\gamma]^{I}\widehat{k},\nabla[\gamma]^{I}u,\nabla[\gamma]^{I}\mathcal{E},\nabla[\gamma]^{I}A)$
are controlled by the same Sobolev norms as $(\widehat{k},u,\mathcal{E},A)$.
Indeed, higher-order energies will only additional contribute terms at higher-
order in the small data due to the need to commute derivatives and apply
Proposition 4.1.2. With this in mind, to study the leading order behavior of
$E_{s}$, we see that it will suffice to estimate $E^{(1)}$ by the appropriate
small data norms.

We will denote $E_{\mathrm{Ein}}=\sum_{i=1}^{s}E^{(i)}_{\mathrm{Ein}}$ and
$E_{\mathrm{YM}}=\sum_{i=1}^{s}E^{(i)}_{\mathrm{YM}}$, and have, by the
immediately preceding discussion, the equivalence

$\displaystyle E_{\mathrm{Ein}}$
$\displaystyle\approx\sum_{i=1}^{s}\lVert\nabla[\gamma]^{i-1}\widehat{k}\rVert_{L^{2}}^{2}+\lVert\nabla[\gamma]^{i}u\rVert_{L^{2}}^{2}\approx\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2},$ $\displaystyle E_{\mathrm{YM}}$
$\displaystyle\approx\sum_{i=1}^{s}\lVert\nabla[\gamma]^{i-1}\mathcal{E}\rVert_{L^{2}}^{2}+\lVert\nabla[\gamma]^{i}A\rVert_{L^{2}}^{2}\approx\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}.$

where $f\approx g$ if there is a constant $C$ such that $C^{-1}f\leq g\leq
Cf$.

To begin our analysis, we will estimate the time evolution of the energy and
show that, to leading order, the energy must decay. It will be convenient to
study the gravitational and Yang-Mills separately, so we compute that the time
derivative of $E^{(1)}_{\mathrm{Ein}}$ is given as

$\displaystyle\partial_{T}E^{(1)}_{\mathrm{Ein}}$
$\displaystyle=\underbrace{2\int_{\Sigma}\gamma^{im}\gamma^{jn}\partial_{T}\widehat{k}_{ij}\widehat{k}_{mn}\>\mu_{g}+\frac{1}{2}\int_{\Sigma}\gamma^{im}\gamma^{jn}\partial_{T}u_{ij}\mathcal{L}_{g,\gamma}u_{mn}\>\mu_{g}}_{\text{Type
I}_{\mathrm{Ein}}}$
$\displaystyle+\underbrace{\frac{1}{4}\int_{\Sigma}\gamma^{im}\gamma^{jn}u_{ij}[\partial_{T},\mathcal{L}_{g,\gamma}]u_{mn}\>\mu_{g}}_{\text{Type
II}_{\mathrm{Ein}}}$
$\displaystyle+\underbrace{2\int_{\Sigma}\partial_{T}\gamma^{im}\gamma^{jn}\widehat{k}_{ij}\widehat{k}_{mn}\>\mu_{g}+\frac{1}{2}\int_{\Sigma}\partial_{T}\gamma^{im}\gamma^{jn}u_{ij}\mathcal{L}_{g,\gamma}u_{mn}\>\mu_{g}}_{\text{Type
III}_{\mathrm{Ein}}}$
$\displaystyle+\underbrace{\frac{1}{2}\int_{\Sigma}\gamma^{im}\gamma^{jn}\widehat{k}_{ij}\widehat{k}_{mn}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}+\frac{1}{8}\int_{\Sigma}\gamma^{im}\gamma^{jn}u_{ij}\mathcal{L}_{g,\gamma}u_{mn}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}}_{\text{Type
IV}_{\mathrm{Ein}}},$

while the time derivative of $E^{(1)}_{\mathrm{YM}}$ is given by

$\displaystyle\partial_{T}E^{(1)}_{\mathrm{YM}}$
$\displaystyle=\underbrace{2\int_{\Sigma}\gamma^{ij}\partial_{T}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}+2\int_{\Sigma}\gamma^{ij}\partial_{T}(A-\widehat{A})^{a}{}_{bi}\mathfrak{L}_{g,\gamma}(A-\widehat{A})^{b}{}_{aj}\>\mu_{g}}_{\text{Type
I}_{\mathrm{YM}}}$
$\displaystyle+\underbrace{\int_{\Sigma}\gamma^{ij}(A-\widehat{A})^{a}{}_{bi}[\partial_{T},\mathfrak{L}_{g,\gamma}](A-\widehat{A})^{b}{}_{aj}\>\mu_{g}}_{\text{Type
II}_{\mathrm{YM}}}$
$\displaystyle+\underbrace{\int_{\Sigma}\partial_{T}\gamma^{ij}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}+\int_{\Sigma}\partial_{T}\gamma^{ij}(A-\widehat{A})^{a}{}_{bi}\mathfrak{L}_{g,\gamma}(A-\widehat{A})^{b}{}_{aj}\>\mu_{g}}_{\text{Type
III}_{\mathrm{YM}}}$
$\displaystyle+\underbrace{\frac{1}{2}\int_{\Sigma}\gamma^{ij}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}+\frac{1}{2}\int_{\Sigma}\gamma^{ij}(A-\widehat{A})^{a}{}_{bi}\mathfrak{L}_{g,\gamma}(A-\widehat{A})^{b}{}_{aj}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}}_{\text{Type
IV}_{\mathrm{YM}}}.$

Meanwhile, the time evolution of the correction terms
$\Gamma^{(1)}_{\mathrm{Ein}}$ and $\Gamma^{(1)}_{\mathrm{YM}}$ will be given
by

$\displaystyle\partial_{T}\Gamma^{(1)}_{\mathrm{Ein}}$
$\displaystyle=\frac{1}{n}\int_{\Sigma}\gamma^{im}\gamma^{jn}\partial_{T}\widehat{k}_{ij}u_{mn}\>\mu_{g}+\frac{1}{n}\int_{\Sigma}\gamma^{im}\gamma^{jn}\widehat{k}_{ij}\partial_{T}u_{mn}\>\mu_{g}$
$\displaystyle\quad+\frac{2}{n}\int_{\Sigma}\partial_{T}\gamma^{im}\gamma^{jn}\widehat{k}_{ij}u_{mn}\>\mu_{g}+\frac{1}{2n}\int_{\Sigma}\gamma^{im}\gamma^{jn}\widehat{k}_{ij}u_{mn}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}$
$\displaystyle\partial_{T}\Gamma^{(1)}_{\mathrm{YM}}$
$\displaystyle=\frac{1}{n}\int_{\Sigma}\gamma^{ij}\partial_{T}\mathcal{E}^{a}{}_{bi}A^{b}{}_{aj}\>\mu_{g}+\frac{1}{n}\int_{\Sigma}\gamma^{ij}\mathcal{E}^{a}{}_{bi}\partial_{T}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad+\frac{1}{n}\int_{\Sigma}\partial_{T}\gamma^{ij}\mathcal{E}^{a}{}_{bi}A^{b}{}_{aj}\>\mu_{g}+\frac{1}{2}\int_{\Sigma}\gamma^{ij}\mathcal{E}^{a}{}_{bi}A^{b}{}_{aj}\operatorname{tr}_{g}(\partial_{T}g)\>\mu_{g}.$

The time derivatives of the small data $(\widehat{k},u,\mathcal{E},A)$ are
given by the evolution equations for $(\widehat{k},g,\mathcal{E},A)$. In
particular, with the time derivative of the background metric $\gamma$—an
element of the vector space $T_{\gamma}\mathcal{N}$—able to be decomposed as
in (15), we find that the time evolution of $u$ is given by

$\displaystyle\partial_{T}u_{ij}$
$\displaystyle=2(\omega+n)\widehat{k}_{ij}-h^{\text{TT}\parallel}-X^{m}\nabla[\gamma]_{m}u_{ij}+\mathcal{F}^{(1)}_{ij}$
(51) $\displaystyle\mathcal{F}^{(1)}_{ij}$ $\displaystyle=\frac{2}{n}\omega
u_{ij}-u_{im}\nabla[\gamma]_{j}X^{m}-u_{jm}\nabla[\gamma]_{i}X^{m}+\frac{2}{n}\omega\gamma_{ij}-L_{X+Y^{\parallel}}\gamma_{ij},$
(52)

where the principle terms that may be problematic for regularity and the
leading-order decay terms have been separated from the remaining terms
$\mathcal{F}^{(1)}_{ij}$.

