# Global existence of the solution to Einstein-Yang-Mills-Higgs equations with
small initial datum

Zonglin Jia Boling Guo

###### Abstract.

The problem involved in this paper is the global existence of the solution to
the $\mathfrak{su}(2)$-Einstein-Yang-Mills-Higgs(EYMH) equation. The approach
we employ stems from H. Lindblad and I. Rodnianski and is dependent of wave
coordinates and Lorentzian gauge conditions. Our main conclusion is that the
EYMH system admits global existence provided the initial datum are
sufficiently small. To the best of our knowledge, there is no similar result
in the area of EYMH equations.

Key words: Einstein-Yang-Mills-Higgs equation, small initial datum, global
existence

##  1\. Introduction

Recently many mathematicians are concerned about the EYMH equation. They,
following the idea of D. Christodoulou (see e.g. [2]), usually reduce this
equation under special metrics to get global existence. In our present
article, we take into account of general metric solving the EYMH system and
are going to get global existence with small initial datum.

This paper is inspired by the work in [12], which give a new proof of the
global stability of Minkowski space originally established by Christodoulou
and Klainerman in [5]. The smart method of H. Lindblad and I. Rodnianski is
based on the wave coordinates, which play a critical role in giving some more
exquisite estimates. Following their idea, we also employ wave coordinates and
Lorentzian gauges. Like [12] our frame of the article is the contradiction
argument.

Throughout the paper, the same indices appearing twice means summing it.
Besides, we also appoint that, when denoting superscripts or subscripts, the
Greek letters such as $\alpha,\beta,\gamma,\cdots$ belong to $\\{0,1,2,3\\}$,
while the Latin letters $i,j,k,\cdots$ are in $\\{1,2,3\\}$.

We consider the following equations on $(\mathbb{R}^{4},g)$

$\displaystyle\left\\{\begin{array}[]{rcl}&Ric_{\alpha\beta}-\frac{1}{2}R\cdot
g_{\alpha\beta}=T_{\alpha\beta}&\quad\quad\text{(Einstein equation)}\\\
&g^{\lambda\mu}(\nabla_{\lambda}\tilde{F}_{\mu\alpha}+[A_{\lambda},\tilde{F}_{\mu\alpha}])=\tilde{J}_{\alpha}&\quad\quad\text{(Yang-
Mills equation)}\\\
&g^{\lambda\mu}\hat{\nabla}_{\lambda}\hat{\nabla}_{\mu}\Phi=V^{\prime}(|\Phi|^{2})\Phi,&\quad\quad\text{(Higgs
equation)}\end{array}\right.$

where $g$ is a Lorentzian metric with signature $(-,+,+,+)$, $Ric$ and $R$ are
Ricci tensor and scalar curvature of $g$ respectively. Furthermore, we assume
that
$\tilde{F}_{\alpha\beta}:=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}+[A_{\alpha},A_{\beta}]$
and $T_{\alpha\beta}$ is given by

$\displaystyle T_{\alpha\beta}$ $\displaystyle:=$
$\displaystyle\langle\tilde{F}_{\alpha}^{\lambda},\tilde{F}_{\beta\lambda}\rangle-\frac{1}{4}g_{\alpha\beta}\langle\tilde{F}^{\lambda\mu},\tilde{F}_{\lambda\mu}\rangle+(\hat{\nabla}_{\alpha}\Phi)^{{\dagger}}\hat{\nabla}_{\beta}\Phi+(\hat{\nabla}_{\beta}\Phi)^{{\dagger}}\hat{\nabla}_{\alpha}\Phi$
$\displaystyle-
g_{\alpha\beta}\\{g^{\rho\sigma}(\hat{\nabla}_{\rho}\Phi)^{{\dagger}}\hat{\nabla}_{\sigma}\Phi+V(\Phi^{{\dagger}}\Phi)\\},$

where we have raised the indices by $(g^{\alpha\beta})$ which is the inverse
of the metric matrix $(g_{\alpha\beta})$. $A$ is a 1-form, called the Yang-
Mills potential, defined on $\mathbb{R}^{4}$ with values in the Lie algebra
$\mathfrak{su}(2)$; $\Phi\in C^{1}(\mathbb{R}^{4},\mathbb{C}^{2})$ is a
scalar-multiplet field, called the Higgs field. $\nabla$ denotes the covariant
derivative with respect to the metric $g$, and $[\cdot,\cdot]$ means the Lie
bracket of the Lie algebra $\mathfrak{su}(2)$. Moreover,
$\hat{\nabla}_{\alpha}\Phi$ is the gauge covariant derivative of the complex
doublet Higgs field, defined by

