# Local well-posedness of the Einstein-Yang-Mills system in CMCSHGC gauge

Puskar Mondal  Puskar Mondal  Center of Methematical Sciences and
Applications, Department of Mathematics, Harvard University
<EMAIL_ADDRESS>
###### Abstract

We study the local well-posedness of the Einstein-Yang-Mills equations in
constant mean extrinsic curvature spatial harmonic generalized Coulomb gauge
(CMCSHGC). In this choice of gauge, the complete Einstein-Yang-Mills equations
reduce to a coupled elliptic-hyperbolic system. Utilizing the method developed
by Andersson and Moncrief [1], we establish the existence of a unique, local,
continuous-in-time solution of this coupled system. This yields an ‘in time’
continuation criteria of the solutions which is to be used in the potential
future proof of an improved continuation criteria for this coupled system
utilizing Moncrief’s light cone estimate technique.

##  1 Introduction

Construction of solutions to classical field equations such as the Einstein
and Yang-Mills equations is of extreme importance in mathematical physics.
This is primarily because the solutions of these equations (or their quantum
analogs) describe the three most important fundamental forces present in
nature- the gravity, strong and weak forces. If we ignore the quantum aspects
for the moment, these equations exhibit interesting dynamics at the pure
classical level due to their rich non-linear structure. For pure gravity, this
non-linear structure gives rise to the formation of black holes through the
concentration of gravitational energy (spacetime curvature). If coupling
between the Einstein and Yang-Mills fields is considered, then interesting
phenomena occur even in very special geometries solely because of the non-
linearities. Striking examples of such phenomena are the existence of soliton-
like solutions [2, 3] and non-trivial dynamics [4] even in spherically
symmetric spacetime a very special geometry. Therefore, from a physical
perspective, it is extremely important to study the dynamics of these coupled
equations. In particular, one is interested in the long-time behavior of these
equations and whether the solutions (lying in suitable function spaces)
persist for all time and preserve the regularity of the initial data. In other
words, regular Cauchy data should extend uniquely and continuously to globally
defined, singularity-free solutions of the associated field equations on the
entire spacetime. If a breakdown of the solutions were to occur that would be
pathological in the sense that the underlying theory then turns out to be non-
physical. In the context of gravity (or gravity coupled to a source), the
question of predictability (or breakdown of such) is closely tied to the
Cosmic Censorship Conjecture [5] of Penrose, which to this day remains open.
In the case of gravitational dynamics, it is slightly more subtle due to the
fact that singularities do occur due to the collapse of matter and such
singularities are in general hidden behind a horizon and therefore cannot be
accessed by a time-like observer located in the domain of outer
communications. Therefore, the occurrence of a naked singularity is considered
pathological (or the breakdown of the theory) in the context of gravity.
However, before studying the long-time behavior of the solutions of these
equations, it is necessary to establish a short-time well-posedness result. In
other words, one should be able to prove that local in time solutions to these
field equations exist and such solutions are unique and depend continuously on
the initial data. In doing so one obtains continuation criteria of the
solutions and therefore the global existence (or finite time breakdown)
depends on satisfying (or not satisfying) this continuation criterion.

The Einstein and Yang-Mills equations (or its abelian version the Maxwell
equations) however contain constraints in addition to evolution equations.
Therefore, it is often necessary to rewrite the system, either by extracting a
hyperbolic system or performing suitable gauge fixing. The gauge fixing may
yield a hyperbolic system such as spacetime harmonic gauge (also called wave
gauge or de Donder gauge; the spacetime coordinates $\\{x^{\mu}\\}$ are forced
to satisfy the wave equation $\Box_{\hat{g}}x^{\mu}=0$, $\hat{g}$ being the
Lorentzian metric of the spacetime under consideration) which turns the
Einstein evolution equations into hyperbolic equations. In particular, Yvonne
Choquet Bruhat’s [6] celebrated result on a local well-posedness theorem for
the vacuum Einstein equations was obtained in this gauge. The analog of
spacetime harmonic gauge in the context of Yang-Mills theory is the Lorentz
gauge, which once again puts the Yang-Mills evolution equations into
hyperbolic form [7, 8]. 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 ones is the constant mean
curvature spatial harmonic gauge (CMCSH gauge) introduced by Andersson and
Moncrief [1] 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 [1]. 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 [9] 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 [10] which used the spacetime harmonic gauge
to obtain a global stability result of Minkowski space. Lefloch and Ma [11]
used the same gauge to establish a small data global existence result of the
Einstein-coupled-to-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 [12, 13]. In addition to these examples,
there are numerous other studies (e.g., [14, 15, 16]) that utilize this
particular choice of gauge. Andersson and Moncrief [18] proved an asymptotic
stability result of the Milne universe utilizing the CMCSH gauge. Later
Andersson and Fajman used it to prove a small data global existence result for
the Einstein-Vlassov system on a Milne spacetime [19]. In addition, [20]
obtained a rough data local well-posedness result for the vacuum Einstein
equations, [21] proved a local well-posedness result of Einstein-coupled-
with-Vlasov-matter, [22] studied the stability of the Milne universe in the
presence of a perfect fluid, [23] studied the asymptotic stability of the
Milne universe coupled to a Klein-Gordon field, etc. There are numerous other
studies where CMCSH gauge is used to study the vacuum gravity problem or
gravity coupled to matter fields (e.g., [24, 25, 26]). A substantial amount of
study has also been done in the Yang-Mills sector. Eardley and Moncrief [27,
28] established the global existence of Yang-Mills fields on a Minkowski
background in temporal gauge. Later, [29] extended the global existence result
to globally hyperbolic curved spacetimes in the same gauge. Using Coulomb
gauge, Klainerman and Machedon [9], as mentioned earlier, established a global
existence result in energy norm. Later Tao [30] 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.

This paper was motivated in part by the desire to adapt the Andersson and
Moncrief [1] argument to including a Yang-Mills source in Einstein’s
equations. This introduces an extra gauge choice for the Yang-Mills sector
namely the generalized Coulomb gauge. In addition, we also want to study three
particular problems in the near future: the fully non-linear stability of the
Milne universe for Yang-Mills fields, the global existence problem for the
Yang-Mills equations on a curved spacetime using Moncrief’s light cone
estimate technique (in prep.), and obtaining an improved breakdown criteria
for the coupled Einstein-Yang-mills equations (similar to the ones provided by
[31, 32]). However, before proceeding to tackle such problems, it is mandatory
to establish a local existence theorem for the fully coupled Einstein-Yang-
Mills equations or the Yang-Mills equations on a curved background. However,
If we solve the more general problem of fully coupled Einstein-Yang-Mills
equations, the latter follows automatically. The local existence theorem
automatically provides a continuation criterion that is crucial to
establishing a global existence result. As an example, [27, 28] first proved a
local existence theorem for the Yang-Mills equations and obtained a
continuation criterion as the non-blow-up of a certain Sobolev norm.
Thereafter they proved, using a light cone estimate technique on the Minkowski
background, that the same Sobolev norm that appears in the continuation
criterion of the local existence theorem can not blow up in finite time
thereby yielding global existence. The current article aims to complete the
first part of the argument in the case of the coupled Einstein-Yang-Mills
equations. There are a few minor difficulties one needs to account for in the
fully coupled system in our choice of gauge. One of the minor problems is the
Gribov degeneracies associated with the spatial harmonic gauge of gravity and
generalized Coulomb gauge of Yang-Mills theory [33, 35, 34, 36, 37]. This
roughly translates to the gauge orbits ceasing to be transversal to the chosen
gauge slice. In addition to this problem, we also need to deal with subtle
issues associated with the regularity and closing of the energy estimates in
the fully coupled system. We will discuss this in relevant chapters. The
remaining is a routine procedure to obtain a local well-posedness theorem.

We start with deriving the gauge fixed coupled Einstein-Yang-Mills equations.
The gauge fixing reduces the system to a coupled elliptic-hyperbolic system.
The gauge variables (lapse function, shift vector field, and the shifted
temporal component of the Yang-Mills connection) are determined as non-local
functionals of the dynamical variables by solving the relevant elliptic
equations. We then obtain energy inequalities which together with a
contraction mapping argument yields the existence of a unique solution. The
in-time continuity and Cauchy stability follow in a standard way. The
resulting theorem provides continuation criteria of the local solution.

##  2 Notations

The ‘$n+1$’ dimensional $C^{\infty}$ spacetime manifold is denoted by
$\hat{M}$. We are interested in spacetime manifolds $\hat{M}$ with the product
topology $\mathbb{R}\times\Sigma$, where $\Sigma$ denotes the $n$-dimensional
closed spatial manifold diffeomorphic to a Cauchy hypersurface (i.e., every
inextendible causal curve intersects $\Sigma$ exactly once). We denote the
space of Riemannian metrics on $\Sigma$ by $\mathcal{M}_{\Sigma}$. We
explicitly work in the $L^{2}$ (with respect to a background metric) Sobolev
space $H^{s}$ for $s>\frac{n}{2}+1$. We will consider tensors as sections of a
rank-$k$ ($k\geq 0$) vector bundle $\mathcal{B}^{k}$ of tensors over
$(\Sigma,g)$, where $g\in\mathcal{M}_{\Sigma}$ is the dynamical metric that
solves the Einstein’s equations written in $n+1$ form. The $L^{2}$ inner
product on a fibre of such a rank-$k$ bundle is defined as

$\displaystyle\langle
u|v\rangle_{L^{2}}:=\int_{\Sigma}u_{i_{1}i_{2}i_{3}....i_{k}}v_{j_{1}j_{2}j_{3}....j_{k}}\gamma^{i_{1}j_{1}}\gamma^{i_{1}j_{2}}\gamma^{i_{3}j_{3}}....\gamma^{i_{k}j_{k}}\mu_{g},$
(1)

where $\gamma\in\mathcal{M}_{\Sigma}$ is a $C^{\infty}$ background metric
lying in a small enough neighbourhood of $g$ in a suitable space (this is
possible since $C^{\infty}$ is dense in Sobolev spaces for compact manifolds)
and $\mu_{g}=\sqrt{\det(g_{ij})}dx^{1}\wedge dx^{2}\wedge
dx^{3}\wedge.........\wedge dx^{n}$ is the volume form associated with the
metric $g$. The standard norms are defined naturally as follows

$\displaystyle||u||_{L^{2}}:=\langle u|u\rangle^{1/2}_{L^{2}}$ (2)
$\displaystyle||u||_{L^{\infty}}:=\sup_{\Sigma}(u_{i_{1}i_{2}i_{3}....i_{k}}u_{j_{1}j_{2}j_{3}....j_{k}}\gamma^{i_{1}j_{1}}\gamma^{i_{1}j_{2}}\gamma^{i_{3}j_{3}}....\gamma^{i_{k}j_{k}})^{1/2}$
(3)

and so on. We adapt the following definition of a twisted rough Laplacian
acting on sections of $\mathcal{B}^{k}$

$\displaystyle\Delta^{\gamma}_{g}u:=-\frac{1}{\mu_{g}}\nabla[\gamma]_{i}(g^{ij}\mu_{g}\nabla[\gamma]_{j}u).$
(4)

This Laplacian is self-adjoint with respect to the following inner product on
derivatives

$\displaystyle\langle\nabla[\gamma]u|\nabla[\gamma]v\rangle_{L^{2}}:=\int_{\Sigma}g^{ij}\nabla[\gamma]_{i}u_{i_{1}i_{2}....i_{k}}\nabla[\gamma]_{j}v_{j_{1}j_{2}....j_{k}}\gamma^{i_{1}j_{1}}\gamma^{i_{1}j_{2}}....\gamma^{i_{k}j_{k}}\mu_{g}.$
(5)

The covariant derivative with respect to the dynamical metric $g$ will be
denoted simply by $\nabla$. We define the ordinary Laplacian of $g$ in the
following way such that it has non-negative spectrum i.e.,

$\displaystyle\Delta_{g}\equiv-g^{ij}\nabla_{i}\nabla_{j}.$ (6)

To formulate the Yang-Mills theory over the spacetime
$\mathbb{R}\times\Sigma$, we first choose a compact semi-simple Lie group $G$.
If a section of the principle $G-$bundle defined over $\mathbb{R}\times\Sigma$
is chosen and the connection is pulled back to the base manifold, then it
yields a $1-$form field on the base which takes values in the Lie algebra
$\mathfrak{g}$ of $G$. Let us consider the dimension of the group $G$ to be
$dim_{G}$ and since $\mathfrak{g}:=T_{e}G$, it has a natural vector space
structure. Assume that the vector space $\mathfrak{g}$ has a basis
$\\{\chi_{A}\\}_{A=1}^{dim_{G}}$ given by a set of $k\times k$ real valued
matrices ($k$ being the dimension of the representation $V$ of the Lie algebra
$\mathfrak{g}$). The connection $1-$form field is then defined to be

$\displaystyle\hat{A}:=\hat{A}^{A}_{\mu}\chi_{A}dx^{\mu}=\hat{A}^{A}_{\mu}(\chi_{A})^{a}_{b}dx^{\mu}=\hat{A}^{a}~{}_{b\mu}dx^{\mu},~{}a,b=1,2,3,...,k.$
(7)

From now on by the connection 1-form field $\hat{A}_{\mu}$, we will always
mean $\hat{A}^{a}~{}_{b\mu}$. In the current setting
$\hat{A}\in\Omega^{1}(\mathbb{R}\times\Sigma;End(V))$, where $End(V)$ denotes
the space of endomorphisms of the vector space $V$. The curvature of this
connection is defined to be the Yang-Mills field
$F\in\Omega^{2}(\mathbb{R}\times\Sigma;End(V))$

$\displaystyle\hat{F}^{a}~{}_{b\mu\nu}:=\partial_{\mu}\hat{A}^{a}~{}_{b\nu}-\partial_{\nu}\hat{A}^{a}~{}_{b\mu}+[\hat{A},\hat{A}]^{a}~{}_{b\mu\nu},$
(8)

where the bracket is defined on the Lie algebra $\mathfrak{g}$ and given by
commutator of matrices under multiplication. The Yang-Mills coupling constant
is set to unity. Since $G$ is compact, it admits a positive definite adjoint
invariant metric on $\mathfrak{g}$. We choose a basis of $\mathfrak{g}$ such
that this adjoint invariant metric takes the Cartesian form $\delta_{AB}$ and
work with representations for which the bases satisfy

$\displaystyle-\tr(\chi_{A}\chi_{B})=(\chi_{A})^{a}_{b}(\chi_{A})^{b}_{a}=\delta_{AB}.$
(9)

This metric allows us to define a gauge invariant (since it is adjoint
invariant) inner product on $\mathfrak{g}$. An $L^{2}$ inner product of $A\in
sections(\mathcal{B}^{k};End(V))$ and $B\in sections(\mathcal{B}^{k};End(V))$
is defined as follows

$\displaystyle\langle
A|B\rangle_{L^{2}}:=\int_{\Sigma}A^{a}~{}_{bi_{1}i_{2}i_{3}....i_{k}}B^{b}~{}_{aj_{1}j_{2}j_{3}....j_{k}}\gamma^{i_{1}j_{1}}\gamma^{i_{1}j_{2}}\gamma^{i_{3}j_{3}}....\gamma^{i_{k}j_{k}}\mu_{g},$
(10)

where the contraction on the gauge indices (or the internal indices) is
performed by means of the adjoint invariant metric on $\mathfrak{g}$. Notice
that this inner product is not gauge invariant unless $A$ and $B$ both
transform as tensors under gauge transformations.

Under a gauge transformation $\mathcal{G}$, the $\mathfrak{g}$ valued 1-form
field $\hat{A}$ transforms as

$\displaystyle\hat{A}_{\mu}\mapsto\mathcal{G}\hat{A}_{\mu}\mathcal{G}+\mathcal{G}\partial_{\mu}\mathcal{G}^{-1}$
(11)

and therefore $\hat{A}_{\mu}$ is not a tensor in the sense that it is not a
$(1,1)$ section of the associated $V-$bundle over $\mathbb{R}\times\Sigma$.
For any $\mathfrak{g}$ valued section
$\mathcal{K}^{a}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}$ of a vector bundle
over $\mathbb{R}\times\Sigma$ that transform as a tensor under the gauge
transformation, the gauge covariant derivative is defined to be

$\displaystyle\mathcal{D}_{\alpha}\mathcal{K}^{a}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}$
(12)
$\displaystyle:=\nabla_{\alpha}\mathcal{K}^{a}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}+A^{a}~{}_{c\alpha}\mathcal{K}^{c}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}-A^{c}~{}_{b\alpha}\mathcal{K}^{a}~{}_{c\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}$
$\displaystyle=\nabla_{\alpha}\mathcal{K}^{a}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}}+[A,\mathcal{K}]^{a}~{}_{b\mu_{1}\mu_{2}\mu_{3}....\mu_{k}},$

where $\nabla_{\alpha}$ is the ordinary spacetime covariant derivative with
respect to a Lorentzian metric on $\mathbb{R}\times\Sigma$. Naturally this
notion of gauge covariant derivative passes down to the spatial manifold
$\Sigma$.
Lastly, for functions $f(t)\in\mathbb{R}_{\geq 0}$ and
$g(t)\in\mathbb{R}_{\geq 0}$ $f(t)\lesssim g(t)$ means $f(t)\leq Cg(t)$ and
$f(t)\gtrsim g(t)$ means $f(t)\geq Cg(t)$, and $f(t)\approx g(t)$ implies
$C_{1}g(t)\leq f(t)\leq C_{2}g(t)$ for $0<C_{1},C_{2},C<\infty$. The involved
constants may depend only on the background geometry. The spaces of symmetric
covariant 2-tensors and vector fields on $\Sigma$ are denoted by
$S^{0}_{2}(\Sigma)$ and $\mathfrak{X}(\Sigma)$, respectively.