It will also be convenient to separate out such terms in the evolution
equations for $\widehat{k}$, $\mathcal{E}$, and $A$. Doing so, we write the
evolution equation for $\widehat{k}$ as

$\displaystyle\partial_{T}\widehat{k}_{ij}$
$\displaystyle=-(n-1)\widehat{k}_{ij}-\frac{1}{2}(\omega+n)\mathcal{L}_{g,\gamma}u_{ij}-X^{m}\nabla[\gamma]_{m}\widehat{k}_{ij}+\mathcal{F}^{(2)}_{ij}$
(53) $\displaystyle\mathcal{F}^{(2)}_{ij}$
$\displaystyle=-\frac{n-2}{n}\omega\widehat{k}_{ij}-\widehat{k}_{im}\nabla[\gamma]_{j}X^{m}-\widehat{k}_{jm}\nabla[\gamma]_{i}X^{m}$
$\displaystyle\qquad+\nabla[\gamma]_{i}\nabla[\gamma]_{j}\omega-\frac{1}{2}g^{mn}\big{(}\nabla[\gamma]_{i}u_{jn}+\nabla[\gamma]_{j}u_{in}-\nabla[\gamma]_{n}u_{ij}\big{)}\nabla[\gamma]_{m}\omega$
$\displaystyle\qquad+\tau_{0}^{2}e^{-2T}(\omega+n)\bigg{\\{}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}-F^{a}{}_{bik}F^{b}{}_{aj}{}^{k}+\bigg{(}\frac{1}{2(n-1)}|F|^{2}-\frac{1}{n-1}|\mathcal{E}|^{2}\bigg{)}(u_{ij}+\gamma_{ij})\bigg{\\}}$
$\displaystyle\qquad-\frac{\omega}{n^{2}}(u_{ij}+\gamma_{ij})+(\omega+n)\big{\\{}2\widehat{k}_{ik}\widehat{k}_{j}{}^{k}-J_{ij}\big{\\}}$
(54) $\displaystyle J_{ij}$
$\displaystyle=\frac{1}{2}(u_{i\ell}u^{mn}R[\gamma]^{\ell}{}_{mjn}+u_{j\ell}u^{mn}R[\gamma]^{\ell}{}_{min})$
$\displaystyle\qquad+\frac{1}{2}g^{mn}\gamma^{k\ell}\bigg{\\{}\nabla[\gamma]_{j}u_{nk}\nabla[\gamma]_{\ell}u_{im}+\nabla[\gamma]_{i}u_{\ell
m}\nabla[\gamma]_{k}u_{jn}-\frac{1}{2}\nabla[\gamma]_{j}u_{nk}\nabla[\gamma]_{i}u_{\ell
m}$
$\displaystyle\qquad+\nabla[\gamma]_{m}u_{i\ell}\nabla[\gamma]_{n}u_{jk}-\nabla[\gamma]_{m}u_{i\ell}\nabla[\gamma]_{k}u_{jn}\bigg{\\}}.$
(55)

For $A$, we have

$\displaystyle\partial_{T}A^{a}{}_{bi}$
$\displaystyle=-(\omega+n)\mathcal{E}^{a}{}_{bi}-X^{j}\nabla[\gamma]_{j}A^{a}{}_{bi}+\mathcal{F}^{(3)}{}^{a}{}_{bi}$
(56) $\displaystyle\mathcal{F}^{(3)}{}^{a}{}_{bi}$
$\displaystyle=-\nabla[\gamma]_{i}\varphi^{a}{}_{b}-[A_{i},\varphi]^{a}{}_{b}-A^{a}{}_{bj}\nabla[\gamma]_{i}X^{j}.$
(57)

Finally, for $\mathcal{E}$ we write the evolution equation in the form

$\displaystyle\partial_{T}\mathcal{E}^{a}{}_{bi}$
$\displaystyle=-(n-3)\mathcal{E}^{a}{}_{bi}+(\omega+n)\mathfrak{L}_{g,\gamma}A^{a}{}_{bi}-X^{k}\nabla[\gamma]_{k}\mathcal{E}^{a}{}_{bi}+\mathcal{F}^{(4)}{}^{a}{}_{bi}$
(58) $\displaystyle\mathcal{F}^{(4)}{}^{a}{}_{bi}$
$\displaystyle=-\frac{n-2}{n}\omega\mathcal{E}^{a}{}_{bi}+g^{jk}\nabla[\gamma]_{j}\omega\big{(}\nabla[\gamma]_{i}A^{a}{}_{bk}-\nabla[\gamma]_{k}A^{a}{}_{bi}+[A_{i},A_{k}]^{a}{}_{b}\big{)}$
$\displaystyle\qquad+\mathcal{E}^{a}{}_{bj}\big{(}2(\omega+n)\widehat{k}_{i}{}^{j}-\nabla[\gamma]_{i}X^{j}\big{)}-[\mathcal{E}_{i},\varphi]^{a}{}_{b}+(\omega+n)S^{a}{}_{bi}$
(59) $\displaystyle S^{a}{}_{bi}$
$\displaystyle=g_{im}g^{j\ell}\nabla[\gamma]_{j}g^{mk}\big{(}\nabla[\gamma]_{k}A^{a}{}_{b\ell}-\nabla[\gamma]_{\ell}A^{a}{}_{bk}+[A_{k},A_{\ell}]^{a}{}_{b}\big{)}$
$\displaystyle\qquad+\nabla[\gamma]_{j}g^{j\ell}\big{(}\nabla[\gamma]_{i}A^{a}{}_{b\ell}-\nabla[\gamma]_{\ell}A^{a}{}_{bi}+[A_{i},A_{\ell}]^{a}{}_{b}\big{)}$
$\displaystyle\qquad+g^{j\ell}\big{(}2[\nabla[\gamma]_{j}A_{i},A_{\ell}]^{a}{}_{b}+[A_{i},\nabla[\gamma]_{j}A_{\ell}]^{a}{}_{b}-[\nabla[\gamma]_{i}A_{j},A_{\ell}]^{a}{}_{b}\big{)}$
$\displaystyle\qquad-g^{j\ell}\big{[}[A_{i},A_{j}],A_{\ell}\big{]}^{a}{}_{b}+\nabla[\gamma]_{i}\big{(}g^{j\ell}\nabla[\gamma]_{j}\widehat{A}^{a}{}_{b\ell}-g^{j\ell}[\widehat{A}_{j},A_{\ell}-\widehat{A}_{\ell}]^{a}{}_{b}\big{)}.$
(60)

###  5.1 Yang-Mills Sector

####  5.1.1 Time Evolution of Energy

Recall we fix $\widehat{A}\equiv 0$. With this condition imposed, we begin by
estimating the Type $\mathrm{I}_{\mathrm{YM}}$ terms. Explicitly, we find that
these terms are given by

$\displaystyle\mathrm{I}_{\mathrm{YM}}$
$\displaystyle=2\langle\partial_{T}\mathcal{E},\mathcal{E}\rangle+2\langle\partial_{T}A,\mathfrak{L}_{g,\gamma}A\rangle$
$\displaystyle=-2(n-3)\langle\mathcal{E},\mathcal{E}\rangle+2\langle(\omega+n)\mathfrak{L}_{g,\gamma}A,\mathcal{E}\rangle-2\langle
X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle+2\langle\mathcal{F}^{(4)},\mathcal{E}\rangle$
$\displaystyle\quad-2\langle(\omega+n)\mathcal{E},\mathfrak{L}_{g,\gamma}A\rangle-2\langle
X^{k}\nabla[\gamma]_{k}A,\mathfrak{L}_{g,\gamma}A\rangle+2\langle\mathcal{F}^{(3)},\mathfrak{L}_{g,\gamma}A\rangle$
$\displaystyle=-2(n-3)\langle\mathcal{E},\mathcal{E}\rangle-2\langle
X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle+2\langle\mathcal{F}^{(4)},\mathcal{E}\rangle$
$\displaystyle\quad-2\langle
X^{k}\nabla[\gamma]_{k}A,\mathfrak{L}_{g,\gamma}A\rangle+2\langle\mathcal{F}^{(3)},\mathfrak{L}_{g,\gamma}A\rangle.$

We see that from the choice of operator $\mathfrak{L}_{g,\gamma}$ to act on
$A$, the principle terms cancel point-wise. We will now also show that only
$\langle\mathcal{E},\mathcal{E}\rangle$ gives a leading order decay.

First we must handle the terms containing derivatives of $\mathcal{E}$ and $A$
in the direction of the shift $X$. Because the total energy is constructed by
repeatedly applying the second-order differential operator
$\mathfrak{L}_{g,\gamma}$ to the small data, additional derivatives of
$\mathcal{E}$ and $A$ will be problematic as they cannot be controlled by the
appropriate Sobolev norm for the highest-order energies. As such, we must show
that the terms containing additional derivatives of $\mathcal{E}$ and $A$ can
indeed be controlled by appropriate norms, a fact which will hold true due to
the higher regularity of $X$. In particular, using integration by parts, we
may explicitly compute $\langle
X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle$ as

$\displaystyle\langle X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle$
$\displaystyle=\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\langle X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle$
$\displaystyle=-\int_{\Sigma}\gamma^{ij}X^{k}\mathcal{E}^{a}{}_{bi}\nabla[\gamma]_{k}\mathcal{E}^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad-\int_{\Sigma}\gamma^{ij}\big{(}\nabla[\gamma]_{k}X^{k}+X^{k}(\Gamma[g]^{\ell}{}_{k\ell}-\Gamma[\gamma]^{\ell}{}_{k\ell})\big{)}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}$
$\displaystyle 2\langle X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle$
$\displaystyle=-\int_{\Sigma}\gamma^{ij}\big{(}\nabla[\gamma]_{k}X^{k}+X^{k}(\Gamma[g]^{\ell}{}_{k\ell}-\Gamma[\gamma]^{\ell}{}_{k\ell})\big{)}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\langle X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle$
$\displaystyle=-\frac{1}{2}\int_{\Sigma}\gamma^{ij}\big{(}\nabla[\gamma]_{k}X^{k}+X^{k}(\Gamma[g]^{\ell}{}_{k\ell}-\Gamma[\gamma]^{\ell}{}_{k\ell})\big{)}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}\>\mu_{g}.$