$\displaystyle\hat{\nabla}_{\alpha}\Phi:=\partial_{\alpha}\Phi-\sqrt{-1}A^{I}_{\alpha}\sigma_{I}\Phi/2,$

where $(\sigma_{I})_{I=1,2,3}$ are the conventional Pauli spin matrices.
$\Phi^{{\dagger}}$ denotes the Hermitian conjugate of $\Phi$ and $|\Phi|$ is
given by $|\Phi|^{2}:=\Phi^{{\dagger}}\Phi$. $V$ is a real function defined on
$\mathbb{R}^{+}$, often called the self-interaction potential, with the $s$-th
derivative $V^{(s)}$. It is worthy to point out that $\mathfrak{su}(2)$ admits
a non-degenerate inner product, denoted by the symbol
$``\langle\cdot,\cdot\rangle"$, satisfying the following property

(1.3) $\displaystyle\langle
f,[k,l]\rangle=\langle[f,k],l\rangle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
f,k,l\in\mathfrak{su}(2).$

$\tilde{J}$ is a $\mathfrak{su}(2)$-valued one-form called the Yang-Mills
current, defined by

$\displaystyle\tilde{J}:=\\{\Phi^{{\dagger}}S^{I}\hat{\nabla}_{\alpha}\Phi-(\hat{\nabla}_{\alpha}\Phi)^{{\dagger}}S^{I}\Phi\\}dx^{\alpha}\otimes
T_{I},$

where $S^{I}:=\sqrt{-1}\sigma_{I}/2$ and $\\{T_{I}\\}_{I=1,2,3}$ is the basis
of $\mathfrak{su}(2)$ given by

$\displaystyle
T_{1}:=\sqrt{-1}\sigma_{1}/2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{2}:=-\sqrt{-1}\sigma_{2}/2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{3}:=\sqrt{-1}\sigma_{3}/2.$

In order to make our main result easier to understand, we have to do some
preparation firstly. Let $h:=g-m$, where $m$ is the Minkowski metric of
$\mathbb{R}^{4}$, and set

$\displaystyle\mathfrak{L}:=\\{\partial_{\alpha},\,\,\,\,\Omega_{\alpha\beta}:=-x_{\alpha}\partial_{\beta}+x_{\beta}\partial_{\alpha},\,\,\,\,\tilde{S}:=t\partial_{t}+r\partial_{r}\\}.$

This series of vector fields plays a vital role in the research of wave
equations in Minkowski space-time. We denote the above vector fields by
$Z^{\iota}$, where $\iota:=(0,\cdots,1,\cdots,0)$. Let
$I:=(\iota_{1},\cdots,\iota_{k})$ with $|\iota_{i}|=1$ for $1\leqslant
i\leqslant k$, be a multi-index of length $|I|=k$ and let
$Z^{I}:=Z^{\iota_{1}}\cdots Z^{\iota_{k}}$ denote a product of $k$ vector
fields from the family $\mathfrak{L}$. By a sum $I_{1}+I_{2}=I$ we mean a sum
over all possible order preserving partitions of the multi-index $I$ into two
multi-indices $I_{1}$ and $I_{2}$, i.e. if $I=(\iota_{1},\cdots,\iota_{k})$,
then $I_{1}=(\iota_{i_{1}},\cdots,\iota_{i_{n}})$ and
$I_{2}=(\iota_{i_{n+1}},\cdots,\iota_{i_{k}})$, where $i_{1},\cdots,i_{k}$ is
any reordering of the integers $1,\cdots,k$ such that $i_{1}<\cdots<i_{n}$ and
$i_{n+1}<\cdots<i_{k}$. The usual wave operator is given by
$\Box:=m^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$. From Section 2 of
[12] it follows that we have the next commutation properties:

$\displaystyle[\Box,\partial_{\alpha}]=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[\Box,\Omega_{\alpha\beta}]=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[\Box,\tilde{S}]=2\Box,$

where $[X,Y]:=XY-YX$ is the commutator. For $Z\in\mathfrak{L}$, denote
$[Z,\Box]:=c_{Z}\cdot\Box$, i.e. $c_{Z}=2$, if $Z=\tilde{S}$, and 0,
otherwise. Moreover, we set

(1.4) $\mathcal{E}_{N}(t):=\sum\limits_{|I|\leqslant
N,Z\in\mathfrak{L}}\left(||\sqrt{w}\partial
Z^{I}h(\cdot,t)||_{L^{2}}+||\sqrt{w}\partial
Z^{I}A(\cdot,t)||_{L^{2}}+||\sqrt{w}\partial
Z^{I}\Phi(\cdot,t)||_{L^{2}}\right),$

where the weight is defined as

$\displaystyle
w(q):=\left\\{\begin{array}[]{rcl}&1+(1+|q|)^{1+2\gamma}&\quad\quad\text{when
$q>0$,}\\\ &1+(1+|q|)^{-2\mu}&\quad\quad\text{when $q<0$}\end{array}\right.$

with $q:=|x|-t$ and two constants $\gamma\in(0,1/2)$ and $\mu\in(0,1/2)$ being
fixed.

Now we state the main result of this paper.

######  Theorem 1.1.

Given an integer $N\geqslant 4$, there exists a constant $\varepsilon_{0}>0$
such that if $\varepsilon\leqslant\varepsilon_{0}$ and the initial datum
$h|_{t=0}$, $\partial_{t}h|_{t=0}$, $A|_{t=0}$, $\partial_{t}A|_{t=0}$,
$\Phi|_{t=0}$ and $\partial_{t}\Phi|_{t=0}$ obey
$\mathcal{E}_{N}(0)\leqslant\varepsilon$, then the solution of EYMH equations
$(g(t)=h(t)+m,A(t),\Phi(t))$ belongs to $\tilde{E}_{N+1}(\infty)$, provided
$|V^{(s)}(x)|\leqslant C_{s}\cdot x^{\tilde{\gamma}}$ for any integer
$s\in[0,N-4]$ and any $x\in[0,1]$. Here $\tilde{\gamma}\in(3/2+\gamma,\infty)$
is a fixed constant.

######  Remark 1.2.

The definition of $\tilde{E}_{N+1}(\infty)$ is given in Subsection 2.6.

In the sequel, we would like to review some previous results. In [3], D. Chae
considered the characteristic initial-value problem of the coupled Einstein
and nonlinear Klein-Gordon system, where data is given on an initial outgoing
null hypersurface, and under spherically symmetric assumption he studied the
global evolution problem toward future of the given initial null hypersurface.
Employing similar tricks, in [4] D. Chae gave a detailed proof of global
existence of solutions to Einstein-Maxwell-Higgs system in the spherical
symmetry. In [16], C. Tadmon and S.B. Tchapnda investigated the initial value
problem for the spherically symmetric $\mathfrak{su}(2)$-EYMH system. They
obtained global existence and decay properties. In addition, people care about
relativistic hydrodynamics. In [13], M. Sango and C. Tadmon considered the
Einstein-Maxwell-Euler and obtained global well-posedness in Bondi
coordinates. All the idea of [3, 4, 13, 16] origins from [2] and is to reduce
the system to a single first order integro-differential equation and then use
the contraction mapping theorem in appropriate function spaces.