##  3 Gauge Fixed Einstein-Yang Mills field equations

We utilize the ADM formalism that splits the spacetime described by an ‘$n+1$’
dimensional $C^{\infty}$ Lorentzian manifold $\hat{M}$ into
$\mathbb{R}\times\Sigma$. As mentioned previously, each level set
$\\{t\\}\times\Sigma$ of the time function $t$ is an orientable $n-$manifold
diffeomorphic to a Cauchy hypersurface (assuming the spacetime to be globally
hyperbolic) and equipped with a Riemannian metric. Such a split may be
executed by introducing a lapse function $N$ and shift vector field $X$
tangent to $\Sigma$ and defined such that

$\displaystyle\partial_{t}$ $\displaystyle=$ $\displaystyle N\mathbf{n}+X,$
(13)

where $t$ and $\mathbf{n}$ are time and a hypersurface orthogonal future
directed timelike unit vector i.e., $\hat{g}(\mathbf{n},\mathbf{n})=-1$,
respectively. By means of the natural projection operator
$\Pi:=\hat{g}+\mathbf{n}\otimes\mathbf{n}$, one may decompose every tensor
field into their $\Sigma-$parallel and $\Sigma-$orthogonal components. This
splitting writes the spacetime metric $\hat{g}$ in local coordinates
$\\{x^{\alpha}\\}_{\alpha=0}^{n}=\\{t,x^{1},x^{2},....,x^{n}\\}$ as

$\displaystyle\hat{g}$ $\displaystyle=$ $\displaystyle-N^{2}dt\otimes
dt+g_{ij}(dx^{i}+X^{i}dt)\otimes(dx^{j}+X^{j}dt)$ (14)

and the stress-energy tensor as

$\displaystyle\mathbf{T}$ $\displaystyle=$ $\displaystyle
E\mathbf{n}\otimes\mathbf{n}+2\mathbf{J}\odot\mathbf{n}+\mathbf{S},$ (15)

where $\mathbf{J}\in\mathfrak{X}(\Sigma)$, $\mathbf{S}\in S^{2}_{0}(\Sigma)$,
and $A\odot B=\frac{1}{2}(A\otimes B+B\otimes A$). Here,
$\mathfrak{X}(\Sigma)$ and $S^{2}_{0}(\Sigma)$ are the space of vector fields
and the space of symmetric covariant 2-tensors, respectively. Here
$E:=\mathbf{T}(\mathbf{n},\mathbf{n})$ is the energy density observed by a
time-like observer with $n+1$-velocity $\mathbf{n}$,
$\mathbf{J}_{i}=-\mathbf{T}(\partial_{i},\mathbf{n})$ is the momentum density,
$\mathbf{J}_{i}=-\mathbf{T}(\mathbf{n},\partial_{i})=$ is the energy flux
density, and $\mathbf{S}_{ij}=\mathbf{T}(\partial_{i},\partial_{j})$ is the
momentum flux density (with respect to the chosen constant $t$ hypersurface
$\Sigma$). Both the momentum density and the energy flux density have the same
value due to the symmetry of the stress energy tensor. The choice of a spatial
slice in the spacetime leads to consideration of the second fundamental form
$k_{ij}$ which describes the extrinsic geometry of the slice while embedded in
the ambient spacetime. The trace of the second fundamental form
($\tau:=\tr_{g}k$) is the mean extrinsic curvature of the slice, which will
play an important role in the analysis. Under such decomposition, the Einstein
equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}R\hat{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle T_{\mu\nu}$ (16)

take the form ($8\pi G=c=1$)

$\displaystyle\partial_{t}g_{ij}$ $\displaystyle=$
$\displaystyle-2Nk_{ij}+L_{X}g_{ij},$ (17) $\displaystyle\partial_{t}k_{ij}$
$\displaystyle=$
$\displaystyle-\nabla_{i}\nabla_{j}N+N\\{R_{ij}+\tr_{g}kk_{ij}-2k_{ik}k^{k}_{j}$
$\displaystyle-\frac{1}{n-1}(E-S)g_{ij}-\mathbf{S}_{ij}\\}+L_{X}k_{ij}$

along with the constraints (Gauss and Codazzi equations)

$\displaystyle R(g)-|k|^{2}+(\tr_{g}k)^{2}$ $\displaystyle=$ $\displaystyle
2E,$ (18) $\displaystyle\nabla_{j}k^{j}_{i}-\nabla_{i}\tr_{g}k$
$\displaystyle=$ $\displaystyle-\mathbf{J}_{i},$ (19)

where $S=g^{ij}\mathbf{S}_{ij}$. The vanishing of the covariant divergence of
the stress energy tensor i.e., $\nabla_{\nu}T^{\mu\nu}=0$ is equivalent to the
continuity equation and equations of motion of the source

$\displaystyle\frac{\partial E}{\partial t}$ $\displaystyle=$ $\displaystyle
L_{X}E+NE\tau-N\nabla_{i}\mathbf{J}^{i}-2\mathbf{J}^{i}\nabla_{i}N+N\mathbf{S}^{ij}k_{ij},$
(20) $\displaystyle\frac{\partial\mathbf{J}^{i}}{\partial t}$ $\displaystyle=$
$\displaystyle
L_{X}\mathbf{J}^{i}+N\tau\mathbf{J}^{i}-\nabla_{j}(N\mathbf{S}^{ij})+2Nk^{i}_{j}\mathbf{J}^{j}-E\nabla^{i}N.$
(21)

We want to study the Einstein-Yang-Mills system. One may for convenience
decompose the $n+1$ $\mathfrak{g}$-valued connection $1-$form field
$A^{a}~{}_{b\mu}$ into its component parallel and perpendicular to a constant
$t$ hypersurface $\Sigma$ as follows

$\displaystyle\hat{A}^{a}~{}_{b}=A^{a}~{}_{b}-\hat{g}(A^{a}~{}_{b},\mathbf{n})\mathbf{n}$
(22)

where $A^{a}~{}_{b}$ is a $\mathfrak{g}$ valued $1-$form field parallel to the
spatial manifold $\Sigma$ i.e., $A\in\Omega^{1}(\Sigma;End(V))$. Importantly
note that $\hat{A}^{a}~{}_{bi}=A^{a}~{}_{bi}$ but $\hat{A}^{a}~{}_{b}^{i}\neq
A^{a}~{}_{b}^{i}$ unless the shift vector field $X$ vanishes. Similarly, we
decompose the Yang-Mills field strength $\hat{F}$ as follows

$\displaystyle\hat{F}^{a}~{}_{b}=\mathcal{E}^{a}~{}_{b}\otimes\mathbf{n}-\mathbf{n}\otimes\mathcal{E}^{a}~{}_{b}+F^{a}~{}_{b},$
(23)

where $\mathcal{E}\in\Omega^{1}(\Sigma;End(V))$ is the electric field and
$F\in\Omega^{2}(\Sigma;End(V))$ is related to the magnetic field. Once again
$\hat{F}^{a}~{}_{bij}=F^{a}~{}_{bij}$ but $\hat{F}^{a}~{}_{b}~{}^{ij}\neq
F^{a}~{}_{b}~{}^{ij}$. The stress-energy tensor of an $n+1-$Yang-Mills field
$\hat{F}$ is given by

$\displaystyle
T_{\mu\nu}=F^{a}~{}_{b\mu\alpha}F^{b}~{}_{a\nu}~{}^{\alpha}-\frac{1}{4}(F^{a}~{}_{b\alpha\beta}F^{b}~{}_{a}~{}^{\alpha\beta})\hat{g}_{\mu\nu}.$
(24)

After projecting onto the spatial hypersurface $\Sigma$, several components of
the stress-energy tensor may be evaluated as follows

$\displaystyle
E=\frac{1}{2}g^{ij}\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}+\frac{1}{4}g^{ik}g^{jl}F^{a}~{}_{bij}F^{b}~{}_{akl},~{}\mathbf{J}_{i}=-g^{jk}F^{a}~{}_{bij}\mathcal{E}^{b}~{}_{ak},$
(25)
$\displaystyle\mathbf{S}_{ij}=-\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}+\left(\frac{1}{2}g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}-\frac{1}{4}g^{km}g^{ln}F^{a}~{}_{bkl}F^{b}~{}_{amn}\right)g_{ij}$
$\displaystyle+g^{kl}F^{a}~{}_{bik}F^{b}~{}_{ajl}$

Using this available information, the complete system of evolution and
constraint equations are expressible as follows

$\displaystyle\partial_{t}A^{a}~{}_{bi}=N\mathcal{E}^{a}~{}_{bi}-F^{a}~{}_{bij}X^{j}+\partial_{i}A^{a}~{}_{b0}-[A_{0},A_{i}]^{a}~{}_{b},$
(26)
$\displaystyle\partial_{t}\mathcal{E}^{a}~{}_{bi}=N\tr_{g}k\mathcal{E}^{a}~{}_{bi}-(\nabla_{j}N)F^{a}~{}_{bi}~{}^{j}-N\nabla_{j}F^{a}~{}_{bi}~{}^{j}+N[F_{i}~{}^{j},A_{j}]^{a}~{}_{b}$
(27)
$\displaystyle+[\mathcal{E}_{i},A_{0}-A_{k}X^{k}]^{a}~{}_{b}+X^{k}\nabla_{k}\mathcal{E}^{a}~{}_{bi}+\mathcal{E}^{a}~{}_{bk}\nabla_{i}X^{k}-2Nk^{j}_{i}\mathcal{E}^{a}~{}_{bj},$
$\displaystyle\partial_{t}g_{ij}=-2Nk_{ij}+L_{X}g_{ij}$ (28)
$\displaystyle\partial_{t}k_{ij}=-\nabla_{i}\nabla_{j}N+N\\{R_{ij}+\tr_{g}kk_{ij}-2k_{ik}k^{k}_{j}+\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}-g^{kl}F^{a}~{}_{bik}F^{b}~{}_{akl}$
$\displaystyle-\frac{g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}}{n-1}g_{ij}+\frac{g^{km}g^{ln}F^{a}~{}_{bkl}F^{b}~{}_{amn}}{2(n-1)}g_{ij}\\}+L_{X}k_{ij},$
$\displaystyle
R(g)-|k|^{2}+(\tr_{g}k)^{2}=g^{ij}\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}+\frac{1}{2}g^{ik}g^{jl}F^{a}~{}_{bij}F^{b}~{}_{akl},$
(29)
$\displaystyle\nabla_{j}k^{j}_{i}-\nabla_{i}\tr_{g}k=g^{jk}F^{a}~{}_{bij}\mathcal{E}^{b}~{}_{ak},$
(30) $\displaystyle
g^{ij}\nabla_{i}\mathcal{E}^{a}~{}_{bj}+g^{ij}[A_{i},\mathcal{E}_{j}]^{a}~{}_{b}=0.$
(31)

This system is not hyperbolic since the Ricci tensor of the spatial metric and
the divergence of the spatial Yang-Mills curvature contain terms that obstruct
the ellipticity of the maps $g\mapsto$Ricci$[g]$ and
$A\mapsto\nabla_{j}F^{a}~{}_{bi}~{}^{j}$. Therefore we need to extract a
hyperbolic system of evolution equations by fixing a suitable gauge, that is,
remove these obstructions. For the gravitational sector, we use the constant
mean extrinsic curvature spatial harmonic gauge. Setting the mean extrinsic
curvature ($\tau$) of the hypersurface $M$ to a time-dependent constant allows
it to be a suitable time function. This choice of temporal gauge yields an
elliptic equation for the lapse function $N$. However, not all spacetimes
admit a constant mean extrinsic curvature (CMC) hypersurface. The existence of
a CMC slice is far from obvious and is a part of active mathematical research
(for details see [38, 39, 40, 41]). This gauge condition reads

$\displaystyle\tau:=\tr_{g}k=monotonic~{}function~{}of~{}time~{}alone.$ (32)

This choice allows $\tau$ to play the role of time i.e.,

$\displaystyle t=monotonic~{}function~{}of~{}\tau.$ (33)

To utilize the property $\partial_{i}\tau=0$, one may simply compute the
entity $\frac{\partial\tau}{\partial t}$ to yield

$\displaystyle\partial_{t}\tau$ (34)
$\displaystyle=-g^{ij}\nabla_{i}\nabla_{j}N+\left(|k|^{2}_{g}+\frac{n-2}{n-1}g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}+\frac{1}{2(n-1)}g^{kl}g^{mn}F^{a}~{}_{bkm}F^{b}~{}_{aln}\right)N$

which is nothing but an elliptic equation for the lapse function $N$. Once we
have fixed the temporal gauge, we need to fix the spatial gauge which would
yield an equation for the shift vector field. Once again, we follow the work
of [17, 18] regarding spatial gauge fixing. Let $\zeta:(M,g)\to(M,\gamma)$ be
a harmonic map with the Dirichlet energy
$\frac{1}{2}\int_{M}g^{ij}\frac{\partial\zeta^{k}}{\partial
x^{i}}\frac{\partial\zeta^{l}}{\partial x^{j}}\gamma_{kl}\mu_{g}$, where
$~{}\gamma$ is a fixed background Riemannian metric. Since the harmonic maps
are the critical points of the Dirichlet energy functional, $\zeta$ satisfies
the following formal Euler-Lagrange equation

$\displaystyle
g^{ij}\left(\partial_{i}\partial_{j}\zeta^{k}-\Gamma[g]_{ij}^{l}\partial_{l}\zeta^{k}+\Gamma[\gamma]_{\alpha\beta}^{k}\partial_{i}\zeta^{\alpha}\partial_{j}\zeta^{\beta}\right)=0.$
(35)

Now, we fix the gauge by imposing the condition that $\zeta=id$, which leads
to the following equation

$\displaystyle-g^{ij}(\Gamma[g]_{ij}^{k}-\Gamma[\gamma]_{ij}^{k})$
$\displaystyle=$ $\displaystyle 0.$ (36)

where $\Gamma[\gamma]_{ij}^{k}$ is the connection with respect to a background
Riemannian metric $\gamma$. Choice of this spatial harmonic gauge yields the
following elliptic equation for the shift vector field $X$ after time
differentiating equation (36)

$\displaystyle\Delta_{g}X^{i}-R^{i}_{j}X^{j}$ $\displaystyle=$
$\displaystyle-2k^{ij}\nabla_{j}N-2Ng^{ik}g^{lj}F^{a}~{}_{bkl}\mathcal{E}^{b}~{}_{aj}+\tau\nabla^{i}N$
$\displaystyle+(2Nk^{jk}-2\nabla^{j}X^{k})(\Gamma[g]^{i}_{jk}-\Gamma[\gamma]^{i}_{jk}).$

The Ricci$[g]$ is expressible as follows

$\displaystyle
R[g]_{ij}=\frac{1}{2}\Delta^{\gamma}_{g}g_{ij}+\alpha_{ij}+\mathcal{N}[g,\partial
g]_{ij},$ (38)

where

$\displaystyle\mathcal{N}_{ij}=\frac{1}{2}\left(g_{il}g^{mn}R[\gamma]^{l}_{mjn}+g_{jl}g^{mn}R[\gamma]^{l}_{min}\right)+\frac{1}{2}g^{mn}\gamma^{ls}\left(\nabla[\gamma]_{j}g_{ns}\nabla[\gamma]_{l}g_{im}\right.$
(39)
$\displaystyle\left.+\nabla[\gamma]_{i}g_{lm}\nabla[\gamma]_{s}g_{jn}-\frac{1}{2}\nabla[\gamma]_{j}g_{ns}\nabla[\gamma]_{i}g_{lm}+\nabla[\gamma]_{m}g_{il}\nabla[\gamma]_{n}g_{js}\right.$
$\displaystyle\left.-\nabla[\gamma]_{m}g_{il}\nabla[\gamma]_{s}g_{jn}\right)-\frac{1}{2}V^{l}\nabla[\gamma]_{l}g_{ij}$

and

$\displaystyle\alpha_{ij}:=\frac{1}{2}(\nabla_{i}V_{j}+\nabla_{j}V_{i}),~{}V^{k}:=g^{ij}(\Gamma[g]_{ij}^{k}-\Gamma[\gamma]_{ij}^{k}).$
(40)

In the SH gauge therefore the term in $\alpha_{ij}$ obstructing the
ellipticity of the map $g\mapsto$ Ricci$[g]$ vanishes yielding a hyperbolic
system.