Here we also use that the covariant derivative with respect to the background
metric $\gamma$ of the measure $\mu_{g}$ is given by
$(\Gamma[g]-\Gamma[\gamma])^{\ell}{}_{k\ell}\mu_{g}$. We may then estimate the
inner product as

$\displaystyle\big{|}\langle
X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle\big{|}$
$\displaystyle\leq C\big{(}\lVert\nabla[\gamma]X\rVert_{L^{\infty}}+\lVert
X\rVert_{L^{\infty}}\lVert\Gamma[g]-\Gamma[\gamma]\rVert_{L^{\infty}}\big{)}\lVert\mathcal{E}\rVert_{L^{2}}^{2}$
$\displaystyle\leq C\big{(}\lVert\nabla[\gamma]X\rVert_{H^{s-1}}+\lVert
X\rVert_{H^{s-1}}\lVert\Gamma[g]-\Gamma[\gamma]\rVert_{H^{s-1}}\big{)}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2},$
$\displaystyle\leq C\big{(}\lVert X\rVert_{H^{s+1}}+\lVert
X\rVert_{H^{s+1}}\lVert
u\rVert_{H^{s}}\big{)}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}$ (61)

where we use the embeddings $H^{s}\hookrightarrow H^{s-1}\hookrightarrow
L^{\infty}$ for $s-1>\frac{n}{2}$, and that to leading order
$\lVert\Gamma[g]-\Gamma[\gamma]\rVert_{H^{s-1}}=\lVert u\rVert_{H^{s}}$. Due
to the higher regularity of $X$, we are thus still able to control the inner
product.

We may perform a similar computation for the term containing a derivative of
$A$ in the direction of $X$, giving

$\displaystyle\langle X^{k}\nabla$
$\displaystyle[\gamma]_{k}A,\mathfrak{L}_{g,\gamma}A\rangle$
$\displaystyle=\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}\mathfrak{L}_{g,\gamma}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle=-\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}\big{(}\nabla[\gamma]_{m}(g^{mn}\mu_{g}\nabla[\gamma]_{n}A^{b}{}_{aj})+g^{\ell
m}R[\gamma]^{n}{}_{m\ell j}A^{b}{}_{an}\>\mu_{g}\big{)}$
$\displaystyle=\int_{\Sigma}\gamma^{ij}(\nabla[\gamma]_{m}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}+X^{k}\nabla[\gamma]_{m}\nabla[\gamma]_{k}A^{a}{}_{bi})g^{mn}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad-\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}g^{\ell
m}R[\gamma]^{n}{}_{m\ell j}A^{b}{}_{an}\>\mu_{g}$
$\displaystyle=\int_{\Sigma}\gamma^{ij}g^{mn}X^{k}\nabla[\gamma]_{k}\nabla[\gamma]_{m}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}+\int_{\Sigma}\gamma^{ij}g^{mn}X^{k}R[\gamma]^{\ell}{}_{imk}A^{a}{}_{b\ell}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}g^{mn}\nabla[\gamma]_{m}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}-\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}g^{\ell
m}R[\gamma]^{n}{}_{m\ell j}A^{b}{}_{an}\>\mu_{g}$
$\displaystyle=\frac{1}{2}\int_{\Sigma}\gamma^{ij}g^{mn}X^{k}\nabla[\gamma]_{k}(\nabla[\gamma]_{m}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj})\>\mu_{g}+\int_{\Sigma}\gamma^{ij}g^{mn}X^{k}R[\gamma]^{\ell}{}_{imk}A^{a}{}_{b\ell}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}g^{mn}\nabla[\gamma]_{m}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}-\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}g^{\ell
m}R[\gamma]^{n}{}_{m\ell j}A^{b}{}_{an}\>\mu_{g}$
$\displaystyle=-\frac{1}{2}\int_{\Sigma}\gamma^{ij}\big{(}\nabla[\gamma]_{k}g^{mn}X^{k}+g^{mn}\nabla[\gamma]_{k}X^{k}+g^{mn}X^{k}(\Gamma[g]^{\ell}{}_{k\ell}-\Gamma[\gamma]^{\ell}{}_{k\ell})\big{)}\nabla[\gamma]_{m}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}g^{mn}X^{k}R[\gamma]^{\ell}{}_{imk}A^{a}{}_{b\ell}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}g^{mn}\nabla[\gamma]_{m}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}\nabla[\gamma]_{n}A^{b}{}_{aj}\>\mu_{g}-\int_{\Sigma}\gamma^{ij}X^{k}\nabla[\gamma]_{k}A^{a}{}_{bi}g^{\ell
m}R[\gamma]^{n}{}_{m\ell j}A^{b}{}_{an}\>\mu_{g}.$

Again, we use integration by parts and Stokes’ theorem on a compact manifold
to move the extraneous derivative on $A$ to one on $X$. Using the same
embeddings as for $\mathcal{E}$, we may estimate $\langle
X^{k}\nabla[\gamma]_{k}A,\mathfrak{L}_{g,\gamma}A\rangle$ in terms of the
appropriate Sobolev norms. In particular, using the estimate for $X$ from
Lemma 4.4, we get that the terms containing extra derivatives of $\mathcal{E}$
and $A$ may be controlled by

$\displaystyle\Big{|}\langle
X^{k}\nabla[\gamma]_{k}\mathcal{E},\mathcal{E}\rangle\Big{|}$
$\displaystyle\leq
C\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}})$
$\displaystyle\quad+Ce^{-2T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+Ce^{-4T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}})^{2}$ (62) $\displaystyle\Big{|}\langle
X^{k}\nabla[\gamma]_{k}A,\mathfrak{L}_{g,\gamma}A\rangle\Big{|}$
$\displaystyle\leq C\lVert
A\rVert_{H^{s}}^{2}(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert u\rVert_{H^{s}})$
$\displaystyle\quad+Ce^{-2T}\lVert
A\rVert_{H^{s}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+Ce^{-4T}\lVert
A\rVert_{H^{s}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}$ (63)

Meanwhile, using the Sobolev embeddings and the elliptic estimates for
$\omega$, $X$, and $\varphi$, we can also estimate the terms containing
$\mathcal{F}^{(4)}$ and $\mathcal{F}^{(3)}$ by

$\displaystyle\big{|}\langle\mathcal{F}^{(4)},\mathcal{E}\rangle\big{|}$
$\displaystyle\leq
C\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}\big{(}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}\big{)}+C\lVert\mathcal{E}\rVert_{H^{s-1}}\lVert
A\rVert_{H^{s}}\big{(}\lVert u\rVert_{H^{s}}+\lVert A\rVert_{H^{s}}\big{)}$
$\displaystyle+Ce^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}+Ce^{-4T}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}$ (64)
$\displaystyle\Big{|}\langle\mathcal{F}^{(3)},\mathfrak{L}_{g,\gamma}A\rangle\Big{|}$
$\displaystyle\leq C\lVert
A\rVert_{H^{s}}(\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})$
$\displaystyle+Ce^{-2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}+Ce^{-4T}\lVert
A\rVert_{H^{s}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}$ (65)

Note that each term in $\mathcal{F}^{(3)}$ has regularity at the level of $A$,
and so we can integrate by parts to move derivatives from the second-order
operator $\mathfrak{L}_{g,\gamma}A$ onto $\mathcal{F}^{(3)}$ and retain the
appropriate regularity. Putting all the estimates together, we have that the
Type $\mathrm{I}_{\mathrm{YM}}$ terms may be written as

$\mathrm{I}_{\mathrm{YM}}=-2(n-3)\lVert\mathcal{E}\rVert_{L^{2}}^{2}+\mathcal{R}_{\mathrm{YM}},$
(66)

where $\mathcal{R}_{\mathrm{YM}}$ satisfies the estimate

$\displaystyle|\mathcal{R}_{\mathrm{YM}}|$ $\displaystyle\leq
C(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}})$ $\displaystyle\quad+C\lVert
A\rVert_{H^{s}}(\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})$
$\displaystyle\quad+Ce^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}$
$\displaystyle\quad+Ce^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{3}.$ (67)

We will now see that, to leading order, the Type $\mathrm{I}_{\mathrm{YM}}$
terms will control the decay of $\partial_{T}E_{\mathrm{YM}}$. That is, the
Type $\mathrm{II}_{\mathrm{YM}}$, $\mathrm{III}_{\mathrm{YM}}$, and
$\mathrm{IV}_{\mathrm{YM}}$ terms will satisfy only third-order estimates. To
begin, by differentiating under the integral sign and using that
$\mathfrak{L}_{g,\gamma}$ is a self-adjoint operator for the given inner
product, we may write the Type $\mathrm{II}_{\mathrm{YM}}$ terms as

Observe that the final two terms of $\mathrm{II}_{\mathrm{YM}}$ will cancel
point-wise with terms in $\mathrm{III}_{\mathrm{YM}}$ and
$\mathrm{IV}_{\mathrm{YM}}$. We will thus only look at how to control the
first two terms in the above expression. In particular, writing out the
definition of $\mathfrak{L}_{g,\gamma}$ and differentiating under the integral
sign, we have