For another Einstein-matter system there is a semi-global result for the
Einstein-Maxwell-Yang-Mills equations for small data due to Friedrich [9]. His
results are based on analysis of the “conformal structure” of the Einstein
field equations, i.e. on a study of the “conformal Einstein equations” which
must be satisfied by the “nonphysical” metric which is obtained from the
“physical” metric by a conformal rescaling. The “semi-global” is in the sense
that all its “physical” null geodesies are past complete. In [6], M. Dafermos
studied the stability and instability of the Cauchy horizon for the
spherically symmetric Einstein-Maxwell-scalar field equations and resolved the
issue of uniqueness in the context of a special, spherically symmetric initial
value problem for a system of gravity coupled with matter, whose relation to
the problem of gravitational collapse is well established in the physics
literature. His result is related to the strong cosmic censorship conjecture
of Roger Penrose. In [10], Z. Jia and B. Guo investigated the Einstein-
Maxwell(EM) equations and got that this system admits a global solution with
small initial datum. In [15], J.A. Smoller, A.G. Wasserman and S.T. Yau shew
that the $\mathfrak{su}(2)$-Einstein-Yang-Mills(EYM) equations admit an
infinite family of “black-hole” solutions having a regular event horizon, for
every choice of the radius of the event horizon. In [14], J.A. Smoller and
A.G. Wasserman proved the following property of spherically symmetric
solutions to the $\mathfrak{su}(2)$-EYM equations: Any solution to the EYM
system which is defined in the far field($r>>1$) and has finite ADM mass, is
defined for all $r\in(0,\infty)$. This fact is surprising, since in general
for nonlinear equations, existence theorems are usually only local. However,
for these equations they got a global existence for all solutions defined in a
neighborhood of infinity.

In the case of non-relativistic, there are also research results. In [17], S.
Yang studied the asymptotic behavior of solutions to the Maxwell-Klein-
Gordon(MKG) equations on $\mathbb{R}^{4}$ with large Cauchy data. In order to
get strong quantitative decay estimates for solutions, he brought in some
weighted energy space. In [18], S. Yang and P. Yu provided a gauge independent
proof of a conjecture, which states that the solutions to MKG equations
disperse as linear waves and enjoy peeling properties for pointwise estimates.
The remarkable works [7] and [8] of Eardley and Moncrief established the
global existence to the general Yang-Mills-Higgs system with sufficiently
smooth initial datum. The key step is the demonstration that the
$L^{\infty}$-norm of the curvature is bounded a priori. Their results apply to
any compact gauge group and to any invariant Higgs self-coupling which is
positive and of no higher than quartic degree.

Now let us briefly introduce the method we use. Firstly, in the process of
getting local well-posedness, we applied wave coordinates and Lorentzian
gauges to transform the EYMH equations into a hyperbolic system called the
reduced EYMH systems. In order to show that the solution to the reduced EYMH
systems also solves the original EYMH equations, we have to require that the
initial datum sets satisfy EYMH constraints. For the details of the above
concepts readers may refer to Section 2 of this paper. Secondly, as soon as we
get a local solution, it is natural to consider the maximal existence time $T$
and assume it to be finite. In the next, we define $T^{*}$ to be

(1.6) $\displaystyle T^{*}:=\sup\\{T_{0}:\exists C=C(T),\,s.t.\,\,\,\,\forall
t\in[0,T_{0}),\,\,\,\,\mathcal{E}_{N}(t)\leqslant 2C\varepsilon\\}$

and suppose that $T^{*}<T$. We will show that if $\varepsilon>0$ is small
enough, then the inequality in (1.6) implies the same inequality with $2C$
replaced by $C$ for all $t<T^{*}$. This contradicts the maximality of $T^{*}$
and we will obtain that the inequality holds for all $t<T$. Moreover, since
the energy $\mathcal{E}_{N}(t)$ is now finite at $t=T$(Note that
$\mathcal{E}_{N}(t)$ is continuous with respect to $t$), we can extend the
solution beyond $T$ to contradict maximality of $T$ and show that $T=\infty$.
Hence, our final task is to get energy estimates. Noting the definition of
$\mathcal{E}_{N}$, we compute $\overset{\sim}{\Box}_{g}Z^{I}h$,
$\overset{\sim}{\Box}_{g}Z^{I}A$ and $\overset{\sim}{\Box}_{g}Z^{I}\Phi$ where
$\overset{\sim}{\Box}_{g}:=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$.
Applying Proposition 6.2 of [12] and Gronwall inequality leads to the needed
results.