The remaining gauge choice is the gauge fixing of the Yang-Mills sector.
Before moving to the gauge fixing, we need to mention a small subtlety that
arises in the spatial harmonic gauge. Notice that in this gauge the shift
vector field $X\neq 0$ in general and therefore, the evolution equation of
$A^{a}~{}_{bi}$ (26) includes a term $F^{a}~{}_{bij}X^{j}$ which contains the
first derivative of $A$. This term is therefore at the level of principal
terms concerning regularity. While writing down the energy expression, one
would ideally want this term to be canceled point-wise. However, that does not
happen in our case. Therefore, this term potentially obstructs the closing of
the energy argument. However, this involves a gauge variable $X$ and one
expects that this term should not create an additional problem. After
carefully expressing the term $F^{a}~{}_{bij}X^{j}$ we observe that one may
obtain the following evolution equation for the vector potential $A$

$\displaystyle\partial_{t}A^{a}~{}_{bi}$ $\displaystyle=$ $\displaystyle
N\mathcal{E}^{a}~{}_{bi}+\partial_{i}(A^{a}~{}_{b0}-A^{a}~{}_{bj}X^{j})+[A_{i},A_{0}-A_{j}X^{j}]^{a}~{}_{b}$
$\displaystyle+X^{j}\nabla[\gamma]_{j}A^{a}~{}_{bi}+A^{a}~{}_{bj}\nabla[\gamma]_{i}X^{j}$

Now instead of treating the temporal component $A^{a}_{b0}$ as the Yang-Mills
gauge variable, it suffices to treat
$\varphi^{a}_{b}:=A^{a}~{}_{b0}-A^{a}~{}_{bj}X^{j}$ as the new gauge variable.
By fixing a suitable gauge, we will obtain a necessary elliptic equation for
$\varphi^{a}_{b}$. This issue of regularity will be very crucial while
studying the elliptic equation. We will use a version of the ordinary Coulomb
gauge that is used in Maxwell theory (or sometimes in Yang-Mills theory). The
ordinary Coulomb gauge condition simply states the vanishing of the divergence
of the vector potential $A^{a}~{}_{bi}$. However, we notice that
$A^{a}~{}_{bi}$ does not transform as a tensor. Therefore, we define the
generalized Coulomb gauge condition in the following way

$\displaystyle
g^{ij}\nabla^{\hat{A}}[\gamma]_{j}(A_{i}-\hat{A}_{i}):=g^{ij}\nabla[\gamma]_{j}(A_{i}-\hat{A}_{i})+g^{ij}[\hat{A}_{j},(A_{i}-\hat{A}_{i})]=0,$
(42)

where $\hat{A}$ is a $C^{\infty}$ background connection and
$\nabla^{\hat{A}}[\gamma]$ is the gauge covariant derivative with respect to
the background connection $\hat{A}$ and the background metric $\gamma$. The
time derivative of this gauge condition yields an elliptic equation for
$\varphi^{a}_{b}$. Explicit calculations yield

$\displaystyle
g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}~{}_{b}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}~{}_{b}+g^{ij}\nabla_{i}N\mathcal{E}^{a}~{}_{bj}-Ng^{ij}[A_{i},\mathcal{E}_{j}]^{a}~{}_{b}$
$\displaystyle+g^{ij}[\hat{A}_{i},N\mathcal{E}_{j}]^{a}~{}_{b}+g^{ij}(\Gamma[g]^{k}_{ij}-\Gamma[\gamma]^{k}_{ij})N\mathcal{E}^{a}~{}_{bk}-X^{k}\nabla[\gamma]_{k}g^{ij}\nabla[\gamma]_{i}(A^{a}~{}_{bj}$
$\displaystyle-\hat{A}^{a}~{}_{bj})-X^{k}\nabla[\gamma]_{k}g^{ij}[\hat{A}_{i},A_{j}-\hat{A}_{j}]^{a}~{}_{b}+g^{ij}\nabla[\gamma]_{i}X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bj}$
$\displaystyle+g^{ij}\nabla[\gamma]_{i}A^{a}~{}_{bk}\nabla[\gamma]_{j}X^{k}+g^{ij}A^{a}~{}_{bk}\nabla[\gamma]_{i}\nabla[\gamma]_{j}X^{k}-g^{ij}R[\gamma]^{l}_{jik}A^{a}~{}_{bl}X^{k}$
$\displaystyle+g^{ij}[\hat{A}_{i},A_{k}]^{a}~{}_{b}\nabla[\gamma]_{j}X^{k}+(2Nk^{ij}-\nabla^{i}X^{j}-\nabla^{j}X^{i})\nabla[\gamma]^{\hat{A}}_{i}(A^{a}~{}_{bj}-\hat{A}^{a}~{}_{bj})=0.$

Note that this equation is consistent in terms of regularity. If we assume
that $(g,k,A,\mathcal{E})\in H^{s}\times H^{s-1}\times H^{s}\times
H^{s-1},~{}s>\frac{n}{2}+1$, then by the elliptic equations at hand, we
automatically obtain $(N,X,\varphi)\in H^{s+1}\times H^{s+1}\times H^{s+1}$.
However, if we had used $A_{0}$ instead of $\varphi:=A_{0}-A_{i}X^{i}$, then
the time derivative of the generalized Coulomb gauge condition would yield an
elliptic equation for $A_{0}$ which would contain the second derivative of $A$
as a source term requiring $A_{0}$ to be in $H^{s+2}$ and therefore the energy
estimates for the evolution equations (where a derivative of $A_{0}$ is
present) would not close. In a sense, construction of the shifted potential
$\varphi$ yields a cancellation which removes the problematic term. The term
$\nabla_{j}F^{a}~{}_{bi}~{}^{j}$ appearing on the right hand side of the
evolution equation for the electric field is not generally elliptic as a map
$A\mapsto\nabla_{j}F^{a}~{}_{bi}~{}^{j}$ since it can be written as follows

$\displaystyle\nabla_{j}F^{a}~{}_{bi}~{}^{j}=\Delta^{\gamma}_{g}A^{a}~{}_{bi}+\nabla_{i}\mathcal{C}^{a}~{}_{b}+\mathcal{SL}^{a}~{}_{bi},$
(43)

where

$\displaystyle\mathcal{C}^{a}~{}_{b}$ $\displaystyle:=$ $\displaystyle
g^{ij}\nabla^{\hat{A}}[\gamma]_{j}(A^{a}~{}_{bi}-\hat{A}^{a}~{}_{bi})$
$\displaystyle=$ $\displaystyle
g^{ij}\nabla[\gamma]_{j}(A^{a}~{}_{bi}-\hat{A}^{a}~{}_{bi})+g^{ij}[\hat{A}_{j},(A_{i}-\hat{A}_{i})]^{a}~{}_{b}$

and $\mathcal{SL}^{a}_{bi}$ is semi-linear in $A$ and $g$ and expressed as
follows

$\displaystyle\mathcal{SL}^{a}~{}_{b}~{}^{i}$ (45)
$\displaystyle:=g^{jl}\nabla[\gamma]_{j}g^{ik}\nabla[\gamma]_{i}A^{a}~{}_{bl}+g^{ik}\nabla_{j}g^{jl}\nabla[\gamma]_{k}A^{a}~{}_{bl}-g^{jl}\nabla[\gamma]_{j}g^{ik}\nabla[\gamma]_{l}A^{a}~{}_{bk}$
$\displaystyle-g^{ik}\nabla[\gamma]_{j}g^{jl}\nabla[\gamma]_{k}A^{a}~{}_{bl}+g^{jl}\nabla[\gamma]_{j}g^{ik}[A_{k},A_{l}]^{a}~{}_{b}+g^{ik}\nabla[\gamma]_{j}g^{jl}[A_{k},A_{l}]^{a}~{}_{b}$
$\displaystyle+g^{ik}g^{jl}[\nabla[\gamma]_{j}A_{k},A_{l}]^{a}~{}_{b}+g^{ik}g^{jl}[A_{k},\nabla[\gamma]_{j}A_{l}]^{a}~{}_{b}+g^{ik}\nabla[\gamma]_{k}(g^{jl}\nabla[\gamma]_{j}\hat{A}^{a}~{}_{bl})$
$\displaystyle-g^{ik}\nabla[\gamma]_{k}\left(g^{jl}[\hat{A}_{j},A_{l}-\hat{A}_{l}]^{a}~{}_{b}\right)+g^{ik}g^{jl}R[\gamma]^{m}~{}_{ljk}A^{a}~{}_{bm}.$

Therefore the map
$A\mapsto\nabla_{j}F^{a}~{}_{bi}~{}^{j}-\nabla_{i}\mathcal{C}^{a}~{}_{b}$ is
elliptic thereby turning the Yang-Mills evolution equations into hyperbolic
form. In order to construct solutions to the Einstein-Yang-Mills coupled
evolution and constraint equations in the chosen gauge, we want to solve the
following modified system of evolution equations

$\displaystyle\partial_{t}A^{a}~{}_{bi}=N\mathcal{E}^{a}~{}_{bi}+\partial_{i}\varphi^{a}~{}_{b}+[A_{i},\varphi]^{a}~{}_{b}+X^{j}\nabla[\gamma]_{j}A^{a}~{}_{bi}+A^{a}~{}_{bj}\nabla[\gamma]_{i}X^{j},$
$\displaystyle\partial_{t}\mathcal{E}^{a}~{}_{bi}=N\tr_{g}k\mathcal{E}^{a}~{}_{bi}-(\nabla_{j}N)F^{a}~{}_{bi}~{}^{j}-\underbrace{N(\nabla_{j}F^{a}~{}_{bi}~{}^{j}-\nabla_{i}\mathcal{C}^{a}~{}_{b})}_{elliptic~{}as~{}an~{}image~{}of~{}a~{}map~{}of~{}A}$
(46)
$\displaystyle+N[F_{i}~{}^{j},A_{j}]^{a}~{}_{b}+[\mathcal{E}_{i},\varphi]^{a}~{}_{b}+X^{k}\nabla_{k}\mathcal{E}^{a}~{}_{bi}+\mathcal{E}^{a}~{}_{bk}\nabla_{i}X^{k}-2Nk^{j}_{i}\mathcal{E}^{a}~{}_{bj},$
$\displaystyle\partial_{t}g_{ij}=-2Nk_{ij}+L_{X}g_{ij}$ (47)
$\displaystyle\partial_{t}k_{ij}=-\nabla_{i}\nabla_{j}N+N\\{\underbrace{R_{ij}-\alpha_{ij}}_{elliptic~{}as~{}an~{}image~{}of~{}a~{}map~{}of~{}g}+\tr_{g}kk_{ij}-2k_{ik}k^{k}_{j}$
(48)
$\displaystyle+\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}-g^{kl}F^{a}~{}_{bik}F^{b}~{}_{akl}-\frac{g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}}{n-1}g_{ij}+\frac{g^{km}g^{ln}F^{a}~{}_{bkl}F^{b}~{}_{amn}}{2(n-1)}g_{ij}\\}+L_{X}k_{ij}$

coupled with the elliptic equations for $N,X$, and $\varphi$ needed to
preserve the imposed gauge conditions

$\displaystyle-g^{ij}\nabla_{i}\nabla_{j}N+\left(|k|^{2}_{g}+\frac{n-2}{n-1}g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}+\frac{1}{2(n-1)}g^{kl}g^{mn}F^{a}~{}_{bkm}F^{b}~{}_{aln}\right)$
$\displaystyle N=1,$ (49)
$\displaystyle\Delta_{g}X^{i}-R^{i}_{j}X^{j}+L_{X}V^{i}=-2k^{ij}\nabla_{j}N-2Ng^{ik}g^{lj}F^{a}~{}_{bkl}\mathcal{E}^{b}~{}_{aj}+\tau\nabla^{i}N$
(50)
$\displaystyle+(2Nk^{jk}-2\nabla^{j}X^{k})(\Gamma[g]^{i}_{jk}-\Gamma[\gamma]^{i}_{jk}),$
$\displaystyle
g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}~{}_{b}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}~{}_{b}+g^{ij}\nabla_{i}N\mathcal{E}^{a}~{}_{bj}-Ng^{ij}[A_{i},\mathcal{E}_{j}]^{a}~{}_{b}$
$\displaystyle+g^{ij}[\hat{A}_{i},N\mathcal{E}_{j}]^{a}~{}_{b}+g^{ij}(\Gamma[g]^{k}_{ij}-\Gamma[\gamma]^{k}_{ij})N\mathcal{E}^{a}~{}_{bk}-X^{k}\nabla[\gamma]_{k}g^{ij}\nabla[\gamma]_{i}(A^{a}~{}_{bj}$
(51)
$\displaystyle-\hat{A}^{a}~{}_{bj})-X^{k}\nabla[\gamma]_{k}g^{ij}[\hat{A}_{i},A_{j}-\hat{A}_{j}]^{a}~{}_{b}+g^{ij}\nabla[\gamma]_{i}X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bj}$
$\displaystyle+g^{ij}\nabla[\gamma]_{i}A^{a}~{}_{bk}\nabla[\gamma]_{j}X^{k}+g^{ij}A^{a}~{}_{bk}\nabla[\gamma]_{i}\nabla[\gamma]_{j}X^{k}-g^{ij}R[\gamma]^{l}_{jik}A^{a}~{}_{bl}X^{k}$
$\displaystyle+g^{ij}[\hat{A}_{i},A_{k}]^{a}~{}_{b}\nabla[\gamma]_{j}X^{k}+(2Nk^{ij}-\nabla^{i}X^{j}-\nabla^{j}X^{i})\nabla[\gamma]^{\hat{A}}_{i}(A^{a}~{}_{bj}-\hat{A}^{a}~{}_{bj})=0.$

The modified system coincides with the actual Einstein-Yang-Mills system in
the CMCSHGC gauge. If $\mathcal{GA}$ denotes the group of gauge
transformations, then the configuration space of the gauge fixed coupled
dynamics (‘true’ dynamics) is identified with the space
$\mathcal{M}_{\Sigma}/diff(\Sigma)\times\mathcal{A}/\mathcal{GA}$, where
$\mathcal{A}$ is the space of connections on $\Sigma\times\mathbb{R}$. This
complete coupled elliptic-hyperbolic system is designated as CMCSHGC Einstein-
Yang-Mills system.

##  4 Important inequalities

Here we recall a few important inequalities which are crucial in obtaining the
required energy estimates. All the Sobolev spaces are defined on the spatial
manifold $\Sigma$ and the corresponding norms are defined in the notations
section. Associated constants depend on the background geometry (i.e., on the
metric $\gamma$). We will consider $n\geq 2$.
1. [42] Let $f\in H^{t_{1}},~{}g\in H^{t_{2}}$ and $k\leq\min(t_{1},t_{2},t_{1}+t_{2}-\frac{n}{2}),~{}t_{i}\geq 0$ and some $t_{i}>0$, then the following estimate holds

$\displaystyle||fg||_{H^{k}}\lesssim||f||_{H^{t_{1}}}||g||_{H^{t_{2}}}.$ (52)

2.[43] For $\mathcal{P}\in\mathcal{OP}^{s},~{}s\in\mathbb{R}$, the following
holds

$\displaystyle||\mathcal{P}u||_{H^{r}}\lesssim||u||_{H^{s+r}},~{}~{}r\in\mathbb{R}.$
(53)

Here $\mathcal{OP}^{s}$ denotes the space of pseudo-differential operators
with symbol in Hormander’s class $S^{s}_{1,0}$. For example
$\nabla[\gamma]\in\mathcal{OP}^{1}$.
3.[43] Assume $\mathcal{P}\in\mathcal{OP}^{s},~{}s>0$, $\sigma\geq 0$ then

$\displaystyle||[P,u]v||_{H^{\sigma}}\lesssim\left(||\nabla[\gamma]u||_{L^{\infty}}||v||_{H^{s-1+\sigma}}+||u||_{H^{s+\sigma}}||v||_{L^{\infty}}\right)$
(54)

and more specifically
$[\nabla[\gamma],\nabla[\gamma]^{s-1}]\in\mathcal{OP}^{s-1}$.
4.(Kato and Ponce [44]) If $s>0$, then $H^{s}\cap L^{\infty}$ is an algebra
i.e.,

$\displaystyle||uv||_{H^{s}}\lesssim||u||_{L^{\infty}}||v||_{H^{s}}+||u||_{H^{s}}||v||_{L^{\infty}}$
(55)

and for $s>\frac{n}{2}+1$ due to Sobolev embedding

$\displaystyle||uv||_{H^{s}}\lesssim||u||_{H^{s}}||v||_{H^{s}}.$ (56)

In addition, we will make use of Sobolev embedding, Holder’s and Minkowski’s
inequality whenever necessary.