Collecting all of the Type $\mathrm{II}_{\mathrm{YM}}$,
$\mathrm{III}_{\mathrm{YM}}$, and $\mathrm{IV}_{\mathrm{YM}}$ terms together,
we find that

$\displaystyle\mathrm{II}_{\mathrm{YM}}+\mathrm{III}_{\mathrm{YM}}+\mathrm{IV}_{\mathrm{YM}}$
$\displaystyle=\frac{1}{2}\int_{\Sigma}\big{(}2\partial_{T}\gamma^{ij}+\gamma^{ij}\operatorname{tr}_{g}(\partial_{T}g)\big{)}\big{(}\mathcal{E}^{a}{}_{bi}\mathcal{E}^{b}{}_{aj}+A^{a}{}_{bi}\mathfrak{L}_{g,\gamma}A^{b}{}_{aj}\big{)}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}\partial_{T}g^{k\ell}\nabla[\gamma]_{k}A^{a}{}_{bi}\nabla[\gamma]_{\ell}A^{b}{}_{aj}\>\mu_{g}+2\int_{\Sigma}\gamma^{ij}g^{k\ell}\big{[}\partial_{T},\nabla[\gamma]_{k}\big{]}A^{a}{}_{bi}\nabla[\gamma]_{\ell}A^{b}{}_{aj}\>\mu_{g}$
$\displaystyle\quad-\int_{\Sigma}\gamma^{ij}g^{k\ell}A^{a}{}_{bi}\partial_{T}R[\gamma]^{m}{}_{\ell
jk}A^{b}{}_{am}\>\mu_{g}.$

Using the embedding $H^{s-1}\hookrightarrow L^{\infty}$ for $s-1>\frac{n}{2}$,
we then have the estimate

$\displaystyle\big{|}\mathrm{II}_{\mathrm{YM}}+\mathrm{III}_{\mathrm{YM}}+\mathrm{IV}_{\mathrm{YM}}\big{|}$
$\displaystyle\leq C\big{\\{}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}\big{\\{}\lVert\partial_{T}\gamma\rVert+\lVert\partial_{T}g\rVert_{H^{s-1}}\big{\\}}$
$\displaystyle\quad+C\lVert
A\rVert_{H^{s}}\lVert[\partial_{T},\nabla[\gamma]]A\rVert_{H^{s-1}}+C\lVert
A\rVert_{H^{s}}^{2}\lVert\partial_{T}\mathrm{Riem}[\gamma]\rVert.$

Observe that $[\partial_{T},\nabla[\gamma]]A=(\partial_{T}\Gamma[\gamma])A$
because partial derivatives commute, and so we may write

$\lVert[\partial_{T},\nabla[\gamma]]A\rVert_{H^{s-1}}=\lVert(D\Gamma[\gamma].\partial_{T}\gamma)\cdot
A\rVert_{H^{s-1}}\leq C\lVert\partial_{T}\gamma\rVert\lVert
A\rVert_{H^{s-1}},$

where $(D\Gamma[\gamma].\partial_{T}\gamma)\cdot A$ is the Fréchet derivative
of $\Gamma[\gamma]$ in the direction of $\partial_{T}\gamma$, contracted with
$A$. We may similarly write
$\partial_{T}\mathrm{Riem}[\gamma]=D\mathrm{Riem}[\gamma].\partial_{T}\gamma$
to see that $\lVert\partial_{T}\mathrm{Riem[\gamma]}\rVert\leq
C\lVert\partial_{T}\gamma\rVert$. Using also the evolution equations for $g$
and Lemma 4.2, we finally obtain the estimate

$\displaystyle\big{|}\mathrm{II}_{\mathrm{YM}}+\mathrm{III}_{\mathrm{YM}}+\mathrm{IV}_{\mathrm{YM}}\big{|}$
$\displaystyle\leq C\big{\\{}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}$
$\displaystyle\quad\times\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}$ (68)

We see then that the Type $\mathrm{II}_{\mathrm{YM}}$,
$\mathrm{III}_{\mathrm{YM}}$, and $\mathrm{IV}_{\mathrm{YM}}$ terms in
$\partial_{T}E_{\mathrm{YM}}$ will satisfy a third-order estimate. In fact, if
we increase the constant in (5.1.1), we will have that the non-Type
$\mathrm{I}_{\mathrm{YM}}$ terms satisfy the same estimate as the Type
$\mathrm{I}_{\mathrm{YM}}$ remainder $\mathcal{R}_{\mathrm{YM}}$. We may thus
write $\partial_{T}E_{\mathrm{YM}}$ as

$\partial_{T}E^{(1)}_{\mathrm{YM}}=-2(n-3)\lVert\mathcal{E}\rVert_{L^{2}}^{2}+\mathcal{R}_{\mathrm{YM}},$
(69)

with $\mathcal{R}_{\mathrm{YM}}$ satisfying bound given in (5.1.1) for a re-
defined constant $C$. Note that we may write the estimate on
$\mathcal{R}_{\mathrm{YM}}$ in terms of
$E_{\mathrm{YM}}\approx\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}$ and
$E_{\mathrm{Ein}}\approx\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}$, giving us

$|\mathcal{R}_{\mathrm{YM}}|\leq
C\big{\\{}(E_{\mathrm{YM}})^{\frac{3}{2}}+(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+E_{\mathrm{Ein}}(E_{\mathrm{YM}})^{\frac{1}{2}}+e^{-2T}(E_{\mathrm{YM}})^{2}+e^{-4T}(E_{\mathrm{YM}})^{3}\big{\\}}.$
(70)

######  Remark.

The failure of the energy estimate argument for $n=3$ spatial dimensions
manifests in (69). In particular we see that the leading-order decay term will
vanish, which will prevent us from obtaining a uniformly decaying bound on the
total energy (50) and hence on the Sobolev norms of the small data. As
mentioned earlier, this is a consequence of the conformal invariance of Yang-
Mills equations in $3+1$ dimensions.

Finally, we must find an appropriate estimate for the time derivative of the
correction term $\Gamma^{(1)}_{\mathrm{YM}}$. Using the embedding of $H^{s-1}$
into $L^{\infty}$ for $s>\frac{n}{2}+1$, we write
$\partial_{T}\Gamma^{(1)}_{\mathrm{YM}}$ as

$\partial_{T}\Gamma^{(1)}_{\mathrm{YM}}=-\frac{(n-3)}{n}\langle\mathcal{E},A\rangle+\langle\mathfrak{L}_{g,\gamma}A,A\rangle-\langle\mathcal{E},\mathcal{E}\rangle+\mathcal{S}_{\mathrm{YM}},$
(71)

where $\mathcal{S}_{\mathrm{YM}}$ satisfies the third-order estimate

$\displaystyle|\mathcal{S}_{\mathrm{YM}}|$ $\displaystyle\leq
C\big{\\{}\lVert\omega\rVert_{H^{s+1}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+\lVert\mathcal{E}\rVert_{H^{s-1}}\lVert
A\rVert_{H^{s}}(\lVert X\rVert_{H^{s+1}}+\lVert\partial_{T}g\rVert_{H^{s-1}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\lVert\mathcal{E}\rVert_{H^{s-1}}\lVert\mathcal{F}^{(3)}\rVert_{H^{s-1}}+\lVert
A\rVert_{H^{s}}\lVert\mathcal{F}^{(4)}\rVert_{H^{s-1}}\big{\\}}$
$\displaystyle\leq
C\lVert\mathcal{E}\rVert_{H^{s-1}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}$
$\displaystyle\quad+C\lVert\mathcal{E}\rVert_{H^{s-1}}\lVert
A\rVert_{H^{s}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}\big{\\}}+C\lVert A\rVert_{H^{s}}^{2}(\lVert
u\rVert_{H^{s}}+\lVert A\rVert_{H^{s}})$
$\displaystyle\quad+Ce^{-2T}\big{\\{}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}^{2}+Ce^{-4T}\big{\\{}\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}^{3}$ $\displaystyle\leq
C\big{\\{}(E_{\mathrm{YM}})^{\frac{3}{2}}+(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+E_{\mathrm{Ein}}(E_{\mathrm{YM}})^{\frac{1}{2}}+e^{-2T}(E_{\mathrm{YM}})^{2}+e^{-4T}(E_{\mathrm{YM}})^{3}\big{\\}}.$

Note that $\mathcal{S}_{\mathrm{YM}}$ satisfies the same estimate as
$\mathcal{R}_{\mathrm{YM}}$, albeit with a different constant $C$. Putting
together the two expressions for $\partial_{T}E^{(1)}_{\mathrm{YM}}$ and
$\partial_{T}\Gamma^{(1)}_{\mathrm{YM}}$, we arrive at the total contribution
from the Yang-Mills sector to $\partial_{T}E^{(1)}$,

$\partial_{T}\big{(}E_{\mathrm{YM}}^{(1)}-c_{Y}\Gamma_{\mathrm{YM}}^{(1)}\big{)}=-\big{(}2(n-3)-c_{Y}\big{)}\langle\mathcal{E},\mathcal{E}\rangle-
c_{Y}\langle\mathfrak{L}_{g,\gamma}A,A\rangle+\frac{(n-3)}{n}c_{Y}\langle\mathcal{E},A\rangle+\mathcal{R}_{\mathrm{YM}},$

for a redefined $\mathcal{R}_{\mathrm{YM}}$ that still satisfies the estimate
(70). To show that the total energy decays, we will want to show that, to
leading order, the time evolution is uniformly bounded as
$\partial_{T}E_{s}\leq-\alpha E_{s}$ for some positive constant $\alpha$. We
will now see that the Yang-Mills time evolution independently satisfies a
bound of this form, using a similar analysis to that in Lemma 6.4 of [2].