This paper is organized as follows:

We devote Section 2 to some preliminaries. In Section 3 we reduce the EYMH
equation to a hyperbolic system under wave coordinates and Lorentzian gauge
conditions and get local existence and uniqueness. In Section 4 we rewrite the
reduced EYMH equation as another one with respect to $(h:=g-m,A,\Phi)$.
Moreover, decay estimates and energy estimates are given in Section 5 and
Section 6 respectively.

##  2\. Notations and Preliminaries

In this article, the symbol “$Q_{1}\lesssim Q_{2}$” means that there exists a
constant $C$ such that $Q_{1}\leqslant C\cdot Q_{2}$ for two given quantities
$Q_{1}$ and $Q_{2}$. Throughout our paper, the constant $C$ may depend upon
the maximal existence time $T$.

###  2.1. Equivalence of the Einstein equation

It is easy to check that the Einstein equation is equivalent to a simpler one

(2.1)
$Ric_{\alpha\beta}=\langle\tilde{F}_{\alpha}^{\lambda},\tilde{F}_{\beta\lambda}\rangle-\frac{1}{4}g_{\alpha\beta}\langle\tilde{F}^{\lambda\mu},\tilde{F}_{\lambda\mu}\rangle+(\hat{\nabla}_{\alpha}\Phi)^{{\dagger}}\hat{\nabla}_{\beta}\Phi+(\hat{\nabla}_{\beta}\Phi)^{{\dagger}}\hat{\nabla}_{\alpha}\Phi+g_{\alpha\beta}V(|\Phi|^{2}).$

Thanks to the above transformation, we can rewrite a complicated equation as a
hyperbolic system in wave coordinates later.

###  2.2. Wave coordinates

We say that a metric $g$ of $\mathbb{R}^{4}$ satisfies the wave coordinates
condition if

$\displaystyle\hat{F}^{\lambda}:=g^{\alpha\beta}\Gamma^{\lambda}_{\alpha\beta}=0,$

where $\Gamma^{\lambda}_{\alpha\beta}$ is the connection coefficient of $g$.

###  2.3. Uniform equivalence of Riemannian metric

Given two Riemannian metric $e_{1}$ and $e_{2}$ on a smooth manifold $M$, we
say they are uniformly equivalent to each other if there exist two constants
$0<A<B<\infty$ such that for all $X\in TM$, the following holds true

$\displaystyle A\cdot e_{1}(X,X)\leqslant e_{2}(X,X)\leqslant B\cdot
e_{1}(X,X).$

###  2.4. Sliced spacetime

A Lorentzian spacetime $(V_{T}:=\mathbb{R}^{3}\times[0,T),g)$ is called sliced
for some $t\in(0,\infty]$ if there exists a time-dependent vector
$\beta(x,t):=\beta^{i}(x,t)\frac{\partial}{\partial x^{i}}(x)$, which is
called the shift, tangent to the space slice
$M_{t}:=\mathbb{R}^{3}\times\\{t\\}$ such that

(1) $\vec{n}(\cdot,t):=\frac{\partial}{\partial t}(\cdot,t)-\beta(\cdot,t)$ is
a normal vector to $M_{t}$ for all $t\in[0,T)$. That is to say, given
$X_{t}\in T(M_{t})$($T(M_{t})$ is the tangent bundle of $M_{t}$), we have

(2.2) $\displaystyle g(\vec{n}(\cdot,t),X_{t})=0.$

It is easy to check that (2.2) is equivalent to $g_{0i}=g_{ij}\beta^{j}$;

(2) $\vec{n}$ is timelike, namely,

(2.3) $\displaystyle g(\vec{n},\vec{n})<0.$

Combining (2.2) and (2.3) we arrive at $g_{ij}\beta^{i}\beta^{j}>g_{00}$.