##  5 Energy Estimates

In the study of hyperbolic PDEs, the use of energy functionals is
indispensable in proving local and global well-posedness results. We define
the following ad-hoc energies for the Yang-Mills and Einstein sectors. These
energies will not be gauge invariant and therefore are only defined in our
choice of gauge. One could in principle define gauge-invariant energies, but,
for the current purpose, this makes no difference. Let us first define a few
relevant entities. Let $\Lambda[g]$ be the ellipticity constant of $g$ defined
as the least $\Lambda$ such that the following holds

$\displaystyle\Lambda^{-1}\gamma(\xi,\xi)\leq
g(\xi,\xi)\leq\Lambda\gamma(\xi,\xi),~{}\forall\xi\in sections\\{T\Sigma\\}.$
(57)

From the expression of the Lorentzian metric $\hat{g}$ in terms of $(g,N,X)$
(14), we define the following entity

$\displaystyle\Lambda[\hat{g}]:=\Lambda[g]+||N||_{L^{\infty}}+||N^{-1}||_{L^{\infty}}+||X||_{L^{\infty}}$
(58)

and for a curve $t\mapsto\hat{g}(t)$,

$\displaystyle\Lambda_{t^{*}}:=\sup_{t\in[0,t^{*}]}\Lambda[\hat{g}(t)].$ (59)

Using these definitions, calculation yields the following

$\displaystyle C(\Lambda[\hat{g}])^{-1}(|\nabla[\gamma]g|+|\nabla
N|+|\nabla[\gamma]X|)\leq|\nabla[\gamma]\hat{g}|$ (60) $\displaystyle\leq
C(\Lambda[\hat{g}])(|\nabla[\gamma]g|+|\nabla N|+|\nabla[\gamma]X|).$

We define the Yang-Mills energy as follows

$\displaystyle
E_{YM}:=\frac{1}{2}\left(\int_{\Sigma}A^{a}~{}_{bi}A^{b}~{}_{aj}\gamma^{ij}\mu_{g}+\sum_{I=1}^{s}\int_{\Sigma}\nabla[\gamma]^{I}A^{a}~{}_{bi}\cdot\nabla[\gamma]^{I}A^{b}~{}_{aj}\gamma^{ij}\mu_{g}\right.$
$\displaystyle\left.+\sum_{I=1}^{s}\int_{\Sigma}\nabla[\gamma]^{I-1}\mathcal{E}^{a}~{}_{bi}\cdot\nabla[\gamma]^{I-1}\mathcal{E}^{b}~{}_{aj}\gamma^{ij}\mu_{g}\right).$
(61)

The energy for the gravitational sector is defined as

$\displaystyle
E_{Ein}:=\frac{1}{2}\left(\int_{\Sigma}g_{ij}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}+\sum_{I=1}^{s}\int_{\Sigma}\nabla[\gamma]^{I}g_{ij}\cdot\nabla[\gamma]^{I}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right.$
$\displaystyle\left.+4\sum_{I=1}^{s}\int_{\Sigma}\nabla[\gamma]^{I-1}k_{ij}\cdot\nabla[\gamma]^{I-1}k_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right),$
(62)

$s>\frac{n}{2}+1$. In view of the fact that $\Lambda^{-1}\gamma(\xi,\xi)\leq
g(\xi,\xi)\leq\Lambda\gamma(\xi,\xi)$, the total energy is equivalent to the
desired norm of the data i.e.,

$\displaystyle
E_{s}:=E_{YM}+E_{Ein}\approx||A||^{2}_{H^{s}}+||\mathcal{E}||^{2}_{H^{s-1}}+||g||^{2}_{H^{s}}+||k||^{2}_{H^{s-1}}.$
(63)

We will show that there is unique solution
$(g,k,A,\mathcal{E})\in\mathcal{C}([0,t^{*}];H^{s}\times H^{s-1}\times
H^{s}\times H^{s-1})$ for $s>\frac{n}{2}+1$ and $t^{*}>0$. The multi index $I$
is defined as follows

$\displaystyle\nabla[\gamma]^{I}:=\nabla[\gamma]_{i_{1}}\nabla[\gamma]_{i_{2}}\nabla[\gamma]_{i_{3}}....\nabla[\gamma]_{i_{I}}$
(64)

and the dot product

$\displaystyle\nabla[\gamma]^{I}u\cdot\nabla[\gamma]^{I}v$ (65)
$\displaystyle:=g^{i_{1}j_{1}}....g^{i_{I}j_{I}}\nabla[\gamma]_{i_{1}}\nabla[\gamma]_{i_{2}}\nabla[\gamma]_{i_{3}}....\nabla[\gamma]_{i_{I}}u\cdot\nabla[\gamma]_{j_{1}}\nabla[\gamma]_{j_{2}}\nabla[\gamma]_{j_{3}}....\nabla[\gamma]_{j_{I}}v,$

where the $\cdot$ indicates the contraction of the remaining indices of $u$
and $v$ with respect to the background metric $\gamma$ and the metric on
$\mathfrak{g}$ (if $u$ and $v$ contain internal indices). We start with the
first (lowest) order energy. Multiple applications of the operator
$\langle\nabla[\gamma]\rangle:=(1+\Delta_{\gamma})^{1/2}$ would produce the
desired higher order energy since
$W^{s,p}(\Sigma)=\langle\nabla[\gamma]\rangle^{s}L^{p}(\Sigma),~{}1<p<\infty,~{}s\in\mathbb{R}$,
($\langle\nabla[\gamma]\rangle^{s}$ acts on the spectral side) with norm
defined as

$\displaystyle||v||_{W^{s,p}(\Sigma)}=||\langle\nabla[\gamma]\rangle^{s}v||_{L^{p}(\Sigma)}.$
(66)

Here $\Delta_{\gamma}$ is defined as
$\Delta_{\gamma}:=-\gamma^{ij}\nabla[\gamma]_{i}\nabla[\gamma]_{j}$. If $s$ is
a non-negative integer, then $W^{s,p},~{}1\leq p\leq\infty$ is the closure of
$C^{\infty}(\Sigma)$ with respect to the equivalent norm $\sum_{|I|\leq
s}||\nabla[\gamma]^{I}v||_{L^{p}(\Sigma)}$. The first order energy is defined
as follows

$\displaystyle E_{1}:=E^{1}_{YM}+E^{1}_{Ein}$ (67)
$\displaystyle=\frac{1}{2}\left(\int_{\Sigma}A^{a}~{}_{bi}A^{b}~{}_{aj}\gamma^{ij}\mu_{g}+\int_{\Sigma}g^{kl}\nabla[\gamma]_{k}A^{a}~{}_{bi}\nabla[\gamma]_{l}A^{b}~{}_{aj}\gamma^{ij}\mu_{g}\right.$
$\displaystyle\left.+\int_{\Sigma}\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}\gamma^{ij}\mu_{g}\right)$
(68)
$\displaystyle+\frac{1}{2}\left(\int_{\Sigma}g_{ij}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}+\int_{\Sigma}g^{mn}\nabla[\gamma]_{m}g_{ij}\nabla[\gamma]_{n}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right.$
$\displaystyle\left.+4\int_{\Sigma}k_{ij}k_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right)$
(69)

The higher order energy $E_{s}$ is equivalent to the first order energy of
$\langle\nabla[\gamma]\rangle^{s-1}$ applied on $(g,k,A,\mathcal{E})$ i.e.,

$\displaystyle E_{s}\approx
E_{1}(\langle\nabla[\gamma]\rangle^{s-1}A,\langle\nabla[\gamma]\rangle^{s-2}\mathcal{E},\langle\nabla[\gamma]\rangle^{s-1}g,\langle\nabla[\gamma]\rangle^{s-2}k),~{}~{}s\geq
1.$ (70)

One important fact we will note here is that $\langle\nabla[\gamma]\rangle$ is
time-independent since the background metric $\gamma$ is time-independent. Let
us denote $\langle\nabla[\gamma]\rangle$ by $\langle D\rangle$. Before moving
to the energy estimate, let us split the evolution equations into the linear
and non-linear parts (in terms of the evolution variables
$(g,k,A,\mathcal{E})$) as follows

$\displaystyle\mathcal{L}[g,k,N,X]\mathcal{V}$ (71)
$\displaystyle=\left[\begin{array}[]{c}\partial_{t}g_{ij}+2Nk_{ij}-X^{k}\nabla[\gamma]_{k}g_{ij}\\\
\partial_{t}k_{ij}+\frac{N}{2}\Delta_{\gamma}g_{ij}-N\tr_{g}kk_{ij}-X^{k}\nabla[\gamma]_{k}k_{ij}\\\
\partial_{t}A^{a}~{}_{bi}-N\mathcal{E}^{a}~{}_{bi}-X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bi}\\\
\partial_{t}\mathcal{E}^{a}~{}_{bi}-Ng^{kl}\nabla[\gamma]_{k}\nabla[\gamma]_{l}A^{a}~{}_{bi}-N\tr_{g}k\mathcal{E}^{a}~{}_{bi}-X^{k}\nabla[\gamma]_{k}\mathcal{E}^{a}~{}_{bi}\end{array}\right],$
$\displaystyle\mathcal{V}:=\left[\begin{array}[]{c}g\\\ k\\\ A\\\
\mathcal{E}\end{array}\right],$ (80)
$\displaystyle\mathcal{F}[g,k,N,X,A,\mathcal{E}]$ (81)
$\displaystyle=\left[\begin{array}[]{c}g_{ik}\nabla[\gamma]_{j}X^{k}+g_{kj}\nabla[\gamma]_{i}X^{k}\\\
\left\\{N\mathcal{N}_{ij}-2Nk_{ik}k^{k}_{j}+N\left(\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}-g^{kl}F^{a}~{}_{bik}\cdot
F^{b}~{}_{ajl}-\frac{g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}_{al}}{n-1}g_{ij}\right.\right.\\\
\left.\left.+\frac{g^{mk}g^{nl}F^{a}~{}_{bmn}F^{b}~{}_{akl}}{2(n-1)}g_{ij}\right)-\nabla[\gamma]_{i}\nabla_{j}N\right.\\\
\left.-\frac{1}{2}g^{kl}(\nabla[\gamma]_{i}g_{lj}+\nabla[\gamma]_{j}g_{il}-\nabla[\gamma]_{l}g_{ij})\partial_{k}N+k_{ik}\nabla[\gamma]_{j}X^{k}+k_{kj}\nabla[\gamma]_{i}X^{k}\right\\}\\\
\partial_{i}\varphi^{a}_{b}+[A_{i},\varphi]^{a}~{}_{b}+A^{a}~{}_{bj}\nabla[\gamma]_{i}X^{j}\\\
N\mathcal{SL}^{a}~{}_{bi}-F^{a}~{}_{ai}~{}^{j}\nabla_{j}N+N[F~{}_{i}~{}^{j},A_{j}]^{a}~{}_{b}+[\mathcal{E}_{i},\varphi]^{a}~{}_{b}+\mathcal{E}^{a}~{}_{bk}\nabla[\gamma]_{i}X^{k}-2Nk_{i}^{j}\mathcal{E}^{a}~{}_{bj}\\\
\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{c}\mathcal{F}_{1}\\\
\mathcal{F}_{2}\\\ (\mathcal{F}_{3})^{a}~{}_{b}\\\
(\mathcal{F}_{4})^{a}~{}_{b}\end{array}\right]$ (91)

and therefore the system of evolution equations read

$\displaystyle\mathcal{L}[\mathcal{V}]\mathcal{V}=F[\mathcal{V}].$ (92)

Let us now derive the necessary energy estimates. First define the following
space

$\displaystyle\mathcal{H}^{s}:=H^{s}(\Sigma)\times H^{s-1}(\Sigma)\times
H^{s}(\Sigma)\times H^{s-1}(\Sigma),~{}~{}s>\frac{n}{2}+1.$ (93)

Lemma 1: Assume $\mathcal{V}\in
L^{\infty}([0,t];\mathcal{H}^{1}\cap\mathcal{W}^{1,\infty})$ solves the EYM
evolution equations. Then we have

$\displaystyle\sqrt{E_{1}(t)}\leq
Ce^{C\int_{0}^{t}(||D\hat{g}(t)||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}}+||DA||_{L^{\infty}})dt^{{}^{\prime}}}(\sqrt{E_{1}(0)}+||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{1})})$

for a constant $C=C(\Lambda_{t})$.
Proof: We now explicitly evaluate the time evolution of the first order energy

$\displaystyle\frac{dE_{1}}{dt}=\int_{\Sigma}\left(\gamma^{ij}(\partial_{t}A^{a}~{}_{bi})A^{b}~{}_{aj}+g^{ij}\nabla[\gamma]_{i}\partial_{t}A^{a}~{}_{bm}\nabla[\gamma]_{j}A^{b}~{}_{an}\gamma^{mn}\right.$
$\displaystyle\left.-\frac{1}{2}g^{ik}g^{jl}\partial_{t}g_{kl}\nabla[\gamma]_{i}A^{a}A_{bm}\nabla[\gamma]_{j}A^{b}~{}_{an}\gamma^{mn}+\gamma^{ij}\partial_{t}\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}\right.$
$\displaystyle\left.+\gamma^{ik}\gamma^{jl}\partial_{t}g_{ij}g_{kl}+g^{ij}\nabla[\gamma]_{i}\partial_{t}g_{kl}\nabla[\gamma]_{j}g_{mn}\gamma^{km}\gamma^{ln}+4\gamma^{ik}\gamma^{jl}\partial_{t}k_{ij}k_{kl}\right.$
$\displaystyle\left.-\frac{1}{2}g^{ir}g^{js}\partial_{t}g_{rs}\nabla[\gamma]_{i}g_{kl}\nabla[\gamma]_{j}g_{mn}\gamma^{km}\gamma^{ln}\right)\mu_{g}$
$\displaystyle+\frac{1}{4}\int_{\Sigma}\left(A^{a}~{}_{bi}A^{b}~{}_{aj}\gamma^{ij}+g^{kl}\nabla[\gamma]_{k}A^{a}~{}_{bi}\nabla[\gamma]_{l}A^{b}~{}_{aj}\gamma^{ij}\right.$
$\displaystyle\left.+\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}\gamma^{ij}+g_{ij}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}+g^{mn}\nabla[\gamma]_{m}g_{ij}\nabla[\gamma]_{n}g_{kl}\gamma^{ik}\gamma^{jl}\right.$
$\displaystyle\left.+4k_{ij}k_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right)g^{pq}\partial_{t}g_{pq}\mu_{g}.$
(94)

Now we use the evolution equations for $\mathcal{V}:=(g,k,A,\mathcal{E})$. A
direct calculation yields

$\displaystyle\frac{dE_{1}}{dt}=\int_{\Sigma}\left(\gamma^{ij}(N\mathcal{E}^{a}~{}_{bi}+X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bi}+(\mathcal{F}~{}_{3})^{a}~{}_{bi})A^{b}~{}_{aj}\right.$
$\displaystyle\left.+g^{ij}\nabla[\gamma]_{i}(N\mathcal{E}^{a}~{}_{bm}+X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bm}+(\mathcal{F}_{3})^{a}~{}_{bm})\nabla[\gamma]_{j}A^{a}~{}_{bn}\gamma^{mn}\right.$
$\displaystyle\left.+\gamma^{ij}(N\Delta^{\gamma}_{g}A^{a}~{}_{bi}+N\tr_{g}k\mathcal{E}^{a}~{}_{bi}+X^{k}\nabla[\gamma]_{k}\mathcal{E}^{a}~{}_{bi}+(\mathcal{F}_{4})_{i})\mathcal{E}^{a}~{}_{bj}\right.$
$\displaystyle\left.+\gamma^{ik}\gamma^{jl}(-2Nk_{ij}+X^{k}\nabla[\gamma]_{k}g_{ij}+(\mathcal{F}_{1})_{ij})g_{kl}\right.$
$\displaystyle\left.+g^{ij}\nabla[\gamma]_{i}(-2Nk_{kl}+X^{m}\nabla[\gamma]_{m}g_{kl}+(\mathcal{F}_{1})_{kl})\nabla[\gamma]_{j}g_{mn}\gamma^{km}\gamma^{ln}\right.$
$\displaystyle\left.+4\gamma^{ik}\gamma^{jl}(-\frac{N}{2}\Delta^{\gamma}_{g}g_{ij}+N\tr_{g}kk_{ij}+X^{k}\nabla[\gamma]_{k}k_{ij}+(\mathcal{F}_{2})_{ij})k_{kl}\right.$
$\displaystyle\left.-\frac{1}{2}g^{ik}g^{jl}(-2Nk_{ij}+X^{m}\nabla[\gamma]_{m}g_{kl}+(\mathcal{F}_{1})_{kl})\nabla[\gamma]_{i}A^{a}~{}_{bm}\nabla[\gamma]_{j}A^{b}~{}_{an}\gamma^{mn}\right.$
$\displaystyle\left.-\frac{1}{2}g^{ir}g^{js}(-2Nk_{rs}+X^{m}\nabla[\gamma]_{m}g_{rs}+(\mathcal{F}_{1})_{rs})\nabla[\gamma]_{i}g_{kl}\nabla[\gamma]_{j}g_{mn}\gamma^{km}\gamma^{ln}\right)\mu_{g}$
$\displaystyle+\frac{1}{4}\int_{\Sigma}\left(A^{a}~{}_{bi}A^{b}~{}_{aj}\gamma^{ij}+g^{kl}\nabla[\gamma]_{k}A^{a}~{}_{bi}\nabla[\gamma]_{l}A^{b}~{}_{aj}\gamma^{ij}\right.$
$\displaystyle\left.+\mathcal{E}^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}\gamma^{ij}+g_{ij}g_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}+g^{mn}\nabla[\gamma]_{m}g_{ij}\nabla[\gamma]_{n}g_{kl}\gamma^{ik}\gamma^{jl}\right.$
$\displaystyle\left.+4k_{ij}k_{kl}\gamma^{ik}\gamma^{jl}\mu_{g}\right)(-2N\tr_{g}k+\nabla_{k}X^{k})\mu_{g}$
(95)

We notice that the principal terms of the type
$\Delta^{\gamma}_{g}A^{a}~{}_{bi}\mathcal{E}^{b}~{}_{aj}\gamma^{ij}$ must
cancel with their negative counter parts point-wise in order for the energy
argument to close. We compute

$\displaystyle\gamma^{ij}g^{kl}(\nabla[\gamma]_{k}\nabla[\gamma]_{l}A^{a}~{}_{bi})(N\mathcal{E}^{b}~{}_{aj})$
(96)
$\displaystyle=\underbrace{\nabla[\gamma]_{k}(\gamma^{ij}g^{kl}N\mathcal{E}^{a}~{}_{bj}\nabla[\gamma]_{l}A^{b}~{}_{ai})}_{I:vanishes~{}by~{}integration}-\underbrace{\gamma^{ij}g^{kl}\nabla[\gamma]_{k}(N\mathcal{E}^{a}~{}_{bj})\nabla[\gamma]_{l}A^{b}~{}_{ai}}_{II}$
$\displaystyle-\gamma^{ij}N\mathcal{E}^{a}~{}_{bj}\nabla[\gamma]_{k}g^{kl}\nabla[\gamma]_{l}A^{b}~{}_{ai}$

and notice that the first term vanishes while integrated since it is a total
derivative while the second term $II$ cancels out point-wise with its
negative. The remaining term is harmless in terms of regularity. An exactly
similar calculation holds for the principal term arising from the
gravitational sector. In addition, we also have terms that contain one extra
derivative (such as
$g^{ij}\nabla[\gamma]_{i}(X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bm})$,
$\gamma^{ik}\gamma^{jl}X^{m}\nabla[\gamma]_{m}k_{ij}k_{kl}$ etc) which are
potential obstructions to closing the energy argument. However, these terms
may be controlled by manipulating the derivatives. For example, compute