To begin, let $Y=\frac{c_{Y}}{n-3}$, such that $0<Y<1$, and define
$\alpha_{Y}=(n-3)\big{[}1-\sqrt[\ ]{1-Y}\big{]}$. We may then write

$\displaystyle\partial_{T}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}$
$\displaystyle=-\alpha_{Y}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}-\big{(}2(n-3)-c_{Y}-\alpha_{Y}\big{)}\langle\mathcal{E},\mathcal{E}\rangle$
$\displaystyle\qquad-(c_{Y}-\alpha_{Y})\langle\mathfrak{L}_{g,\gamma}A,A\rangle+\frac{c_{Y}}{n}\big{(}(n-3)-\alpha_{Y}\big{)}\langle\mathcal{E},A\rangle+\mathcal{R}_{\mathrm{YM}},$

For the time evolution of
$E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}$ to satisfy the desired
inequality to leading order, we must have that

$-\big{(}2(n-3)-c_{Y}-\alpha_{Y}\big{)}\langle\mathcal{E},\mathcal{E}\rangle-(c_{Y}-\alpha_{Y})\langle\mathfrak{L}_{g,\gamma}A,A\rangle+\frac{c_{Y}}{n}\big{(}(n-3)-\alpha_{Y}\big{)}\langle\mathcal{E},A\rangle<0.$
(72)

Since $\Sigma$ is compact, $\mathfrak{L}_{g,\gamma}$ will have discrete
spectrum. If we perform a spectral decomposition of $\mathfrak{L}_{g,\gamma}$,
we then have the requirement that

$\displaystyle\sum_{k=1}^{\infty}-\big{(}2(n-3)-c_{Y}-\alpha_{Y}\big{)}\big{\langle}\langle\mathcal{E},e_{k}\rangle,\langle\mathcal{E},e_{k}\rangle\big{\rangle}$
$\displaystyle-(c_{Y}-\alpha_{Y})\lambda_{k}\big{\langle}\langle
A,e_{k}\rangle,\langle A,e_{k}\rangle\big{\rangle}$
$\displaystyle+\frac{c_{Y}}{n}\big{(}(n-3)-\alpha_{Y}\big{)}\big{\langle}\langle\mathcal{E},e_{k}\rangle
e_{k},\langle A,e_{k}\rangle e_{k}\rangle<0,$

where $\lambda_{k}>0$ and $e_{k}$ are the $k^{\text{th}}$ eigenvalue and
eigentensor of $\mathfrak{L}_{g,\gamma}$. However, we see that each term in
the summation is equal to the quadratic form

$M_{k}=\begin{bmatrix}c_{Y}+\alpha_{Y}-2(n-3)&\frac{c_{Y}}{2n}\big{(}(n-3)-\alpha_{Y}\big{)}\\\
\frac{c_{Y}}{2n}\big{(}(n-3)-\alpha_{Y}\big{)}&-\lambda_{k}(c_{Y}-\alpha_{Y})\end{bmatrix}=(n-3)\begin{bmatrix}Y-1-\sqrt[\
]{1-Y}&-\frac{Y}{2n}\sqrt[\ ]{1-Y}\\\ -\frac{Y}{2n}\sqrt[\
]{1-Y}&-\lambda_{k}(Y-1+\sqrt[\ ]{1-Y})\end{bmatrix}$

acting on $\big{(}(\langle\mathcal{E},e_{k}\rangle e_{k},\langle
A,e_{k}\rangle e_{k}),(\langle\mathcal{E},e_{k}\rangle e_{k},\langle
A,e_{k}\rangle e_{k})\big{)}$. Hence we will have (72) hold if and only if
$M_{k}$ is negative definite for each $k$. We may compute the trace and
determinant of $M_{k}$ as

$\displaystyle\operatorname{Tr}M_{k}$ $\displaystyle=(Y-1-\sqrt[\
]{1-Y})-\lambda_{k}\sqrt[\ ]{1-Y}(1-\sqrt[\ ]{1-Y}),$ $\displaystyle\det
M_{k}$
$\displaystyle=(1-Y)Y\bigg{[}\lambda_{k}-\frac{c_{Y}}{4n^{2}(n-3)}\bigg{]}.$

Since $0<Y<1$, we have that $\operatorname{Tr}M_{k}<0$. Since $n>3$ and
$c_{Y}<\sqrt[\ ]{\lambda_{Y}}\leq\lambda_{k}$, we have $\det M_{k}>0$. Hence
each $M_{k}$ is negative definite, and so we get the following lemma.

######  Lemma 5.1.

Fix $\widehat{A}=0$ and $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of
sufficiently small radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$
that satisfy the Einstein-Yang-Mills system. Then the time evolution of
$E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}$ is given by

$\partial_{T}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}=-\big{(}2(n-3)-c_{Y}\big{)}\langle\mathcal{E},\mathcal{E}\rangle-
c_{Y}\langle\mathfrak{L}_{g,\gamma}A,A\rangle+\frac{(n-3)}{n}c_{Y}\langle\mathcal{E},A\rangle+\mathcal{R}_{\mathrm{YM}},$
(73)

where $\mathcal{R}_{\mathrm{YM}}$ satisfies the third-order estimate

$|\mathcal{R}_{\mathrm{YM}}|\leq
C\big{\\{}(E_{\mathrm{YM}})^{\frac{3}{2}}+(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+E_{\mathrm{Ein}}(E_{\mathrm{YM}})^{\frac{1}{2}}+e^{-2T}(E_{\mathrm{YM}})^{2}+e^{-4T}(E_{\mathrm{YM}})^{3}\big{\\}}$
(74)

for a constant $C$ depending only on the background geometry and $\tau_{0}$.
In particular, the time evolution of the Yang-Mills energies will satisfy the
bound

$\partial_{T}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}\leq-\alpha_{Y}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}+\mathcal{R}_{\mathrm{YM}},$
(75)

where

$\alpha_{Y}=(n-3)\bigg{(}1-\sqrt[\ ]{1-\frac{c_{Y}}{n-3}}\bigg{)}.$ (76)

####  5.1.2 Positive-Definiteness of Energy

Having the time evolution of the Yang-Mills sector, we also wish to show that
$E_{\mathrm{YM}}-c_{Y}\Gamma_{\mathrm{YM}}$ is positive definite, and in
particular is bounded from below by the Sobolev norms of the small data
$\mathcal{E}$ and $A$. Doing so will allow us to later bound these norms by
the total energy, which we will show will decay and thus force
$\lVert\mathcal{E}\rVert_{H^{s-1}}$ and $\lVert A\rVert_{H^{s}}$ to zero. We
give the result of this section in the following lemma.

######  Lemma 5.2.

Fix $\widehat{A}=0$ and $s>\frac{n}{2}+1$. Then there is a $\delta>0$ and a
constant $C>0$, such that $(\widehat{k},u,\mathcal{E},A)\in B_{\delta}(0)$
satisfy the inequality

$\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert A\rVert_{H^{s}}^{2}\leq
C\sum_{i=1}^{s}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}}).$ (77)

###### Proof.

To begin, note that $(\widehat{k},u,\mathcal{E},A)=(0,0,0,0)$ is a fixed point
of the total energy $E_{s}$ and hence the Yang-Mills energy
$\sum_{i=1}^{s}E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}}$. Since
the Yang-Mills energy is a smooth functional of $\mathcal{E}$ and $A$, to
determine if it is positive-definite we only need to see if the Hessian at
$(\widehat{k},g,\mathcal{E},A)=(0,\gamma,0,0)$ is positive-definite. The
Hessian at this fixed point may be written in the form

$D^{2}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}})\big{(}(\mathcal{E},A),(\mathcal{E},A)\big{)}=2\langle\mathcal{E},\mathfrak{L}_{\gamma,\gamma}^{i-1}\mathcal{E}\rangle+2\langle
A,\mathfrak{L}_{\gamma,\gamma}^{i}A\rangle-\frac{4c_{Y}}{n}\langle\mathcal{E},\mathfrak{L}_{\gamma,\gamma}^{i-1}A\rangle$

Performing a spectral decomposition of $\mathfrak{L}_{\gamma,\gamma}$, we then
see that the Hessian will be positive definite if and only if the quadratic
form