Using the above inequality we define a positive function $N$ called the lapse
which is given by $N:=\sqrt{g_{ij}\beta^{i}\beta^{j}-g_{00}}$. Then we can
write $g$ as

(2.4) $\displaystyle g=-N^{2}dt\otimes
dt+g_{ij}(dx^{i}+\beta^{i}dt)\otimes(dx^{j}+\beta^{j}dt).$

Because $g$ is Lorentzian, $(g_{ij})$ is positive definite. In order to
represent $(g^{\alpha\beta})$ via $(N,\beta,g_{ij})$, we denote the inverse of
$(g_{ij})$ by $(g_{*}^{ij})$. It is easy to check

$\displaystyle
g^{00}=-N^{-2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g^{0i}=N^{-2}\beta^{i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g^{ij}=g_{*}^{ij}-N^{-2}\beta^{i}\beta^{j}.$

Thanks to the above discussion, now we can give the following lemma without
proof.

######  Lemma 2.1.

A spacetime $(V_{T},g)$ is sliced and Lorentzian if and only if
$g_{t}:=i_{t}^{*}g$, which is induced by the embedding
$i_{t}:\mathbb{R}^{3}\longrightarrow V_{T}$, $x\mapsto(x,t)$, is positive
definite and

$\displaystyle g_{0i}\cdot g_{*}^{ij}\cdot g_{0j}>g_{00}.$

######  Remark 2.2.

Lemma 2.1 tells us that determining a sliced Lorentzian metric $g$ on $V_{T}$
is equivalent to determining the following quantities:

(1) a Riemannian metric $g_{t}$ on $M_{t}$;

(2) a positive function $N$ on $M_{t}$;

(3) a tangent vector field $\beta$ to $M_{t}$.

######  Remark 2.3.

In case $g_{t}$ is positive definite, the inverse of $(g_{t})_{ij}$ which is
denoted by $(g_{t})^{ij}$ is just $g_{*}^{ij}$. Hence, throughout this article
we always use the symbol $g_{*}^{ij}$.

######  Remark 2.4.

Let ${}^{t}\Gamma^{k}_{ij}$ be the coefficient of the Levi-Civita connection
${}^{t}\nabla$ on $(M_{t},g_{t})$. Now we are going to give the specific
expression of $\Gamma^{\theta}_{\lambda\mu}$ via ${}^{t}\Gamma^{k}_{ij}$,
$\beta^{i}$ and $N$. They are

$\displaystyle\Gamma^{k}_{ij}=\frac{1}{2}N^{-2}\beta^{k}(g_{jp}\cdot{}^{t}\nabla_{i}\beta^{p}+g_{ip}\cdot{}^{t}\nabla_{j}\beta^{p}-\partial_{t}g_{ij})+{}^{t}\Gamma^{k}_{ij};$
$\displaystyle\Gamma^{0}_{ij}=\frac{1}{2}N^{-2}\partial_{t}g_{ij}-\frac{1}{2}N^{-2}(g_{kj}\cdot{}^{t}\nabla_{i}\beta^{k}+g_{ki}\cdot{}^{t}\nabla_{j}\beta^{k});$
$\displaystyle\Gamma^{k}_{0j}=\frac{1}{2}(g_{*}^{kl}-N^{-2}\beta^{k}\beta^{l})\cdot(\partial_{t}g_{lj}+{}^{t}\nabla_{j}\beta^{q}g_{ql}-{}^{t}\nabla_{l}\beta^{q}g_{qj})+\frac{\beta^{k}}{2N^{2}}\partial_{j}\\{g(\beta,\beta)-N^{2}\\};$
$\displaystyle\Gamma^{k}_{00}=\frac{1}{2}\left(g_{*}^{kl}-\frac{\beta^{k}\beta^{l}}{N^{2}}\right)\\{2\partial_{t}(g_{lp}\beta^{p})-\partial_{l}[g(\beta,\beta)-N^{2}]\\}+\frac{\beta^{k}}{2N^{2}}\partial_{t}\\{g(\beta,\beta)-N^{2}\\};$
$\displaystyle\Gamma^{0}_{i0}=N^{-1}\partial_{i}N-\frac{1}{4}N^{-2}\partial_{i}\\{g(\beta,\beta)\\}-\frac{1}{2}N^{-2}\beta^{l}\cdot{}^{t}\nabla_{l}\beta^{p}g_{pi}+\frac{1}{2}N^{-2}\beta^{l}\partial_{t}g_{il};$
$\displaystyle\Gamma^{0}_{00}=\frac{\beta^{l}\beta^{p}\partial_{t}g_{pl}}{2N^{2}}+\frac{(\partial_{t}+\beta)(N)}{N}-\frac{(L_{\beta}g)(\beta,\beta)}{2N^{2}},$

where $L_{\beta}$ is the Lie derivative with respect to $\beta$.