$\displaystyle\int_{\Sigma}\gamma^{ik}\gamma^{jl}X^{m}\nabla[\gamma]_{m}k_{ij}k_{kl}\mu_{g}=\frac{1}{2}\int_{\Sigma}X^{m}\nabla[\gamma]_{m}(\gamma^{ik}\gamma^{jl}k_{ij}k_{kl})\mu_{g}$
(97)
$\displaystyle=\frac{1}{2}\int_{\Sigma}\nabla[\gamma]_{m}\left(X^{m}\gamma^{ik}\gamma^{jl}k_{ij}k_{kl}\mu_{g}\right)-\frac{1}{2}\int_{\Sigma}\nabla[\gamma]_{m}X^{m}\gamma^{ik}\gamma^{jl}k_{ij}k_{kl}\mu_{g}$
$\displaystyle-\frac{1}{2}\int_{\Sigma}X^{m}\gamma^{ik}\gamma^{jl}k_{ij}k_{kl}(\Gamma[g]^{k}_{km}-\Gamma[\gamma]^{k}_{km})\mu_{g}$
$\displaystyle=-\frac{1}{2}\int_{\Sigma}\nabla[\gamma]_{m}X^{m}\gamma^{ik}\gamma^{jl}k_{ij}k_{kl}\mu_{g}-\frac{1}{2}\int_{\Sigma}X^{m}\gamma^{ik}\gamma^{jl}k_{ij}k_{kl}(\Gamma[g]^{k}_{km}-\Gamma[\gamma]^{k}_{km})\mu_{g},$

and therefore

$\displaystyle|\int_{\Sigma}\gamma^{ik}\gamma^{jl}X^{m}\nabla[\gamma]_{m}k_{ij}k_{kl}\mu_{g}|$
(98)
$\displaystyle\lesssim||\nabla[\gamma]X||_{L^{\infty}}\int_{\Sigma}|k|^{2}_{\gamma}\mu_{g}+||k||_{L^{\infty}}||X||_{L^{\infty}}\int_{\Sigma}\left(|k|^{2}_{\gamma}+|\nabla[\gamma]g|^{2}_{\gamma}\right)\mu_{g}.$

Other terms of this type may be handled in a similar way. Collecting all the
terms and applying Holder’s inequality, we obtain the following differential
inequality for the energy

$\displaystyle\frac{dE_{1}}{dt}\leq
C(\Lambda_{t})\left(E_{1}+\nabla[\gamma]g||_{L^{\infty}}E_{1}+|\tr_{g}k|E_{1}+||\nabla[\gamma]g||_{L^{\infty}}E_{1}\right.$
$\displaystyle\left.+||\nabla[\gamma]X||_{L^{\infty}}E_{1}+||Riem[\gamma]||_{L^{\infty}}E_{1}+||\nabla[\gamma]A||_{L^{\infty}}E_{1}+||\mathcal{E}||_{L^{\infty}}E_{1}\right.$
$\displaystyle\left.+||k||_{L^{\infty}}E_{1}+||\nabla[\gamma]g||_{L^{\infty}}E_{1}+||\mathcal{F}_{3}||_{L^{2}}\sqrt{E_{1}}+||\mathcal{F}_{1}||_{H^{1}}\sqrt{E_{1}}\right.$
$\displaystyle\left.+||\mathcal{F}_{4}||_{L^{2}}\sqrt{E_{1}}+||\mathcal{F}_{1}||_{L^{2}}\sqrt{E_{1}}+||\mathcal{F}_{2}||_{L^{2}}\sqrt{E_{1}}\right)$

integration of which yields

$\displaystyle\sqrt{E_{1}(t)}-\sqrt{E_{1}(0)}\leq
C(\Lambda_{t})\int_{0}^{t}\left((1+||\nabla[\gamma]g||_{L^{\infty}}+|\tr_{g}k|)\sqrt{E}_{1}(t^{{}^{\prime}})\right.$
$\displaystyle\left.+(||k||_{L^{\infty}}+||Riem[\gamma]||_{L^{\infty}}+||\nabla[\gamma]g||_{L^{\infty}})\sqrt{E(t^{{}^{\prime}})}+(||\nabla[\gamma]X||_{L^{\infty}}\right.$
$\displaystyle\left.+||\nabla[\gamma]A||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}})\sqrt{E(t^{{}^{\prime}})}\right)dt^{{}^{\prime}}+C(\Lambda_{t})||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{1})}.$

Noting $\gamma$ is a $C^{\infty}$ background metric,
$||Riem[\gamma]||_{L^{\infty}}\lesssim 1$ and applying Grönwall’s inequality,
we obtain

$\displaystyle\sqrt{E_{1}(t)}\leq\left(\sqrt{E_{1}(0)}+C(\Lambda_{t})||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{1})}\right)e^{C(\Lambda_{t})\int_{0}^{t}\mathcal{B}(t^{{}^{\prime}})dt^{{}^{\prime}}},$

where $\mathcal{B}(t)$ is defined as follows

$\displaystyle\mathcal{B}(t)$ $\displaystyle:=$
$\displaystyle\left(||\nabla[\gamma]X||_{L^{\infty}}+||\nabla[\gamma]g||_{L^{\infty}}+||k||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}}+||\nabla[\gamma]A||_{L^{\infty}}\right)$
$\displaystyle\leq$
$\displaystyle(||D\hat{g}(t)||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}}+||DA||_{L^{\infty}}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

Now in the view of Sobolev embedding for $s>\frac{n}{2}+1$, $\mathcal{B}$ is
dominated by the available Sobolev norm of $(g,k,X,A,\mathcal{E})$. We need to
obtain higher order energy estimates.
Lemma 2: Assume $\mathcal{V}\in
L^{\infty}([0,t];\mathcal{H}^{s}\cap\mathcal{W}^{1,\infty}),~{}s>\frac{n}{2}+1$
solves the EYM evolution equations. Then we have

$\displaystyle\sqrt{E_{s}(t)}$ $\displaystyle\leq$ $\displaystyle
Ce^{C(\Lambda_{t})\int_{0}^{t}(||\hat{g}||_{H^{s}}+||k||_{H^{s-1}}+||A||_{H^{s}}+||\mathcal{E}||_{H^{s-1}})dt^{{}^{\prime}}}$
(100)
$\displaystyle(\sqrt{E_{s}(0)}+||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{s})}),$

where
$||\hat{g}||_{H^{s}}\approx||g||_{H^{s}}+||N||_{H^{s+1}}+||X||_{H^{s+1}}$ .

Proof: Now we need to prove a higher order estimate. We want to estimate the
energy corresponding to $\langle D\rangle^{s-1}\mathcal{V}$. Use the
commutator

$\displaystyle\mathcal{L}\langle
D\rangle^{s-1}\mathcal{V}=[\mathcal{L},\langle
D\rangle^{s-1}]\mathcal{V}+\langle\mathcal{D}\rangle^{s-1}\mathcal{F}$ (101)

Using the energy estimate of lemma $1$, one obtains

$\displaystyle\sqrt{E_{s}(t)}$ $\displaystyle\leq$ $\displaystyle
Ce^{\int_{0}^{t}C(||D\hat{g}(t)||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}}+||DA||_{L^{\infty}})dt^{{}^{\prime}}}$
$\displaystyle(\sqrt{E_{s}(0)}+||\langle
D\rangle^{s-1}\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{1})}+||[L,\langle
D\rangle^{s-1}]\mathcal{V}||_{L^{1}([0,t];\mathcal{H}^{1})})$

Now we have to show that the energy argument closes, that is, $||[L,\langle
D\rangle^{s-1}]\mathcal{V}||_{\mathcal{H}^{1}}$ can be estimated using the
maximum available Sobolev norms. We will use the inequalities listed in the
previous section (section 4).

$\displaystyle[\partial_{t},\langle D\rangle^{s-1}]=0$ (103)
$\displaystyle||[N,\langle D\rangle^{s-1}]k||_{H^{1}}\leq
C\left(||\nabla[\gamma]N||_{L^{\infty}}||k||_{H^{s-1}}+||N||_{H^{s}}||k||_{L^{\infty}}\right)$
(104) $\displaystyle[\langle D\rangle^{s-1},X^{k}\nabla[\gamma]_{k}]g=[\langle
D\rangle^{s-1},X^{k}]\nabla[\gamma]_{k}g+X^{k}[\langle
D\rangle^{s-1},\nabla[\gamma]_{k}]g$ (105)

Note $[\langle D\rangle^{s-1},\nabla[\gamma]_{k}]\in\mathcal{OP}^{s-1}$ and
following inequality (52)

$\displaystyle||[\langle D\rangle^{s-1},X^{k}\nabla[\gamma]_{k}]g||_{H^{1}}$
(106) $\displaystyle\leq||[\langle
D\rangle^{s-1},X^{k}]\nabla[\gamma]_{k}g||_{H^{1}}+||X^{k}[\langle
D\rangle^{s-1},\nabla[\gamma]_{k}]g||_{H^{1}}$
$\displaystyle\lesssim||\nabla[\gamma]X||_{L^{\infty}}||g||_{H^{s}}+||X||_{H^{s}}||\nabla[\gamma]g||_{L^{\infty}}.$

We need to estimate the term of the following type

$\displaystyle||[N\Delta^{\gamma}_{g},\langle D\rangle^{s-1}]g||_{L^{2}}$
(107)

where

$\displaystyle\Delta^{\gamma}_{g}:=g^{ij}\nabla[\gamma]_{i}\nabla[\gamma]_{j}$
(108)

is of the type $g^{-1}D^{2}$. We perform a series of manipulations of the
commutators

$\displaystyle[N\Delta_{\gamma},\langle D\rangle^{s-1}]g=[Ng^{-1}D^{2},\langle
D\rangle^{s-1}]g$ (109) $\displaystyle=Ng^{-1}[D^{2},\langle
D\rangle^{s-1}]g+\underbrace{[Ng^{-1},\langle D\rangle^{s-1}]D^{2}g}_{II}$

Now we notice that the second term $II$ is potentially dangerous and therefore
requires further attention. We further simplify it as follows

$\displaystyle[Ng^{-1},\langle D\rangle^{s-1}]D^{2}g=Ng^{-1}\langle
D\rangle^{s-1}D^{2}g-\langle D\rangle^{s-1}(Ng^{-1}D^{2}g)$ (110)
$\displaystyle=Ng^{-1}\langle D\rangle^{s-1}DDg-\langle
D\rangle^{s-1}\left(D(Ng^{-1}Dg)-D(Ng^{-1})Dg\right)$ (111)
$\displaystyle=[Ng^{-1},\langle D\rangle^{s-1}D]Dg+\langle
D\rangle^{s-1}\left(D(Ng^{-1})Dg\right)$ (112)

and therefore see that all the terms are estimated by the maximum available
Sobolev norms. Using the commutator estimate, we obtain

$\displaystyle||[N\Delta_{\gamma},\langle
D\rangle^{s-1}]g||_{L^{2}}\lesssim||N||_{H^{s}}||\nabla[\gamma]g||_{L^{\infty}}+||\nabla[\gamma]N||_{L^{\infty}}||g||_{H^{s}}.$
(113)

The remaining terms are estimated similarly by means of the commutator
estimate

$\displaystyle||[\langle
D\rangle^{s-1},X^{k}\nabla[\gamma]_{k}]k||_{L^{2}}\lesssim||X||_{L^{\infty}}||k||_{H^{s-1}}+||X||_{H^{s}}||k||_{L^{\infty}},$
(114) $\displaystyle||[N,\langle
D\rangle^{s-1}]\mathcal{E}||_{H^{1}}\lesssim\left(||\nabla[\gamma]N||_{L^{\infty}}||\mathcal{E}||_{H^{s-1}}+||N||_{H^{s}}||\mathcal{E}||_{L^{\infty}}\right),$
(115) $\displaystyle||[\langle
D\rangle^{s-1},X^{k}\nabla[\gamma]_{k}]A||_{H^{1}}\lesssim||\nabla[\gamma]X||_{L^{\infty}}||A||_{H^{s}}+||X||_{H^{s}}||\nabla[\gamma]A||_{L^{\infty}},$
(116) $\displaystyle||[N\Delta_{\gamma},\langle
D\rangle^{s-1}]A||_{L^{2}}\lesssim||N||_{H^{s}}||\nabla[\gamma]A||_{L^{\infty}}+||\nabla[\gamma]N||_{L^{\infty}}||A||_{H^{s}},$
(117) $\displaystyle||[\langle
D\rangle^{s-1},X^{k}\nabla[\gamma]_{k}]\mathcal{E}||_{L^{2}}\lesssim||X||_{L^{\infty}}||\mathcal{E}||_{H^{s-1}}+||X||_{H^{s}}||\mathcal{E}||_{L^{\infty}}$
(118)

Consider the equation

$\displaystyle L\langle D\rangle^{s-1}\mathcal{V}=[L,\langle
D\rangle^{s-1}]\mathcal{V}+\langle\mathcal{D}\rangle^{s-1}\mathcal{F}$ (119)

and apply the energy inequality established in the previous lemma to obtain

$\displaystyle\sqrt{E_{s}(t)}-\sqrt{E_{s}(0)}\leq
C(\Lambda_{t})\int_{0}^{t}\left((1+||\nabla[\gamma]g||_{L^{\infty}}+|\tr_{g}k|)\sqrt{E}_{s}(t^{{}^{\prime}})\right.$
$\displaystyle\left.+(||k||_{L^{\infty}}+||Riem[\gamma]||_{L^{\infty}}+||\mathcal{E}||_{L^{\infty}}+||\nabla[\gamma]A||_{L^{\infty}}+||\nabla[\gamma]g||_{L^{\infty}})\sqrt{E_{s}(t^{{}^{\prime}})}\right.$
$\displaystyle\left.+||\nabla[\gamma]X||_{L^{\infty}}\sqrt{E_{s}(t^{{}^{\prime}})}\right)dt^{{}^{\prime}}+C(\Lambda_{t})||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{s})}+C(\Lambda_{t})\int_{0}^{t}||[L,\langle
D\rangle^{s-1}]\mathcal{V}(t^{{}^{\prime}})||_{\mathcal{H}^{1}}dt^{{}^{\prime}}.$

Here we have used the fact that $||\langle
D\rangle^{s-1}\mathcal{F}(t)||_{\mathcal{H}^{1}}\leq
C(\Lambda_{t})||\mathcal{F}(t)||_{\mathcal{H}^{s}}$. Now, using the previous
calculations, we have

$\displaystyle\int_{0}^{t}||[L,\langle
D\rangle^{s-1}]\mathcal{V}(t^{{}^{\prime}})||_{\mathcal{H}^{1}}dt^{{}^{\prime}}\leq
C(\Lambda_{t})\int_{0}^{t}||D\hat{g}||_{L^{\infty}}||\mathcal{V}(t^{{}^{\prime}})||_{\mathcal{H}^{s}}dt^{{}^{\prime}}$
$\displaystyle\leq
C(\Lambda_{t})\int_{0}^{t}(||D\hat{g}||_{L^{\infty}})\sqrt{E_{s}(t^{{}^{\prime}})}dt^{{}^{\prime}}$

and therefore

$\displaystyle\sqrt{E_{s}(t)}-\sqrt{E_{s}(0)}\leq\int_{0}^{t}C(\Lambda_{t})\left((1+||\nabla[\gamma]g||_{L^{\infty}}+|\tr_{g}k|)\sqrt{E}_{s}(t^{{}^{\prime}})\right.$
$\displaystyle\left.+(||k||_{L^{\infty}}+||\nabla[\gamma]g||_{L^{\infty}})\sqrt{E_{s}(t^{{}^{\prime}})}+(||\nabla[\gamma]X||_{L^{\infty}}+||\nabla[\gamma]g||_{L^{\infty}}\right.$
$\displaystyle\left.+||k||_{L^{\infty}})\sqrt{E_{s}(t^{{}^{\prime}})}\right)dt^{{}^{\prime}}+C(\Lambda_{t})||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{s})}$
$\displaystyle+C(\Lambda_{t})\int_{0}^{t}||D\hat{g}||_{L^{\infty}}\sqrt{E_{s}(t^{{}^{\prime}})}dt^{{}^{\prime}}.$

Using Gronwall’s inequality and Sobolev embedding with $s>\frac{n}{2}+1$
yields the desired estimate (notice that the energy argument closes using
Sobolev embedding)

$\displaystyle\sqrt{E_{s}(t)}$ $\displaystyle\leq$ $\displaystyle
Ce^{C(\Lambda_{t})\int_{0}^{t}(||\hat{g}||_{H^{s}}+||k||_{H^{s-1}}+||A||_{H^{s}}+||\mathcal{E}||_{H^{s-1}})dt^{{}^{\prime}}}$
$\displaystyle(\sqrt{E_{s}(0)}+||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{s})}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

Let us now perform a bit more analysis of the source term $\mathcal{F}$. We
have to show that $||\mathcal{F}||_{L^{1}([0,t];\mathcal{H}^{s})}$ can be
controlled by the maximum available Sobolev norms. We claim that the non-
linear functional $\mathcal{F}$ is locally Lipschitz as the following map

$\displaystyle\mathcal{F}:\mathcal{H}^{s}\to\mathcal{H}^{s}$ (121)

provided that $(N,X,\varphi)\in H^{s+1}(\Sigma)\times H^{s+1}(\Sigma)\times
H^{s+1}(\Sigma)$. First we show that the energy argument closes i.e., if
$\mathcal{V}\in\mathcal{H}^{s}$, then
$\mathcal{F}(\mathcal{V})\in\mathcal{H}^{s}$. From this calculation, the local
Lipschitz property of the map $\mathcal{F}:\mathcal{H}^{s}\to\mathcal{H}^{s}$
follows.
Lemma 3: Let $\mathcal{V}:=(g,k,A,\mathcal{E})\in\mathcal{H}^{s}$. If
$N,X,\varphi\in H^{s+1}\times H^{s+1}\times H^{s+1},~{}s>\frac{n}{2}+1$, then
$\mathcal{F}$ as a map

$\displaystyle\mathcal{F}:\mathcal{H}^{s}\to\mathcal{H}^{s},$ (122)
$\displaystyle\mathcal{V}\mapsto\mathcal{F}(\mathcal{V})$

is continuous.