$M_{k}=\begin{bmatrix}1&-c_{Y}/n\\\ -c_{Y}/n&\lambda_{k}\end{bmatrix}$

is. However, this follows directly from the construction of $c_{Y}$ and the
fact that $\mathfrak{L}_{\gamma,\gamma}$ has non-negative spectrum. We see
that $\operatorname{Tr}M=1+\lambda_{k}>0$ and $\det
M=\lambda_{k}-\frac{c_{Y}^{2}}{n^{2}}>0$, as $c_{Y}$ has been chosen such that
$c_{Y}<\sqrt[\ ]{\lambda_{Y}}<\sqrt[\ ]{\lambda_{k}}$. Thus $M$ is positive
definite and so, it follows, is the Hessian of the $i^{\text{th}}$ Yang-Mills
energy. Since this is true for all $1\leq i\leq s$, we then have that
$\sum_{i=1}^{s}D^{2}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}})\geq
0$. However, because each of the quadratic forms $M_{k}$ are positive definite
and thus invertible, we have that
$\sum_{i=1}^{s}D^{2}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}})$
will give an isomorphism of subsets of function spaces. From this isomorphism,
we may conclude that for some constant $C$,

$\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert A\rVert_{H^{s}}^{2}\leq
C\sum_{i=1}^{s}D^{2}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}}).$
(78)

Finally, we may apply the Morse-Palais lemma, giving there exists a
neighborhood of the origin $B_{\delta}(0)$ such that, up to a diffeomorphism
$B_{\delta}(0)\rightarrow B_{\delta}(0)$, we have

$\sum_{i=1}^{s}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}})=\sum_{i=1}^{s}D^{2}(E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}}).$
(79)

Combining (78) and (79), we obtain the desired estimate. ∎

Having obtained an estimate on the time evolution of the Yang-Mills energy,
and seeing that the Yang-Mills energy bounds the Sobolev norms of
$\mathcal{E}$ and $A$, we are now ready to study the gravitational sector and
prove similar results for $\widehat{k}$ and $u$.

###  5.2 Gravitational Sector

####  5.2.1 Time Evolution of Energy

We will now study the time evolution of $\partial_{T}E^{(1)}_{\mathrm{Ein}}$.
In particular, we will see that Type $\mathrm{I}_{\mathrm{Ein}}$ terms
control, to leading order, the decay of $\partial_{T}E^{(1)}_{\mathrm{Ein}}$,
with the other type terms contributing only to higher order. Many of the
calculations are done explicitly in [36], with only the estimates changing
here due to the differing evolution equations and elliptic estimates. As such,
we will present only the necessary equations and estimates here, and refer
readers to the relevant sections of [36] for additional calculations when
appropriate.

To begin, we may write the Type $\mathrm{I}_{\mathrm{Ein}}$ terms as

$\displaystyle\mathrm{I}_{\mathrm{Ein}}$
$\displaystyle=2\langle\partial_{T}\widehat{k},\widehat{k}\rangle+\frac{1}{2}\langle\partial_{t}u,\mathcal{L}_{g,\gamma}u\rangle$
$\displaystyle=-2(n-1)\langle\widehat{k},\widehat{k}\rangle-\langle(\omega+n)\mathcal{L}_{g,\gamma}u,\widehat{k}\rangle-2\langle
X^{m}\nabla[\gamma]_{m}\widehat{k},\widehat{k}\rangle+2\langle\mathcal{F}^{(2)},\widehat{k}\rangle$
$\displaystyle\quad+\langle(\omega+n)\widehat{k},\mathcal{L}_{g,\gamma}u\rangle-\frac{1}{2}\langle
h^{\text{TT}\parallel},\mathcal{L}_{g,\gamma}u\rangle-\frac{1}{2}\langle
X^{m}\nabla[\gamma]_{m}u,\mathcal{L}_{g,\gamma}u\rangle+\langle\mathcal{F}^{(1)},\mathcal{L}_{g,\gamma}u\rangle$
$\displaystyle=-2(n-1)\langle\widehat{k},\widehat{k}\rangle-2\langle
X^{m}\nabla[\gamma]_{m}\widehat{k},\widehat{k}\rangle+2\langle\mathcal{F}^{(2)},\widehat{k}\rangle$
$\displaystyle\quad-\frac{1}{2}\langle
h^{\text{TT}\parallel},\mathcal{L}_{g,\gamma}u\rangle-\frac{1}{2}\langle
X^{m}\nabla[\gamma]_{m}u,\mathcal{L}_{g,\gamma}u\rangle+\frac{1}{2}\langle\mathcal{F}^{(1)},\mathcal{L}_{g,\gamma}u\rangle,$

where we see the principle terms cancel point-wise. As in the Yang-Mills
sector, we have potentially problematic terms containing additional
derivatives of $\widehat{k}$ and $u$. However, using integration by parts, we
can move the extra derivative shift field $X\in H^{s+1}$. Doing so, we obtain

$\langle
X^{k}\nabla[\gamma]_{k}\widehat{k},\widehat{k}\rangle=-\frac{1}{2}\int_{\Sigma}\gamma^{ij}\gamma^{mn}\big{(}\nabla[\gamma]_{k}X^{k}+X^{k}(\Gamma[g]^{\ell}{}_{k\ell}-\Gamma[\gamma]^{\ell}{}_{k\ell})\big{)}\widehat{k}_{in}\widehat{k}_{jm}\>\mu_{g}$

and

$\displaystyle\langle X^{k}\nabla$
$\displaystyle[\gamma]_{k}u,\mathcal{L}_{g,\gamma}u\rangle$
$\displaystyle=-\frac{1}{2}\int_{\Sigma}\gamma^{ij}\gamma^{mn}\big{(}\nabla[\gamma]_{k}g^{\ell
r}X^{k}+g^{\ell r}\nabla[\gamma]_{k}X^{k}+g^{\ell
r}X^{k}(\Gamma[g]^{s}{}_{ks}-\Gamma[\gamma]^{s}{}_{ks})\big{)}\nabla[\gamma]_{\ell}u_{in}\nabla[\gamma]_{r}u_{jm}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}\gamma^{mn}g^{\ell
r}X^{k}\big{(}R[\gamma]^{s}{}_{i\ell k}u_{sn}+R[\gamma]^{s}{}_{n\ell
r}u_{is}\big{)}\nabla[\gamma]_{r}u_{jm}\>\mu_{g}$
$\displaystyle\quad+\int_{\Sigma}\gamma^{ij}\gamma^{mn}g^{\ell
r}\nabla[\gamma]_{\ell}X^{k}\nabla[\gamma]_{k}u_{in}\nabla[\gamma]_{r}u_{jm}\>\mu_{g}-2\int_{\Sigma}\gamma^{ij}\gamma^{mn}X^{k}\nabla[\gamma]_{k}u_{in}g^{ab}g^{cd}\gamma_{js}R[\gamma]^{s}{}_{amc}u_{bd}\>\mu_{g}.$

Using the Sobolev embeddings $H^{s-1}\hookrightarrow L^{\infty}$ for
$s-1>\frac{n}{2}$, we then have the estimates

$\displaystyle\Big{|}\langle
X^{k}\nabla[\gamma]_{k}\widehat{k},\widehat{k}\rangle\Big{|}$
$\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}})\big{\\}}$
$\displaystyle\quad+C\big{\\{}e^{-2T}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}$ (80) $\displaystyle\Big{|}\langle
X^{k}\nabla[\gamma]_{k}u,\mathcal{L}_{g,\gamma}u\rangle\Big{|}$
$\displaystyle\leq C\big{\\{}\lVert
u\rVert_{H^{s}}^{2}(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}})\big{\\}}$ $\displaystyle\quad+C\big{\\{}e^{-2T}\lVert
u\rVert_{H^{s}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}\lVert
u\rVert_{H^{s}}^{2}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}$ (81)

We must also estimate the term containing the transverse-traceless
$h^{\text{TT}\parallel}$, which is potentially problematic as we have no a
priori bound for $h^{\text{TT}\parallel}$. However by an explicit calculation,
as done in equations (165)—(172) of [36], we have that
$h^{\text{TT}\parallel}$ will be bounded by

$\lVert h^{\text{TT}\parallel}\rVert\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert\omega\rVert_{H^{s+1}}\big{\\}},$
(82)

where because $h^{\text{TT}\parallel}$ is an element of the finite-dimensional
deformation space, all its norms are equivalent. This also means that we need
not worry about the extra derivative acting on $u$, as we may freely move it
to $h^{\text{TT}\parallel}$ via integration by parts to achieve the necessary
regularity of $u$. Using the bound on $h^{\text{TT}\parallel}$ and the Sobolev
embeddings, we have the estimate

$\Big{|}\langle
h^{\text{TT}\parallel},\mathcal{L}_{g,\gamma}u\rangle\Big{|}\leq C\lVert
u\rVert_{H^{s}}^{2}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})\big{\\}}.$ (83)

To finish the estimates of the Type $\mathrm{I}_{\mathrm{Ein}}$ terms, we may
estimate those containing $\mathcal{F}^{(2)}$ and $\mathcal{F}^{(1)}$ as

$\displaystyle\Big{|}\langle\mathcal{F}^{(2)},\widehat{k}\rangle\Big{|}$
$\displaystyle\leq
C\lVert\widehat{k}\rVert_{H^{s-1}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}\big{\\}}$
$\displaystyle\quad+C\big{\\{}e^{-2T}\lVert\widehat{k}\rVert_{H^{s-1}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}\lVert\widehat{k}\rVert_{H^{s-1}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}},$ (84)
$\displaystyle\Big{|}\langle\mathcal{F}^{(1)},\mathcal{L}_{g,\gamma}u\rangle\Big{|}$
$\displaystyle\leq C\lVert
u\rVert_{H^{s}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}\big{\\}}$ $\displaystyle\quad+C\big{\\{}e^{-2T}\lVert
u\rVert_{H^{s}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}\lVert
u\rVert_{H^{s}}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}.$ (85)

We note that each term in $\mathcal{F}^{(1)}$ has regularity at least at the
level of $u$, and so we may integrate by parts to move excess derivatives from
$\mathcal{L}_{g,\gamma}u$ onto $\mathcal{F}^{(1)}$ without any issue.