###  2.5. Regular sliced spacetime

A sliced spacetime $(V_{T},g,\beta)$ is called regular with respect to
$(\mathbb{R}^{3},e)$(where $e$ is the standard Euclidean metric of
$\mathbb{R}^{3}$) if

(1) The metrics $g_{t}$ are uniformly equivalent to $e$, i.e. there exist
continuous strictly positive functions $B_{1}(t)$, $B_{2}(t)$ such that for
all $t\in[0,T)$ and each tangent vector $X$ to $\mathbb{R}^{3}$ it holds true
on $M_{t}$

$\displaystyle B_{1}(t)\cdot e(X,X)\leqslant g_{t}(X,X)\leqslant B_{2}(t)\cdot
e(X,X);$

(2) The lapse $N$ is such that there exist continuous strictly positive
functions $C_{1}(t)$, $C_{2}(t)$ on $[0,T)$ such that on each $M_{t}$ it holds
true

$\displaystyle C_{1}(t)\leqslant N(x,t)\leqslant
C_{2}(t)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x\in\mathbb{R}^{3};$

(3) The shift $\beta$ is uniformly bounded in $e$-norm on each $M_{t}$ by a
number $b(t)$.

###  2.6. Sobolev space on $V_{T}$

We denote by $E_{s}(T)$ the following Banach space

$E_{s}(T):=\bigcap\limits_{0\leqslant k\leqslant
s}C^{s-k}([0,T),H_{k}(\mathbb{R}^{3})).$

The Sobolev space $\tilde{E}_{s}(T)$ is the space of functions $u$, such that
$u\in\bar{C}(V_{T})$, space of continuous and bounded functions on $V_{T}$,
while $\partial u\in E_{s-1}(V_{T})$, where $\partial$ is the Levi-Civita
connection on $(V_{T},m)$(Recall that $m$ is the standard Minkowski metric of
$V_{T}\subseteq\mathbb{R}^{4}$).

###  2.7. EYMH initial data set

A EYMH initial data set is a ten-tuple
$(\mathbb{R}^{3};\bar{g},K,\bar{A}^{\mbox{space}},\bar{A}_{0},\bar{E},\bar{\beta},\bar{N},\bar{\Phi},\bar{\Psi})$.
$\bar{g}$ and $K$ are the Riemannian metric and symmetric 2-tensor on
$\mathbb{R}^{3}$ respectively. Meanwhile, $\bar{A}^{\mbox{space}}$ is a
$\mathfrak{su}(2)$-valued one-form on $\mathbb{R}^{3}$; $\bar{A}_{0}$ is a
$\mathfrak{su}(2)$-valued function on $\mathbb{R}^{3}$; $\bar{E}$ is a
$\mathfrak{su}(2)$-valued vector field on $\mathbb{R}^{3}$; $\bar{\beta}$ is a
vector field on $\mathbb{R}^{3}$. Moreover, $\bar{N}$ is a positive function
on $\mathbb{R}^{3}$. $\bar{\Phi}$ and $\bar{\Psi}$ are $\mathbb{C}^{2}$-valued
functions on $\mathbb{R}^{3}$, where $\mathbb{C}$ is the set of all the
complex numbers.