Proof: Using the expression of $\mathcal{F}(\mathcal{V})$ in conjunction with
Minkowski’s inequality and the algebra property of the Sobolev space $H^{k}$
for $k>\frac{n}{2}$, we obtain

$\displaystyle||\mathcal{F}_{1}||_{H^{s}}\leq C||g||_{H^{s}}||X||_{H^{s+1}},$
(123) $\displaystyle||\mathcal{F}_{2}||_{H^{s-1}}\leq
C\left(||N||_{H^{s-1}}||\mathcal{N}||_{H^{s-1}}+||N||_{H^{s-1}}||k||^{2}_{H^{s-1}}+||N||_{H^{s-1}}||\mathcal{E}||^{2}_{H^{s-1}}\right.$
$\displaystyle\left.+||N||_{H^{s-1}}||A||^{2}_{H^{s-1}}+||N||_{H^{s+1}}+||N||_{H^{s}}||g||_{H^{s}}+||k||_{H^{s-1}}||X||_{H^{s}}\right).$

Now from the explicit expression of $\mathcal{N}_{ij}$ (39), we obtain

$\displaystyle||\mathcal{N}||_{H^{s-1}}\lesssim||g||^{2}_{H^{s}}$ (124)

and therefore

$\displaystyle||\mathcal{F}_{2}||_{H^{s-1}}\leq
C\left(||N||_{H^{s-1}}||g||^{2}_{H^{s}}+||N||_{H^{s-1}}||k||^{2}_{H^{s-1}}+||N||_{H^{s-1}}||\mathcal{E}||^{2}_{H^{s-1}}\right.$
$\displaystyle\left.+||N||_{H^{s-1}}||A||^{2}_{H^{s-1}}+||N||_{H^{s+1}}+||N||_{H^{s}}||g||_{H^{s}}+||k||_{H^{s-1}}||X||_{H^{s}}\right).$

The remaining components of $\mathcal{F}$ satisfy

$\displaystyle||\mathcal{F}_{3}||_{H^{s}}\leq
C\left(||\varphi||_{H^{s+1}}+||A||_{H^{s}}||\varphi||_{H^{s}}+||A||_{H^{s}}||X||_{H^{s+1}}\right),$
(125) $\displaystyle||\mathcal{F}_{4}||_{H^{s-1}}\leq
C\left(||\mathcal{E}||_{H^{s-1}}||\varphi||_{H^{s-1}}+||\mathcal{E}||_{H^{s-1}}||X||_{H^{s}}+||N||_{H^{s-1}}||\mathcal{SL}||_{H^{s-1}}\right.$
$\displaystyle\left.+||N||_{H^{s-1}}||k||_{H^{s-1}}||\mathcal{E}||_{H^{s-1}}+||N||_{H^{s-1}}||F||_{H^{s-1}}+||N||_{H^{s-1}}||F||_{H^{s-1}}||A||_{H^{s-1}}\right).$

Once again we need to make sure that the nonlinear term $\mathcal{SL}$ is
under control. Recalling the expression of $\mathcal{SL}$ (45), we obtain

$\displaystyle||\mathcal{SL}||_{H^{s-1}}\lesssim||g||_{H^{s}}||\hat{A}||_{H^{s}}+||\hat{A}||_{H^{s+1}}+||g||_{H^{s}}||\hat{A}||_{H^{s-1}}||A-\hat{A}||_{H^{s-1}}$
$\displaystyle+||\hat{A}||_{H^{s}}||A-\hat{A}||_{H^{s-1}}+||\hat{A}||_{H^{s-1}}||A-\hat{A}||_{H^{s}}+||g||_{H^{s}}||A||_{H^{s}}$
$\displaystyle+||R[\gamma]||_{H^{s-1}}||A||_{H^{s-1}}+||g||_{H^{s}}||A||^{2}_{H^{s-1}}+||A||_{H^{s}}||A||_{H^{s-1}}+||g||_{H^{s}}||F||_{H^{s-1}}.$

Now note that $\hat{A}\in C^{\infty}(\mathfrak{X}(\Sigma))$, $\gamma\in
C^{\infty}(\mathcal{M}_{\Sigma})$, and
$||F||_{H^{s-1}}\lesssim||A||_{H^{s}}+||A||^{2}_{H^{s-1}}$. Since
$N,X,\varphi\in H^{s+1}$, this yields
$||\mathcal{F}(\mathcal{V})||_{\mathcal{H}^{s}}\lesssim||\mathcal{V}||_{\mathcal{H}^{s}}$
which establishes continuity of $\mathcal{F}$.
Lemma 4: Let $\mathcal{V}:=(g,k,A,\mathcal{E})\in\mathcal{H}^{s}$. If
$N,X,\varphi\in H^{s+1}\times H^{s+1}\times H^{s+1},~{}s>\frac{n}{2}+1$, then
$\mathcal{F}$ as a map

$\displaystyle\mathcal{F}:\mathcal{H}^{k}\to\mathcal{H}^{k},$ (126)
$\displaystyle\mathcal{V}\mapsto\mathcal{F}(\mathcal{V})$

is locally Lipschitz for $1\leq k\leq s$.
Proof: We recall the following product estimate (4.1)

$\displaystyle||fg||_{H^{k}}\lesssim||f||_{H^{t_{1}}}||g||_{H^{t_{2}}}$ (127)

for $f\in H^{t_{1}},~{}g\in H^{t_{2}}$ and
$k\leq\min(t_{1},t_{2},t_{1}+t_{2}-\frac{n}{2}),~{}t_{i}\geq 0$ and some
$t_{i}>0$.This estimate yields that the multiplication maps $H^{s}\times
H^{1}\to H^{1}$ and $H^{s}\times L^{2}\to L^{2}$ are continuous for
$s>\frac{n}{2}+1$. Now repeating the calculation exactly as the lemma 3 and
using this product estimate, we obtain

$\displaystyle||\mathcal{F}(\mathcal{V}_{1})-\mathcal{F}(\mathcal{V}_{2})||_{\mathcal{H}^{k}}$
(128) $\displaystyle\leq
C(1+||\mathcal{V}_{1}||_{\mathcal{H}^{s}}+||\mathcal{V}_{2}||_{H^{s}}+||\mathcal{V}_{1}||^{2}_{\mathcal{H}^{s}}+||\mathcal{V}_{2}||^{2}_{\mathcal{H}^{s}})||\mathcal{V}_{1}-\mathcal{V}_{2}||_{\mathcal{H}^{k}}.$

Therefore the map $\mathcal{F}:\mathcal{H}^{k}\to\mathcal{H}^{k}$ is also
Lipschitz on $\mathcal{B}^{k}_{1/C_{L}}(\mathcal{V}^{0})$ with Lipschitz
constant $C_{L}$, that is, there exists a constant $C_{L}$ such that the
following holds

$\displaystyle||\mathcal{F}(\mathcal{V}_{1})-\mathcal{F}(\mathcal{V}_{2})||_{\mathcal{H}^{k}}\leq
C_{L}||\mathcal{V}_{1}-\mathcal{V}_{2}||_{\mathcal{H}^{k}}$ (129)

for $1\leq k\leq s.$ $\square$

###  5.1 Existence of a Solution

In this section, we will establish the existence of a unique solution of the
gauge fixed Einstein-Yang-Mills equations in the function space
$\mathcal{C}\left([0,t^{*}];\mathcal{H}^{s}\right),~{}s>\frac{n}{2}+1$ for a
suitable $t^{*}>0$ utilizing the technique developed by Anderson and Moncrief
[1]. Their technique is capable of handling the gauge fixed Einstein equations
where the gauge group is the diffeomorphism group naturally arising from
gravity. Leaving out a few subtleties discussed in the overview section, the
major difference between the work of [1] and the present article is the
inclusion of Yang-Mills source terms.

Recall that we have data on the initial hypersurface at $t=0$ and given by
$\mathcal{V}^{0}:=\mathcal{V}(0)=[g(0),k(0),A(0),\mathcal{E}(0)]^{T}$. Let us
now construct a sequence $\mathcal{V}^{0}_{k}\in C^{\infty}\cap
B^{s}_{R}(\mathcal{V}^{0}),~{}k\geq 1$ by applying an approximation to the
identity on the initial data $\mathcal{V}^{0}$ i.e.,

$\displaystyle\lim_{k\to\infty}||\mathcal{V}^{0}-\mathcal{V}^{0}_{k}||_{\mathcal{H}^{s}}=0.$
(130)

Here $B^{s}_{R}(\mathcal{V}^{0})$ is a ball of radius $R$ centered at
$\mathcal{V}^{0}$ in $\mathcal{H}^{s}$. Since $C^{\infty}$ is dense in Sobolev
spaces, this mollification procedure is standard. Now we construct a sequence
of approximate solutions $\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}\in
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$ with initial data given by
$\mathcal{V}^{0}_{k}$ and prove that this sequence converges in
$\mathcal{C}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$ for a suitable $t^{*}<1$
(we choose $t^{*}<1$ so that the involved constants do not depend on time). We
determine the sequence $\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}$ by solving the
following set of linear hyperbolic PDEs

$\displaystyle\mathcal{L}_{k}\mathcal{V}_{k+1}=\mathcal{F}_{k},$ (131)

where $\mathcal{L}_{k}:=\mathcal{L}[\mathcal{V}_{k}]$ and
$\mathcal{F}_{k}=\mathcal{F}[\mathcal{V}_{k}]$. Notice
$N_{k}=N[\mathcal{V}_{k}]$, $X_{k}=X[\mathcal{V}_{k}]$, and
$\varphi_{k}=\varphi[\mathcal{V}_{k}]$ are determined by solving the
associated elliptic equations (3,50,3). Set
$\mathcal{V}_{1}=\mathcal{V}^{0}_{1}$ and $\mathcal{F}_{1}=0$. Now we invoke
the energy estimates of lemma (2) and apply them to the above set of linear
hyperbolic PDEs. As long as
$\\{\mathcal{V}_{k^{{}^{\prime}}}\\}_{k^{{}^{\prime}}=1}^{k}$ takes values in
$B^{s}_{R}(\mathcal{V}^{0})$, the following energy estimate holds

$\displaystyle\sqrt{E_{s}(\mathcal{V}_{k+1};t)}\leq
C_{R}\left(\sqrt{E_{s}(\mathcal{V}_{k+1};0)}+||\mathcal{F}_{k}||_{L^{1}([0,t];\mathcal{H}^{s}}\right)$
(132)

for a $t<1$ and a constant $C_{R}$ dependent on
$||\mathcal{V}_{k}||_{\mathcal{H}^{s}}$. Now, we have control on the initial
sequence $\\{\mathcal{V}^{0}_{k}\\}_{k=1}^{\infty}$. In particular it is a
Cauchy sequence in $B^{s}_{R}(\mathcal{H}^{s})$. In fact we restrict it within
a smaller ball of radius $R/4$. More specifically

$\displaystyle\mathcal{V}^{0}_{k}\in B^{s}_{R/4}(\mathcal{V}^{0}),~{}\forall
k\geq 1,$ (133) $\displaystyle
C_{R}||\mathcal{V}^{0}_{k}-\mathcal{V}^{0}_{k^{{}^{\prime}}}||_{\mathcal{H}^{s}}<\frac{R}{4}~{}\forall
k,k^{{}^{\prime}}\geq 1.$ (134)

We need to prove that the sequence $\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}$
converges in $\mathcal{C}\left([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0})\right)$
and the limit solves the gauge fixed Einstein-Yang-Mills evolution equations
for a suitable $t^{*}<1$. The major difficulty in the PDE setting is that the
closed and bounded balls are not compact in infinite dimensions. We accomplish
this goal by the following three steps.
Lemma 5: There exists a time $t^{*}\in(0,1)$ such that
$\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}\subset
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$.

Proof: This is equivalent to proving that there exists a time $t^{*}\in(0,1)$
such that if
$\\{\mathcal{V}_{k^{{}^{\prime}}}\\}_{k^{{}^{\prime}}=1}^{k}\subset
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$, then $\mathcal{V}_{k+1}\in
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$. For this purpose we use the
following difference equation

$\displaystyle\mathcal{L}_{k}(\mathcal{V}_{k+1}-\mathcal{V}_{1})=\mathcal{F}_{k}-\mathcal{L}_{k}\mathcal{V}_{1}$
(135)

apply the energy estimate (132) to yield

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}$
(136) $\displaystyle\leq
C_{R}\left(||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{1}||_{\mathcal{H}^{s}}+||\mathcal{F}_{k}||_{L^{1}([0,t^{*}];\mathcal{H}^{s})}+||\mathcal{L}_{k}\mathcal{V}_{1}||_{L^{1}([0,t^{*}];\mathcal{H}^{s})}\right)$
$\displaystyle=C_{R}\left(||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{1}||_{\mathcal{H}^{s}}+||\mathcal{F}_{k}||_{L^{1}([0,t^{*}];\mathcal{H}^{s})}+||\mathcal{L}_{k}\mathcal{V}^{0}_{1}||_{L^{1}([0,t^{*}];\mathcal{H}^{s})}\right)$

Now, by construction, we have
$C_{R}||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{1}||_{\mathcal{H}^{s}}<R/4$.
Now we notice that
$\int_{0}^{t^{*}}||\mathcal{F}_{k}||_{\mathcal{H}^{s}}dt\leq
t^{*}||\mathcal{F}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}$ and
$\int_{0}^{t^{*}}||\mathcal{L}_{k}\mathcal{V}^{0}_{1}||_{\mathcal{H}^{s}}dt\leq
t^{*}||\mathcal{L}_{k}\mathcal{V}^{0}_{1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s}})$.
Therefore using the argument of lemma $3$ ($\mathcal{F}$ is a norm preserving
map in a sense that
$||\mathcal{F}(\mathcal{V}_{k})||_{\mathcal{H}^{s}}\lesssim||\mathcal{V}_{k}||_{\mathcal{H}^{s}}$)
there exists a time $t^{*}$ satisfying $0<t^{*}<1$ and such that the following
holds

$\displaystyle
C_{R}t^{*}||\mathcal{F}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}<\frac{R}{4},~{}C_{R}t^{*}||\mathcal{L}_{k}\mathcal{V}^{0}_{1}||_{\mathcal{H}^{s}}<R/4$
(137)

since $\\{\mathcal{V}_{k^{{}^{\prime}}}\\}_{k^{{}^{\prime}}=1}^{k}\subset
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$. This yields

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}<3R/4.$
(138)

However by construction $\mathcal{V}_{1}=\mathcal{V}^{0}_{1}$ and therefore
$\mathcal{V}_{1}\in B^{s}_{R/4}(\mathcal{V}^{0})$ . This yields
$\mathcal{V}_{k+1}\in L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$ for a
suitable time
$t^{*}\in(0,1)$.$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

Lemma 6: There exists a time $t^{*}\in(0,1)$ such that
$\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}$ converges sub-sequentially in
$L^{\infty}([0,t^{*}];\mathcal{H}^{1})$.