It now remains to estimate the Type $\mathrm{II}_{\mathrm{Ein}}$,
$\mathrm{III}_{\mathrm{Ein}}$, and $\mathrm{IV}_{\mathrm{Ein}}$ terms. We may
do so via a similar calculation to that done Section 5.1, as well as in [36].
We thus omit this computation, giving the result as

$\displaystyle\mathrm{II}_{\mathrm{Ein}}+\mathrm{III}_{\mathrm{Ein}}+\mathrm{IV}_{\mathrm{Ein}}$
$\displaystyle=\frac{1}{8}\int_{\Sigma}\big{(}4\partial_{T}\gamma^{im}\gamma^{jn}+\gamma^{im}\gamma^{jn}\operatorname{tr}_{g}(\partial_{T}g)\big{)}\big{(}4\widehat{k}_{ij}\widehat{k}_{mn}+u_{ij}\mathcal{L}_{g,\gamma}u_{mn}\big{)}\>\mu_{g}$
$\displaystyle\quad+\frac{1}{4}\int_{\Sigma}\gamma^{im}\gamma^{jn}\partial_{T}g^{k\ell}\nabla[\gamma]_{k}u_{ij}\nabla[\gamma]_{\ell}u_{mn}\>\mu_{g}$
$\displaystyle\quad+\frac{1}{2}\int_{\Sigma}\gamma^{im}\gamma^{jn}g^{k\ell}\big{[}\partial_{T},\nabla[\gamma]_{k}\big{]}u_{ij}\nabla[\gamma]_{\ell}u_{mn}\>\mu_{g}$
$\displaystyle\quad-\frac{1}{2}\int_{\Sigma}\gamma^{im}\gamma^{jn}g^{kp}g^{\ell
q}u_{ij}R[\gamma]_{mkn\ell}u_{pq}\>\mu_{g}.$

We may then obtain an estimate for the remaining terms of

$\displaystyle\big{|}\mathrm{II}_{\mathrm{Ein}}+\mathrm{III}_{\mathrm{Ein}}+\mathrm{IV}_{\mathrm{Ein}}\big{|}$
$\displaystyle\leq C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}\big{\\}}\lVert\partial_{T}g\rVert_{H^{s-1}}$
$\displaystyle\leq C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}\big{\\}}$
$\displaystyle\quad\times\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\;\,+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}.$ (86)

Combining all the estimates and keeping only the leading order terms as we are
working in the small data regime, we find the time evolution of
$E^{(1)}_{\mathrm{Ein}}$ is given by

$\partial_{T}E^{(1)}_{\mathrm{Ein}}=-2(n-1)\langle\widehat{k},\widehat{k}\rangle+\mathcal{R}_{\mathrm{Ein}},$
(87)

where $\mathcal{R}_{\mathrm{Ein}}$ satisfies the bound

$\displaystyle|\mathcal{R}_{\mathrm{Ein}}|$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}\big{\\}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}\big{\\}}$
$\displaystyle\qquad+C(\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}})\big{\\{}e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}})^{2}\big{\\}}$ $\displaystyle\leq
C\big{\\{}(E_{\mathrm{Ein}})^{\frac{3}{2}}+e^{-2T}(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+e^{-4T}(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}^{2}\big{\\}}.$
(88)

We must finally estimate the time derivative of the correction to the
gravitational energy, $\partial_{T}\Gamma^{(1)}_{\mathrm{Ein}}$. By an
explicit calculation using the embeddings $H^{s}\hookrightarrow L^{\infty}$
for $s>\frac{n}{2}+1$ and $H^{s}\hookrightarrow L^{2}$ for any $s$, as well as
standard inequalities, we find $\partial_{T}\Gamma^{(1)}_{\mathrm{Ein}}$ may
be written in the form

$\partial_{T}\Gamma^{(1)}_{\mathrm{Ein}}=-\frac{(n-1)}{n}\langle\widehat{k},u\rangle-\frac{1}{2}\langle\mathcal{L}_{g,\gamma}u,u\rangle+2\langle\widehat{k},\widehat{k}\rangle+\mathcal{S}_{\mathrm{Ein}},$
(89)

where $\mathcal{S}_{\mathrm{Ein}}$ satisfies the estimate

$\displaystyle|\mathcal{S}_{\mathrm{Ein}}|$ $\displaystyle\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}+\lVert
u\rVert_{H^{s}}\big{\\}}\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+e^{-2T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})+e^{-4T}(\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2})^{2}\big{\\}}$ $\displaystyle\leq
C\big{\\{}(E_{\mathrm{Ein}})^{\frac{3}{2}}+e^{-2T}(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+e^{-4T}(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}^{2}\big{\\}}.$

Combining (87) and (89), we then have that the contribution of the
gravitational sector to the time evolution of the first-order energy will be
given as

$\partial_{T}(E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}})=-2\big{(}(n-1)-c_{E}\big{)}\langle\widehat{k},\widehat{k}\rangle-\frac{c_{E}}{2}\langle\mathcal{L}_{g,\gamma}u,u\rangle-\frac{(n-1)}{n}c_{E}\langle\widehat{k},u\rangle+\mathcal{R}_{\mathrm{Ein}}.$

However, as with the Yang-Mills energy, we want to bound, to leading order,
the time evolution of the gravitational sector energy by a term of the form
$-\alpha_{E}(E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}})$. We
note that [2] shows such a result, though the energies we have defined differ
by a constant multiple factor of $\frac{1}{2n^{2}}$. A calculation analogous
to that in Section 5.1.1 shows this does not change the result of Andersson
and Moncrief, and so we obtain the following lemma.

######  Lemma 5.3.

Let $s>\frac{n}{2}+1$ and take $B_{\delta}(0)$ to be a ball of sufficiently
small radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$ that satisfy
the Einstein-Yang-Mills system. Then the time evolution of
$E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}}$ is given by

$\partial_{T}(E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}})=-2\big{(}(n-1)-c_{E}\big{)}\langle\widehat{k},\widehat{k}\rangle-\frac{c_{E}}{2}\langle\mathcal{L}_{g,\gamma}u,u\rangle-\frac{(n-1)}{n}c_{E}\langle\widehat{k},u\rangle+\mathcal{R}_{\mathrm{Ein}},$
(90)

where $\mathcal{R}_{\mathrm{Ein}}$ satisfies the third-order estimate

$|\mathcal{R}_{\mathrm{Ein}}|\leq
C\big{\\{}(E_{\mathrm{Ein}})^{\frac{3}{2}}+e^{-2T}(E_{\mathrm{Ein}})^{\frac{1}{2}}E_{\mathrm{YM}}+e^{-4T}(E_{\mathrm{Ein}})^{\frac{1}{2}}(E_{\mathrm{YM}})^{2}\big{\\}}.$
(91)

In particular, the gravitational energy will be bounded by

$\partial_{T}(E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}})\leq-\alpha_{E}(E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}})+\mathcal{R}_{\mathrm{Ein}},$
(92)

for $\alpha_{E}$ defined as

$\alpha_{E}=(n-1)\bigg{(}1-\sqrt[\ ]{1-\frac{2c_{E}}{n-1}}\bigg{)}.$ (93)

####  5.2.2 Positive-Definiteness of Energy

As done for the Yang-Mills sector, we will need to show we have a bound of the
form $\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert u\rVert_{H^{s}}^{2}\leq
C\sum_{i=s}^{s}(E^{(i)}_{\mathrm{Ein}}+c_{E}\Gamma^{(i)}_{\mathrm{Ein}})$ so
that a decay in the total energy will imply that the perturbations vanish. An
estimate of this form, however, has been shown in Section 7.1 of [2]. While
the energies we have defined differ by a positive constant factor of
$\frac{1}{2n^{2}}$, we note that this will not affect the proof presented by
Andersson and Moncrief. We thus restate here the result of Theorem 7.4 from
their work, which will still hold, in the following form.

######  Lemma 5.4.

[2] Suppose that $\gamma$ satisfies the shadow metric condition for $g$, and
take $s>\frac{n}{2}+1$. Then there is a $\delta>0$ and a constant $C>0$, such
that for $(\widehat{k},u,\mathcal{E},A)\in B_{\delta}(0)$, the inequality

$\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert u\rVert_{H^{s}}^{2}\leq
C\sum_{i=s}^{s}(E^{(i)}_{\mathrm{Ein}}+c_{E}\Gamma^{(i)}_{\mathrm{Ein}}).$

is satisfied.