Proof: Now we construct the following equation

$\displaystyle\mathcal{L}_{k}(\mathcal{V}_{k+1}-\mathcal{V}_{k^{{}^{\prime}}+1})=\mathcal{F}_{k}-\mathcal{F}_{k^{{}^{\prime}}}+(\mathcal{L}_{k^{{}^{\prime}}}-\mathcal{L}_{k})\mathcal{V}_{k^{{}^{\prime}}+1}$
(139)

and apply the lowest order energy estimate (lemma 1) to yield

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
$\displaystyle\leq
C_{R}\left(||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k^{{}^{\prime}}+1}||_{\mathcal{H}^{1}}+||\mathcal{F}_{k}-\mathcal{F}_{k^{{}^{\prime}}}||_{L^{1}([0,t^{*}];\mathcal{H}^{1})}+||(\mathcal{L}_{k^{{}^{\prime}}}-\mathcal{L}_{k})\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{1}([0,t^{*}];\mathcal{H}^{1})}\right).$

Now we use the Lipsitz property of the map
$\mathcal{F}:\mathcal{H}^{1}\to\mathcal{H}^{1}$ from lemma 4 to yield

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
(140) $\displaystyle\leq
C_{R}\left(||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k^{{}^{\prime}}+1}||_{\mathcal{H}^{1}}+t^{*}C_{L}||\mathcal{V}_{k}-\mathcal{V}_{k^{{}^{\prime}}}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}\right.$
$\displaystyle\left.+t^{*}||(\mathcal{L}_{k^{{}^{\prime}}}-\mathcal{L}_{k})\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}\right).$

Now, given
$N_{k}=N(\mathcal{V}_{k}),X_{k}=X(\mathcal{V}_{k}),\varphi_{k}=\varphi(\mathcal{V}_{k})\in
L^{\infty}([0,t^{*}];H^{s+1})$, using the product estimate (52) with
$t_{1}=1$, $t_{2}=s-2$ and $k=0$, we may estimate
$||(\mathcal{L}_{k^{{}^{\prime}}}-\mathcal{L}_{k})\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
as

$\displaystyle||(\mathcal{L}_{k^{{}^{\prime}}}-\mathcal{L}_{k})\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}\leq
C^{{}^{\prime}}_{L}||\mathcal{V}_{k}-\mathcal{V}_{k^{{}^{\prime}}}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}||\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}.$

This works out in a consistent manner since we have established in the
previous lemma (lemma 5) that $\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}\subset
L^{\infty}([0,t^{*}];B^{s}_{R}(\mathcal{V}^{0}))$. Now we can choose a
$t^{*}\in(0,1)$ (may decrease it if required) such that
$t^{*}C_{L}C^{2}_{R}<\frac{1}{2}$. The previous inequality becomes

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{k^{{}^{\prime}}+1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
(141) $\displaystyle\leq
C_{R}||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k^{{}^{\prime}}+1}||_{\mathcal{H}^{1}}+\frac{1}{2}||\mathcal{V}_{k}-\mathcal{V}_{k^{{}^{\prime}}}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}.$

Setting $k^{{}^{\prime}}=k-1$ we obtain

$\displaystyle||\mathcal{V}_{k+1}-\mathcal{V}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
(142) $\displaystyle\leq
C_{R}||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k}||_{\mathcal{H}^{1}}+\frac{1}{2}||\mathcal{V}_{k}-\mathcal{V}_{k-1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$

which yields

$\displaystyle\sum_{k=2}^{\infty}||\mathcal{V}_{k+1}-\mathcal{V}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
(143) $\displaystyle\leq
2C_{R}\sum_{k=1}^{\infty}||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k}||_{\mathcal{H}^{1}}+||\mathcal{V}_{2}-\mathcal{V}_{1}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}.$

Now we have control over the initial sequence (a Cauchy sequence) and
therefore we can extract a sub-sequence to yield
$2C_{R}\sum_{k=1}^{\infty}||\mathcal{V}^{0}_{k+1}-\mathcal{V}^{0}_{k}||_{\mathcal{H}^{1}}<\frac{R}{2}$
which leads to

$\displaystyle\sum_{k=2}^{\infty}||\mathcal{V}_{k+1}-\mathcal{V}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}\leq
R$ (144)

concluding the sub-sequential convergence of
$\\{\mathcal{V}_{k}\\}_{k=1}^{\infty}$ in
$L^{\infty}([0,t^{*}];\mathcal{H}^{1})$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

The remaining steps are showing that the limit is a solution of the gauge
fixed Einstein-Yang-Mills field equations and that it lies in the space
$\mathcal{C}([0,t^{*}];\mathcal{H}^{s}),~{}s>\frac{n}{2}+1$. To accomplish
this we need to use the notion of weak continuity. The space of weakly
continuous functions on $[0,t^{*}]$ with values in $H^{s}$ is defined as
follows

$\displaystyle C_{W}([0,t^{*}];H^{s}):=\left\\{f\in H^{s}|\langle
f,\Psi\rangle\in C([0,t^{*}])~{}\forall\Psi\in H^{-s}\right\\},$ (145)

Here $H^{-s}$ is the dual space of $H^{s}$. We invoke the following lemma from
[1] proof of which follows using a standard ‘$\epsilon/3$’ argument.

Lemma [1]: Let $\\{f_{k}\\}_{k=1}^{\infty}\subset
L^{\infty}([0,t^{*}];H^{s}),~{}s>0$ be a bounded sequence and assume
$\\{f_{k}\\}_{k=1}^{\infty}$ is Cauchy in
$L^{\infty}([0,t^{*}];H^{s^{{}^{\prime}}})$ for some
$s^{{}^{\prime}}\in[0,s)$. Then $\exists f\in C_{W}([0,t^{*}];H^{s})\cap
L^{\infty}([0,t^{*}];H^{s})$ such that $\\{f_{k}\\}_{k=1}^{\infty}$ converges
weakly to $f$ uniformly in $L^{\infty}([0,t^{*}]$.

Utilizing this lemma together with lemma 6, we obtain
$\mathcal{V}_{k}\to\mathcal{V}\in C_{W}([0,t^{*}];H^{s})$. Now using this
result, we show that this limit $\mathcal{V}$ satisfies the evolution equation
$\mathcal{L}\mathcal{V}=\mathcal{F}$.

Lemma 7: Weak limit $\mathcal{V}$ of $\mathcal{V}_{k}$ solves
$\mathcal{L}\mathcal{V}=\mathcal{F}$

Proof: Let us write the following

$\displaystyle\mathcal{L}[\mathcal{V}]\mathcal{V}-\mathcal{F}[\mathcal{V}]$
$\displaystyle=$
$\displaystyle(\mathcal{L}[\mathcal{V}]\mathcal{V}-\mathcal{L}[\mathcal{V}_{k}]\mathcal{V})+\mathcal{L}[\mathcal{V}_{k}](\mathcal{V}-\mathcal{V}_{k+1})+(\mathcal{L}[\mathcal{V}_{k}]\mathcal{V}_{k+1}-\mathcal{F}[\mathcal{V}])$
$\displaystyle=$
$\displaystyle(\mathcal{L}[\mathcal{V}]\mathcal{V}-\mathcal{L}[\mathcal{V}_{k}]\mathcal{V})+\mathcal{L}[\mathcal{V}_{k}](\mathcal{V}-\mathcal{V}_{k+1})+(\mathcal{F}[\mathcal{V}_{k}]-\mathcal{F}[\mathcal{V}])$

Now similar to the calculations associated with the previous lemma, we may
estimate the first term as

$\displaystyle||\mathcal{L}[\mathcal{V}]\mathcal{V}-\mathcal{L}[\mathcal{V}_{k}]\mathcal{V}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{0})}\lesssim||\mathcal{V}-\mathcal{V}_{k}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{s})}||\mathcal{V}||_{L^{\infty}([0,t^{*}];\mathcal{H}^{1})}$
(146)

but $\mathcal{V}_{k}$ converges to $\mathcal{V}$ in
$L^{\infty}([0,t^{*}];\mathcal{H}^{s})$ as $k\to\infty$. Therefore
$(\mathcal{L}[\mathcal{V}]\mathcal{V}-\mathcal{L}[\mathcal{V}_{k}]\mathcal{V})\to
0$ in $L^{\infty}([0,t^{*}];\mathcal{H}^{0})$ as $k\to\infty$. Similarly using
Lipsitz property of $\mathcal{F}$, it follows that
$(\mathcal{F}[\mathcal{V}_{k}]-\mathcal{F}[\mathcal{V}])\to 0$ in
$L^{\infty}([0,t^{*}];\mathcal{H}^{1})$ as $k\to\infty$. There is a little
subtlety in the term
$\mathcal{L}[\mathcal{V}_{k}](\mathcal{V}-\mathcal{V}_{k+1})$. We want to show
that this term approaches 0 in $L^{\infty}([0,t^{*}];\mathcal{H}^{0})$. In
order for this to happen terms of the type
$g^{ij}_{k}\partial_{i}\partial_{j}(g-g_{k+1})$ and
$g^{ij}_{k}\partial_{i}\partial_{j}(A-A_{k+1})$ should converge in $H^{-1}$.
But this is immediate using the product estimates (52,55) and the duality of
$H^{1}$ and $H^{-1}$ i.e., the multiplication $H^{s}\times H^{-1}\to H^{-1}$
is continuous for $s>\frac{n}{2}+1$. On the other hand, we notice that the
left hand side is independent of $k$. Therefore,
$\mathcal{L}[\mathcal{V}](\mathcal{V})=\mathcal{F}[\mathcal{V}]$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$
Now we prove the ‘in time’ continuity of the solutions.

Lemma 8: Solution $\mathcal{V}$ is continuous as a map
$t\mapsto\mathcal{V}(t)$.

Proof: We have established that $t\mapsto\mathcal{V}(t)$ is weakly continuous.
In order to prove strong continuity we need to prove left and right continuity
at every point in the interval of local existence i.e., $[0,t^{*}]$. In other
words, we must show that

$\displaystyle\lim_{h\to
0}\sup||\mathcal{V}(t+h)-\mathcal{V}(t)||_{\mathcal{H}^{s}}=0=\lim_{h\to
0}\sup||\mathcal{V}(t-h)-\mathcal{V}(t)||_{\mathcal{H}^{s}}.$ (147)

In this case, we need only show $\lim_{h\to
0}\sup||\mathcal{V}(t+h)-\mathcal{V}(t)||_{\mathcal{H}^{s}}=0$ since the
argument can be repeated by reversing the orientation of time. By a re-
parametrization of the time, we need only show the right continuity at $t=0$
i.e., $\lim_{t\to
0}\sup||\mathcal{V}(t)-\mathcal{V}(0)||_{\mathcal{H}^{s}}=0$. This may be done
using a lemma of functional analysis ([45]) which states that if
$y_{n}\rightharpoonup y$ and $||y_{n}||\to||y||$, then $||y_{n}-y||\to 0$.
Now, we have shown earlier that $\mathcal{V}(t)\in
C_{W}([0,t^{*}];\mathcal{H}^{s})$. Therefore we need only show that
$\lim_{t\to
0}\sup||\mathcal{V}(t)||_{\mathcal{H}^{s}}=||\mathcal{V}(0)||_{\mathcal{H}^{s}}$.
But this holds by the energy estimates derived in lemma 1 and 2. Therefore, we
obtain
$\mathcal{V}\in\mathcal{C}([0,t^{*}];\mathcal{H}^{s}),~{}s>\frac{n}{2}+1$.$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

###  5.2 Uniqueness of a solution and its continuous dependence on the data

It is straightforward to establish the uniqueness of the solution
$\mathcal{V}$. Let us assume that there exists two solutions $\mathcal{V}_{1}$
and $\mathcal{V}_{2}$ such that $\mathcal{V}_{1}(0)=\mathcal{V}_{2}(0)$. We
need to show that $\mathcal{V}_{1}(t)=\mathcal{V}_{2}(t)~{}\forall
t\in[0,t^{*}]$. But this follows by the energy inequalities and the
calculations of the lemma 5 since
$||\mathcal{V}_{1}(0)-\mathcal{V}_{2}(0)||_{\mathcal{H}^{s}}=0$. More
precisely, one may obtain an energy inequality $\partial_{t}E_{12}\lesssim
E_{12}$ for $\mathcal{V}_{1}(t)-\mathcal{V}_{2}(t)$ which yields
$E_{12}(t)=0~{}\forall t\in[0,t^{*}]$ if $E_{12}(0)=0$ ($E_{12}(t)$ is the
energy associated with the difference
$\mathcal{V}_{1}(t)-\mathcal{V}_{2}(t)$). For the continuous dependence of the
unique solution on the initial data, we need to show that the map
$\mathcal{V}(0)\mapsto\mathcal{V}(t;\mathcal{V}(0))$ is continuous. For this,
we refer to [1] instead of repeating it once again since it will be exactly
same. This works out since the extra terms that arise due to the inclusion of
the Yang-Mills fields preserve the crucial characteristics of the functional
$\mathcal{F}$ e.g., $\mathcal{F}$ remains Lipschitz as shown in lemma 3 and 4.
Instead, we focus on the elliptic estimates and conservation of gauges and
constraints since now there are one additional constraint (Gauss law
constraint for the electric field) and one additional gauge condition
(generalized Coulomb gauge condition).

##  6 Elliptic Estimates

The remaining task is to show that the gauge variables $g,N,$ and $\varphi$
are determined in terms of $(g,k,A,\mathcal{E})$ and satisfy the regularity
$H^{s+1}\times H^{s+1}\times H^{s+1}$. Let us recall the elliptic equations
satisfied by these gauge variables. The lapse function $N$, the shift vector
field $X$, and the shifted potential $\varphi$ satisfy

$\displaystyle-g^{ij}\nabla_{i}\nabla_{j}N+\underbrace{\left(|k|^{2}_{g}+\frac{n-2}{n-1}g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}+\frac{1}{2(n-1)}g^{kl}g^{mn}F^{a}~{}_{bkm}F^{b}~{}_{aln}\right)}_{\geq
0}N$ $\displaystyle=1,$
$\displaystyle\Delta_{g}X^{i}-R^{i}_{j}X^{j}+L_{X}V^{i}=-2k^{ij}\nabla_{j}N-2Ng^{ik}g^{lj}F^{a}~{}_{bkl}\mathcal{E}^{b}~{}_{aj}+\tau\nabla^{i}N$
(148)
$\displaystyle+(2Nk^{jk}-2\nabla^{j}X^{k})(\Gamma[g]^{i}_{jk}-\Gamma[\gamma]^{i}_{jk}),$

and

$\displaystyle
g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}~{}_{b}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}~{}_{b}+g^{ij}\nabla_{i}N\mathcal{E}^{a}~{}_{bj}-Ng^{ij}[A_{i},\mathcal{E}_{j}]^{a}~{}_{b}$
$\displaystyle+g^{ij}[\hat{A}_{i},N\mathcal{E}_{j}]^{a}~{}_{b}+g^{ij}(\Gamma[g]^{k}_{ij}-\Gamma[\gamma]^{k}_{ij})N\mathcal{E}^{a}~{}_{bk}-X^{k}\nabla[\gamma]_{k}g^{ij}\nabla[\gamma]_{i}(A^{a}~{}_{bj}$
$\displaystyle-\hat{A}^{a}~{}_{bj})-X^{k}\nabla[\gamma]_{k}g^{ij}[\hat{A}_{i},A_{j}-\hat{A}_{j}]^{a}~{}_{b}+g^{ij}\nabla[\gamma]_{i}X^{k}\nabla[\gamma]_{k}A^{a}~{}_{bj}$
$\displaystyle+g^{ij}\nabla[\gamma]_{i}A^{a}~{}_{bk}\nabla[\gamma]_{j}X^{k}+g^{ij}A^{a}~{}_{bk}\nabla[\gamma]_{i}\nabla[\gamma]_{j}X^{k}-g^{ij}R[\gamma]^{l}_{jik}A^{a}~{}_{bl}X^{k}$
$\displaystyle+g^{ij}[\hat{A}_{i},A_{k}]^{a}~{}_{b}\nabla[\gamma]_{j}X^{k}+(2Nk^{ij}-\nabla^{i}X^{j}-\nabla^{j}X^{i})\nabla[\gamma]^{\hat{A}}_{i}(A^{a}~{}_{bj}-\hat{A}^{a}~{}_{bj})=0,$

respectively. With $u=(N,X,\varphi)$, this system is of the following type

$\displaystyle\mathcal{J}(g,k,A,Dg,DA)u=\mathcal{R}(g,k,A,Dg,DA),$ (149)

where the second order elliptic operator $\mathcal{J}$ is of the form

$\displaystyle\mathcal{J}(g,k,A,Dg,DA)u=g^{ij}\partial_{i}\partial_{j}u+b^{i}(g,k,A,Dg,DA)\partial_{i}u+c(g,k,A,Dg,DA)u,$

where $b$ and $c$ are smooth functionals of their arguments. If $\mathcal{J}$
is is an isomorphism between the function spaces $H^{s+1}$ and $H^{s-1}$, then
the standard elliptic theory yields an estimate of the type

$\displaystyle||u||_{H^{s+1}}\leq
C(||\mathcal{J}u||_{H^{s-1}}+||u||_{H^{s-1}}).$ (150)

Following the isomorphism property of the operator $\mathcal{J}$ and the
compact embedding of $H^{s+1}$ into $H^{s-1}$ (for compact manifolds), one may
obtain an estimate of the type (see [1] for the logic behind this argument)

$\displaystyle||u||_{H^{s+1}}\leq C||\mathcal{J}u||_{H^{s-1}}.$ (151)

Counting the number of derivatives and using the multiplication property of
the Sobolev spaces, one controls the source term
$||\mathcal{J}u||_{H^{s-1}}=||\mathcal{R}||_{H^{s-1}}$ yielding $u\in H^{s+1}$
i.e., $(N,X,\varphi)\in H^{s+1}\times H^{s+1}\times H^{s+1}$ for
$s>\frac{n}{2}+1$.

The remaining task is to argue that the operator $\mathcal{J}$ is an
isomorphism between the suitable function spaces. We consider each equation
separately. First we show that the kernel of the map
$\mathcal{Q}:N\mapsto-g^{ij}\nabla_{i}\nabla_{j}N+\left(|k|^{2}_{g}+\frac{n-2}{n-1}g^{kl}\mathcal{E}^{a}~{}_{bk}\mathcal{E}^{b}~{}_{al}+\frac{1}{2(n-1)}g^{kl}g^{mn}F^{a}~{}_{bkm}F^{b}~{}_{aln}\right)N$
is trivial. A straightforward calculation by integration by parts yields
$\ker(\mathcal{Q})=\\{0\\}$ since the entity under the bracket has good sign.
The case of the shift vector field is slightly more subtle. Notice that
$\Delta_{g}X^{i}-R^{i}_{j}X^{j}$ contains a second derivative term of the
metric of the type $L_{X}V^{i}$ and therefore requires a $H^{s+2}$ regularity
of $X$ for the metric to be in $H^{s}$. However, in spatial harmonic gauge
(36), this dangerous second derivative (of metric) term vanishes point-wise
sense since $V^{i}\equiv 0$ in this gauge. For the detailed calculations, see
lemma $3.2$ of [1]. The isomorphism property of the map
$X^{i}\mapsto\Delta_{g}X^{i}-R^{i}_{j}X^{j}+L_{X}V^{i}+2\nabla^{j}X^{k}(\Gamma[g]^{i}_{jk}-\Gamma[\gamma]^{i}_{jk})$
holds if one assumes $||g-\gamma||_{H^{s}}<\epsilon$ (this construction is
possible since $C^{\infty}$ is dense in Sobolev spaces). This was proved in
[1] by considering the pull back of the tensor fields along the flow of the
shift vector field and assuming $\gamma$ to have negative sectional curvature.
Later [24] sketched an argument (see their appendix) relaxing the restriction
of $\gamma$ to have negative sectional curvature. Therefore, in the current
context, the map
$X^{i}\mapsto\Delta_{g}X^{i}-R^{i}_{j}X^{j}+L_{X}V^{i}+2\nabla^{j}X^{k}(\Gamma[g]^{i}_{jk}-\Gamma[\gamma]^{i}_{jk})$
is an isomophism between $H^{s+1}$ and $H^{s-1}$ for
$||g-\gamma||_{H^{s}}<\epsilon$ for $\epsilon>0$.