###  5.3 Total Energy

Recall in (50) that we have defined the total energy as the quantity

$E_{s}=\sum_{i=1}^{s}E^{(i)}=\sum_{i=s}^{s}E^{(i)}_{\mathrm{Ein}}+c_{E}\Gamma^{(i)}_{\mathrm{Ein}}+E^{(i)}_{\mathrm{YM}}-c_{Y}\Gamma^{(i)}_{\mathrm{YM}}.$

Now, as a direct result of Lemmas 5.2 and 5.4, we obtain the following theorem
related to the positive-definiteness of the total energy.

######  Theorem 5.5.

Fix $s>\frac{n}{2}+1$. Suppose that $\gamma$ satisfies the shadow metric
condition for $g$, and take $\widehat{A}=0$. Then there exists a $\delta>0$
and a constant $C>0$ such that, for $(\widehat{k},u,\mathcal{E},A)\in
B_{\delta}(0)$, the inequality

$\lVert\widehat{k}\rVert_{L^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\leq CE_{s}$

holds.

In light of Theorem 5.5, if we can show, with sufficiently small initial data
at $T_{0}$, that $E_{s}$ decays to zero as $T\rightarrow\infty$, then we will
have shown that the perturbations also decay and hence the Einstein-Yang-Mills
system converges to a background solution of the form (27). Furthermore, we
will have that the maximal existence interval of the Cauchy problem for the
Einstein-Yang-Mills system in the chosen gauges is then $[T_{0},\infty)$,
i.e., solutions of the small data Cauchy problem for the Einstein-Yang-Mills
system will exist globally.

To show the decay of $E_{s}$, we first see that, by combining the results of
Lemmas 5.1 and 5.3, we have that $\partial_{T}E^{(1)}$ is bounded from above
by

$\displaystyle\partial_{T}E^{(1)}$
$\displaystyle\leq-\alpha_{E}\big{(}E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}}\big{)}-\alpha_{Y}\big{(}E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}\big{)}+\mathcal{R},$
(94)

where $\mathcal{R}$ satisfies the leading-order estimate

$\displaystyle|\mathcal{R}|$
$\displaystyle\leq|\mathcal{R}_{\mathrm{YM}}|+|\mathcal{R}_{\mathrm{Ein}}|$
$\displaystyle\leq
C\big{\\{}\big{(}(E_{\mathrm{Ein}})^{\frac{1}{2}}+(E_{\mathrm{YM}})^{\frac{1}{2}}\big{)}(E_{\mathrm{Ein}}+E_{\mathrm{YM}})+(E_{\mathrm{Ein}})^{\frac{1}{2}}\big{(}e^{-2T}E_{\mathrm{YM}}+e^{-4T}(E_{\mathrm{YM}})^{2}\big{)}\big{\\}}$
$\displaystyle\leq
C\big{\\{}(E_{\mathrm{Ein}}+E_{\mathrm{YM}})^{\frac{3}{2}}+(E_{\mathrm{Ein}}+E_{\mathrm{YM}})^{\frac{1}{2}}\big{(}e^{-2T}(E_{\mathrm{Ein}}+E_{\mathrm{YM}})+e^{-4T}(E_{\mathrm{Ein}}+E_{\mathrm{YM}})^{2}\big{)}\big{\\}}.$

From Theorem 5.5 and the equivalence of $E_{\mathrm{Ein}}$ and
$E_{\mathrm{YM}}$ with the Sobolev norms of the small data we have that

$E_{\mathrm{Ein}}+E_{\mathrm{YM}}\leq
C\big{\\{}\lVert\widehat{k}\rVert_{H^{s-1}}^{2}+\lVert
u\rVert_{H^{s}}^{2}+\lVert\mathcal{E}\rVert_{H^{s-1}}^{2}+\lVert
A\rVert_{H^{s}}^{2}\big{\\}}\leq CE_{s}.$

This means that $\mathcal{R}$ will satisfy the bound

$|\mathcal{R}|\leq
C\big{\\{}(E_{s})^{\frac{3}{2}}+e^{-2T}(E_{s})^{\frac{3}{2}}+e^{-4T}(E_{s})^{\frac{5}{2}}\big{\\}}.$

As we have finally obtained a bound for the higher-order terms on our total
energy, we may make the estimate $e^{-T}\leq e^{-T_{0}}$ for $T>T_{0}$ without
worry that dangerous terms of the form $e^{T}$ will appear. Using this, we
then find the leading-order estimate of $\mathcal{R}$ to simply be

$|\mathcal{R}|\leq C(E_{s})^{\frac{3}{2}}.$ (95)

Now, define $\Lambda=\frac{1}{2}\min\\{\alpha_{E},\alpha_{Y}\big{\\}}$.
Because both $E^{(1)}_{\mathrm{Ein}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}}$ and
$E^{(1)}_{\mathrm{YM}}-c_{Y}\Gamma^{(1)}_{\mathrm{YM}}$ are positive-definite,
we then have that the time evolution of the first-order total energy is
bounded by

$\displaystyle\partial_{T}E^{(1)}$
$\displaystyle\leq-2\Lambda\big{(}E^{(1)}_{\mathrm{Ein}}+E^{(1)}_{\mathrm{YM}}+c_{E}\Gamma^{(1)}_{\mathrm{Ein}}-\Gamma^{(1)}_{\mathrm{YM}}\big{)}+\mathcal{R}=-\Lambda
E^{(1)}+\mathcal{R}.$ (96)

Furthermore, because $E^{(i)}(\widehat{k},u,\mathcal{E},A)\approx
E^{(1)}\big{(}\nabla[\gamma]^{i}\widehat{k},\nabla[\gamma]^{i}u,\nabla[\gamma]^{i}\mathcal{E},\nabla[\gamma]^{i}A\big{)}$,
we obtain an analogous inequality for each $E^{(i)}$. We remark that in
obtaining these higher-order inequalities, we will need to commute
differential operators in order to apply the same arguments as for the first-
order energies. However by applying Proposition 4.1.2, we will be able to
bound these commutator terms by the appropriate Sobolev norms and avoid any
issues with regularity; c.f. Section 2.1 of [5] or Section 5 of [1] for
similar, detailed calculations.

These higher-order energy estimates, along with the estimate of $\mathcal{R}$
in (95), leads us to the follow theorem.

######  Theorem 5.6.

Fix $\widehat{A}=0$ and $s>\frac{n}{2}+1$. Let $B_{\delta}(0)$ be a ball of
sufficiently small radius $\delta$ containing $(\widehat{k},u,\mathcal{E},A)$
that satisfy the Einstein-Yang-Mills system. Fix $T_{0}>-\infty$. Then there
exists a constant $C>0$ such that for $T\geq T_{0}$, the time evolution of
$E_{s}$ satisfies the differential inequality

$\partial_{T}E_{s}\leq-2\Lambda E_{s}+2C(E_{s})^{\frac{3}{2}}.$ (97)

We may now integrate up this inequality. In particular, let
$Y=(E_{s})^{\frac{1}{2}}$. We then have the differential inequality

$\partial_{T}Y\leq-\Lambda Y+CY^{2}.$

Integrating up and substituting back $E_{s}$, we find that

$E_{s}(T)\leq\Lambda^{2}\Bigg{[}C+\bigg{(}\frac{\Lambda}{\sqrt[\
]{E_{s}(T_{0})}}-C\bigg{)}e^{\Lambda(T-T_{0})}\Bigg{]}^{-2}.$ (98)

When $T>T_{0}$ and $E_{s}(T_{0})<\frac{\Lambda^{2}}{C^{2}}$, we will have that
$E_{s}(T)$ is bounded from above by an exponentially decaying function and
from below by $0$. Hence $E_{s}$ must tend to zero as $T$ tends to infinity.
We must simply require that the initial energy $E_{s}(T_{0})$, determined by
the initial values of $(\widehat{k},u,\mathcal{E},A)$, is sufficiently small.
However, with sufficiently small initial data, we have that the $E_{s}$ will
be bounded for all $T\in[T_{0},\infty)$ and thus, in light of Theorem 5.5, the
data $(\widehat{k},u,\mathcal{E},A)$ will decay to a stable solution of the
vacuum Einstein’s equations. This leads us to the main theorem for this paper.

######  Theorem 5.7.

Suppose that $\gamma_{0}$ has a smooth deformation space $\mathcal{N}$ and
that the operator $\mathcal{L}_{\gamma_{0},\gamma_{0}}$ has non-negative
spectrum. Fix $s>\frac{n}{2}+1$ for $n>3$. There is then a $\delta>0$ such
that if $(\widehat{k}_{0},g_{0}-\gamma_{0},\mathcal{E}_{0},A_{0})\in
B_{\delta}(0)\subset H^{s-1}\times H^{s}\times H^{s-1}\times H^{s}$ with
$\gamma_{0}$ a shadow metric of $g_{0}$, then the Cauchy problem for the
Einstein-Yang-Mills system with initial data
$(\widehat{k}_{0},g_{0},\gamma_{0},\mathcal{E}_{0},A_{0})$ is globally well-
posed to the future.

In addition, if $T\mapsto(\widehat{k},g,\gamma,\mathcal{E},A)$ is the maximal
development to the Cauchy problem for the Einstein-Yang-Mills system, then
there is a $\gamma_{*}\in\mathcal{N}$ such that