A similar problem arises in the case of the elliptic equation for the Yang-
Mills gauge variable $\varphi$. In order to obtain a unique
$\varphi=\varphi(g,k,A,\mathcal{E})$, one must establish the isomorphism
property of the map $\varphi^{a}~{}_{b}\mapsto
g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}~{}_{b}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}~{}_{b}$.
If this map ceases to be an isomorphism, then the gauge orbit develops
directions of tangency to the gauge slice or the gauge orbit intersects the
slice more than once. This is turn introduces degeneracies and defies the
whole purpose of gauge fixing. This is the so called Gribov ambiguity for
which a substantial amount of study exists in the literature [33, 37, 35].
Exactly similar problems occur in the spatial harmonic gauge of gravity as
well [34]. Similar to the gravity case if the background connection $\hat{A}$
is chosen close to the dynamical connection $A$ in $H^{s}$, then the map
$\varphi^{a}~{}_{b}\mapsto
g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}\varphi^{a}~{}_{b}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}\varphi]^{a}~{}_{b}$
is proven to be an isomorphism (see [36, 33]). This can roughly be seen from
the principal part of the equation. Notice that the operator
$g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}$ has a definite sign
(negative) but the operator $g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}]$ does
not. Therefore for small enough $(A-\hat{A})$ (in a suitable function space),
the operator
$g^{ij}\nabla[\gamma]^{A}_{i}\nabla[\gamma]^{A}_{j}+g^{ij}[\hat{A}_{i}-A_{i},\nabla^{A}_{j}]$
has an overall sign and consequently is invertible. Let us now rationalize the
choice of $\hat{A}$ (or $\gamma$) such that
$||A(t)-\hat{A}||_{H^{s}}<\epsilon$ (or $||g(t)-\gamma||_{H^{s}}<\epsilon$)
for a sufficiently small $\epsilon$. Let us consider that the initial data
$(g_{0},k_{0},A_{0},\mathcal{E}_{0})$ solves the constraint equations.
Initially, the background connection and metric are chosen to be $A_{0}$ and
$g_{0}$, respectively. This trivially satisfies the smallness conditions
$||A(t)-\hat{A}||_{H^{s}}<\epsilon$ and $||g(t)-\gamma||_{H^{s}}<\epsilon$.
Therefore, solving the elliptic equations, we also obtain the gauge variables
$(N_{0},X_{0},\varphi_{0})$. Now we evolve the fields up to a time $\hat{t}$
within which the smallness condition is preserved i.e., the evolution time
$\hat{t}$ is obtained by imposing the condition that
$||g(\hat{t})-\gamma||_{H^{s}}\leq\epsilon/2$ and
$||A(\hat{t})-\hat{A}||_{H^{s}}\leq\epsilon/2$. Once this range is exhausted,
we choose the background connection and metric to be the instantaneous values
of the dynamical connection and the metric at time $\hat{t}$ and repeat the
evolution in the same way. The elliptic regularity of these equations yields
unique
$N,X,\varphi=N(g,k,A,\mathcal{E}),X(g,k,A,\mathcal{E}),\varphi(g,k,A,\mathcal{E})$
belonging to $H^{s+1}\times H^{s+1}\times H^{s+1}$.

##  7 Conservation of constraints and gauges along the solution curve

In addition to proving the local well-posedness of the Einstein-Yang-Mills
evolution equations, we must show that the constraints and the gauge
conditions are preserved along the solution curve
$t\mapsto(g(t),k(t),A(t),\mathcal{E}(t),N(t),X(t),\varphi(t))$. In general,
the equations satisfied by the constraints (see [46]) do not exhibit a
particular type in an arbitrary gauge. This hinders one to prove the
preservation of the constraints in a straightforward gauge-invariant way.
However, one would expect that such preservation of the constraints must hold
due to the Bianchi identities. Andersson and Moncrief [1] showed that in CMCSH
gauge, the constraints and the associated entities defining the gauge
conditions (that vanish if the gauge conditions are imposed) satisfy a coupled
nonlinear hyperbolic equation. Using standard energy arguments, they proved if
the gauges and constraints are satisfied initially, then they are preserved
along any solution curve. In the current case, we have additional terms
arising from the Yang-Mills sector. The full set of gauges and constraints are
defined as follows

$\displaystyle T:=\tr_{g}k-t,$ (152) $\displaystyle
S^{i}:=g^{kl}(\Gamma[g]^{i}_{kl}-\Gamma[\gamma]^{i}_{kl}),$ (153)
$\displaystyle
H:=R(g)-|k|^{2}_{g}+(tr_{g}k)^{2}-\mathcal{E}\cdot\mathcal{E}-\frac{1}{2}F\cdot
F-\nabla_{i}S^{i},$ (154) $\displaystyle
M_{i}:=\nabla_{i}tr_{g}k-2\nabla^{j}k_{ij}+2F_{ij}\cdot\mathcal{E}^{j},$ (155)
$\displaystyle
G^{a}_{b}:=g^{ij}\nabla[\gamma]^{\hat{A}}_{j}(A^{a}~{}_{bi}-\hat{A}^{a}~{}_{bi}),$
(156) $\displaystyle
L^{a}~{}_{b}:=g^{ij}\nabla_{i}\mathcal{E}^{a}~{}_{bj}+g^{ij}[A_{i},\mathcal{E}_{j}]^{a}~{}_{b}$
(157)

Explicit calculations using the equations of motion yield the following set of
evolution equations for $(T,S^{i},H,M_{i},G,L)$

$\displaystyle\partial_{t}T=L_{X}T+NH,$ (158)
$\displaystyle\partial_{t}H=L_{X}H+\nabla_{i}NM^{i}+2NH\tr_{g}k+Nk^{ij}(\nabla_{i}S_{j}+\nabla_{j}S_{i})+L_{S}(N\tr_{g}k)$
$\displaystyle-N\Delta_{g}T+2\nabla_{i}N\nabla^{i}T,$
$\displaystyle\partial_{t}S^{i}=L_{X}S^{i}+NM^{i},$ (159)
$\displaystyle\partial_{t}M_{i}=L_{X}M_{i}+H\nabla_{i}N+N\tr_{g}kM_{i}-N(\Delta_{g}S_{i}-\underbrace{R_{ij}S^{j}}_{I})$
(160)
$\displaystyle+N\tr_{g}k\nabla_{i}T+\nabla^{j}N(\nabla_{i}S_{j}+\nabla_{j}S_{i}),$
$\displaystyle\partial_{t}G^{a}~{}_{b}=X^{k}\nabla_{k}G^{a}~{}_{b}+NL^{a}~{}_{b},$
(161)
$\displaystyle\partial_{t}L^{a}~{}_{b}=X^{k}\nabla_{k}L^{a}~{}_{b}-N\Delta_{g}G^{a}~{}_{b}+N\tr_{g}kL^{a}~{}_{b}+\nabla^{i}N\nabla_{i}G^{a}~{}_{b}$
(162) $\displaystyle+N[A_{i},\nabla^{i}G]^{a}~{}_{b}+[L,\varphi]^{a}~{}_{b}$

Notice that this set of equations describe a coupled hyperbolic system of
equations for the variables $(T,S^{i},H,M_{i},G,L)$. Notice that these
equations are exactly similar to those derived by [1]. The only difference is
the appearance of two additional equations from the Yang-Mills sector.
Remarkably, the evolution equations for the Yang-Mills gauge conditions and
constraints are decoupled from the gravity sector. The evolution equations are
of the standard hyperbolic form. We may use the standard energy argument to
show that if $(T(0)=0,S^{i}(0)=0,H(0)=0,M_{i}(0)=0,G(0)=0,L(0)=0)$, then
$(T(t),S^{i}(t),H(t),M_{i}(t),G(t),L(t))\equiv(0,0,0,0,0,0)$ along the
solution curve $t\mapsto(g(t),k(t),A(t),\mathcal{E}(t),N(t),X(t),\varphi(t))$.
Let us define a wave equation type of energy for the above system of evolution
equations as follows

$\displaystyle E_{cg}:=$ (163)
$\displaystyle\frac{1}{2}\int_{\Sigma}\left(T^{2}+|\nabla
T|^{2}_{g}+H^{2}+G^{a}~{}_{b}G^{b}~{}_{a}+g^{ij}g^{kl}\nabla_{i}G^{a}~{}_{bk}\nabla_{j}G^{b}~{}_{al}\right.$
$\displaystyle\left.+L^{a}~{}_{b}L^{b}~{}_{a}+|M|^{2}_{g}+|S|^{2}_{g}+|\nabla
S|^{2}_{g}\right)\mu_{g},$

where the dot indicates a positive definite inner product on the fibers of the
gauge bundle. In the evaluation of the time derivative of the energy
functional, there will be one borderline problematic term that requires
careful control using a product estimate. Naturally, the principal terms
cancel each other via integration by parts. The Lie derivative terms may be
handled by the integration by parts argument as well (similar to the procedure
used in the proof of lemma 1). Assuming $(g,N)\in H^{s}\times
H^{s+1},~{}s>\frac{n}{2}+1$ and the estimate satisfied the by Ricci tensor

$\displaystyle||Ric||_{H^{s-2}}\lesssim||g||_{H^{s}}(1+||g||_{H^{s}}),$ (164)

we may control the borderline problematic term $||NR_{ij}S^{i}M^{j}||_{L^{1}}$
as follows

$\displaystyle||NR_{ij}S^{i}M^{j}||_{L^{1}}\lesssim||N||_{L^{\infty}}||Ric~{}S||_{L^{2}}||M||_{L^{2}}$
$\displaystyle\lesssim||N||_{L^{\infty}}||Ric||_{H^{s-2}}||S||_{H^{1}}||M||_{L^{2}}$
$\displaystyle\lesssim||N||_{L^{\infty}}||g||_{H^{s}}(1+||g||_{H^{s}}||S||_{H^{1}}||M||_{L^{2}}.$

Here we have once again used the product estimate (52). Using Holder’s
inequality and the fact that $(k,A,X,\varphi)\in H^{s-1}\times H^{s}\times
H^{s+1}\times H^{s+1},~{}s>\frac{n}{2}+1$, we obtain following energy
inequality

$\displaystyle\partial_{t}E_{cg}\lesssim E_{cg}$ (165)

which yields

$\displaystyle E_{cg}(0)=0\Rightarrow E_{cg}(t)=0~{}\forall t\in[0,t^{*}].$
(166)

This completes the proof of the conservation of gauges and constraints. Now we
can state the main theorem as follows
Local Well-posedness Theorem: Let $(g_{0},k_{0},A_{0},\mathcal{E}_{0})\in
H^{s}\times H^{s-1}\times H^{s}\times H^{s-1},~{}s>\frac{n}{2}+1$ be the
initial data for the Cauchy problem of the Einstein-Yang-Mills evolution
equations (3-48) in Constant Mean extrinsic Curvature Spatial Harmonic
Generalized Coulomb (CMCSHGC) gauge satisfying the constraint equations
(29-31). This CMCSHGC Cauchy problem is well posed in
$\mathcal{C}\left([0,t^{*}];H^{s}\times H^{s-1}\times H^{s}\times
H^{s-1}\right)$. In particular, there exists a time $t^{*}>0$ dependent on
$||g_{0}||_{H^{s}},||k_{0}||_{H^{s-1}},||A_{0}||_{H^{s}},||\mathcal{E}_{0}||_{H^{s-1}}$
such that the solution map
$(g_{0},k_{0},A_{0},\mathcal{E}_{0})\mapsto(g(t),k(t),A(t),\mathcal{E}(t),N(t),X(t),\varphi(t))$
is continuous as a map

$\displaystyle H^{s}\times H^{s-1}\times H^{s}\times H^{s-1}$ (167)
$\displaystyle\to H^{s}\times H^{s-1}\times H^{s}\times H^{s-1}\times
H^{s+1}\times H^{s+1}\times H^{s+1}.$

Let $t^{*}$ be the maximal time of existence of a solution to the CMCSHGC
Cauchy problem with data $(g_{0},k_{0},A_{0},\mathcal{E}_{0})$, then either
$t^{*}=\infty$ or

$\displaystyle\lim_{t\to
t^{*}}\sup\max\left(C(\Lambda_{t}),||D\hat{g}(t)||_{L^{\infty}},||DA(t)||_{L^{\infty}},||\mathcal{E}(t)||_{L^{\infty}}\right)=\infty.$
(168)

In addition the gauges and constraints are preserved along the solution curve
$t\mapsto(g(t),k(t),A(t),\mathcal{E}(t),N(t),X(t),\varphi(t))$ if they are
satisfied initially.

##  8 Concluding Remarks

We have established a local well-posedness theorem for the coupled Einstein-
Yang-Mills equations without any symmetry assumption utilizing techniques
developed by [1]. This is a routine procedure before studying the global
properties of solutions. There are, however, a few subtleties present in the
current context of the Yang-Mills fields coupled to gravity. Firstly, the
divergence of the Yang-Mills curvature $\nabla^{j}F^{a}~{}_{bij}$ that appears
in the equation of motion should be expressed in terms of the background
metric $\gamma$ instead of the dynamical metric $g$ since the latter produces
a Ricci curvature term in the evolution equation of the electric field
obstructing the closure of the energy estimates. Secondly, it is necessary to
choose the gauge variable of the Yang-Mills sector to be the shifted potential
$\varphi:=A_{0}-A\cdot X$ instead of the usual $A_{0}$ (even in the temporal
gauge one must impose $\varphi=0$ instead of $A_{0}=0$) in order to close the
energy estimates. Lastly, the Gribov degeneracies associated with the spatial
harmonic and generalized Coulomb gauge conditions needed to be addressed. The
remaining arguments of proving a local well-posedness result are handled by
the technique developed by [1].

The continuation criteria of the coupled Einstein-Yang-Mills system is the
boundedness of the entity
$\mathcal{W}(t):=\max\left(C(\Lambda_{t}),||D\hat{g}(t)||_{L^{\infty}},||DA(t)||_{L^{\infty}},||\mathcal{E}(t)||_{L^{\infty}}\right)$
i.e., as long as $\mathcal{W}(t)$ remains finite, the required Sobolev norms
remain finite allowing one to continue the solution in the future time
direction. Notice that $\mathcal{W}$ contains all the derivatives of the
spacetime metric $\hat{g}$ and the connection $A$. However, one may be able to
relax this condition up to a certain extent. In the pure gravity case, [31]
was able to obtain continuation criteria that did not require all derivatives
of the metric $\hat{g}$ but rather only time derivative of the spatial metric
and spatial derivatives of the lapse function and the shift vector field. In
the pure Yang-Mills case on the Minkowski background, Eardley and Moncrief
[28] showed that the boundedness of a point-wise gauge-invariant norm of the
Yang-Mills curvature was sufficient to continue the solutions. Later they
utilized a light-cone estimate technique to show that the point-wise norm of
the Yang-Mills curvature can not blow up in finite time and therefore obtained
the global existence; in fact, a similar result persists on a globally
hyperbolic background as well (work in progress with Prof. Moncrief).
Recently, [47] proved the global existence of critically non-linear wave
fields on background curved spacetimes which is similar in spirit with our
current course of action, that is, first prove a local existence theorem and
then obtain a point-wise bound for the wavefield using the light cone estimate
technique thereby satisfying the continuation criteria. Unlike pure Yang-Mills
or wavefields propagating on a Minkowski spacetime (or a background
spacetime), gravity coupled to the Yang-Mills system is substantially more
delicate. One can not in general prove a global existence theorem for the
coupled system but rather improve the continuation criteria. Utilizing
Moncrief’s light cone estimate technique for the coupled system, we hope to
obtain a point-wise bound on the spacetime curvature (or its trace-less part
the Weyl curvature) and Yang-Mills curvature given a bound on a certain entity
such as the strain tensor associated with a unit normal field to a CMC
hypersurface. The wave equations for space-time curvature or the Yang-Mills
curvature contain non-linearities that satisfy the so-called null condition (a
cancellation occurs between terms that cannot be controlled by the associated
energies) and therefore such an improved continuation criterion is naturally
expected. We note that such results for the vacuum Einstein equations [31] and
Einstein coupled to Klein-Gordon and Maxwell equations [32] exist in the
literature. These results were obtained by a method that is substantially
different from Moncrief’s light cone estimate. We hope that we will be able to
apply the light cone estimate technique to obtain an improved continuation
criterion of the coupled Einstein-Yang-Mills system where the current work
marks the completion of the first step.

##  9 Acknowledgement

P.M would like to thank Prof. Vincent Moncrief for numerous useful discussions
related to this project and for his help improving the manuscript. P.M would
also like to thank Prof. Yau for igniting the interest in coupled Einstein-
Yang-Mills system. This work was supported by CMSA at Harvard University.

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