###  2.8. EYMH development

The development of the initial data set
$(\mathbb{R}^{3};\bar{g},K,\bar{A}^{\mbox{space}},\bar{\Phi})$ is a tetrad
$(\mathbb{R}^{4},g,A,\Phi)$ with a Lorentzian metric $g$, a
$\mathfrak{su}(2)$-valued one-form $A$ on $\mathbb{R}^{4}$ and a
$\mathbb{C}^{2}$-valued function $\Phi$ on $\mathbb{R}^{4}$, such that the
embedding $i_{0}$ of $\mathbb{R}^{3}$ into $\mathbb{R}^{4}$(Recall that
$i_{0}(x):=(x,0)$ for all $x\in\mathbb{R}^{3}$) enjoys the following
properties:

(a) The metric $\bar{g}$ is the pullback of $g$ by $i_{0}$, i.e.
$\bar{g}=i_{0}^{*}g$;

(b) $K$ is the second fundamental form of $i_{0}(\mathbb{R}^{3})$ as a
submanifold of $(\mathbb{R}^{4},g)$;

(c) The $\mathfrak{su}(2)$-valued one-form $\bar{A}^{\mbox{space}}$ is the
pullback of $A$ by $i_{0}$, i.e. $\bar{A}^{\mbox{space}}=i_{0}^{*}A$;

(d) $\bar{\Phi}(x)=\Phi(x,0)$ for all $x\in\mathbb{R}^{3}$.

Furthermore, $(g,A,\Phi)$ satisfies on $\mathbb{R}^{4}$ the EYMH equations and
$(\mathbb{R}^{4},g)$ is a sliced spacetime.

###  2.9. EYMH constraints

Restricting EYMH equations to the initial data set
$(\mathbb{R}^{3};\bar{g},K,\bar{A}^{\mbox{space}},\bar{A}_{0},\bar{E},\bar{\beta},\bar{N},\bar{\Phi})$,
which is embedded into $\mathbb{R}^{4}$, leads to the following identities
called the constraints

(2.5)
$\displaystyle\bar{R}-|K|^{2}_{\bar{g}}+(\mbox{tr}_{\bar{g}}K)^{2}=2\bar{N}^{-2}\cdot
T(\partial_{t}-\beta,\partial_{t}-\beta)|_{t=0},$

(2.6)
$\displaystyle(\mbox{div}_{\bar{g}}K)_{i}-\partial_{i}(\mbox{tr}_{\bar{g}}K)=-\bar{N}^{-1}\cdot
T(\partial_{t}-\beta,\partial_{i})|_{t=0},$

$\displaystyle\tilde{J}(\beta-\partial_{t})|_{t=0}=\bar{g}^{ij}(\bar{\nabla}_{i}\bar{F}_{jk}+[\bar{A}^{\mbox{space}}_{i},\bar{F}_{jk}])\bar{\beta}^{k}+\bar{N}\cdot[\bar{A}^{\mbox{space}}_{i},\bar{E}^{i}]+\bar{F}(\bar{\beta},\bar{\nabla}_{\bar{\beta}}\bar{\beta})/\bar{N}^{2}$
(2.7)
$\displaystyle-2\bar{\beta}(\bar{N})\bar{g}(\bar{\beta},\bar{E})/\bar{N}^{2}+\bar{E}(\bar{N})+\bar{N}\mbox{div}_{\bar{g}}\bar{E}+2\bar{g}(\bar{E},\bar{\beta})\mbox{tr}_{\bar{g}}K+\bar{g}^{ij}\bar{F}_{ik}\bar{\nabla}_{j}\bar{\beta}^{k}$
$\displaystyle+\bar{F}(\bar{\beta},\mbox{grad}_{\bar{g}}\bar{N})/\bar{N}-2\bar{F}_{ik}\bar{\beta}^{i}\bar{\beta}^{j}K_{jl}\bar{g}^{kl}/\bar{N},$

(2.8)
$\displaystyle\partial_{t}A_{i}|_{t=0}-\partial_{i}\bar{A}_{0}+[\bar{A}_{0},\bar{A}^{\mbox{space}}_{i}]:=\bar{N}\cdot\bar{E}^{j}\cdot\bar{g}_{ji}$

(2.9)
$\displaystyle(\partial_{t}\Phi)|_{t=0}-\sqrt{-1}(\bar{A}_{0})^{I}\sigma_{I}\bar{\Phi}/2:=\bar{\Psi}$

where
$\bar{F}:=d\bar{A}^{\mbox{space}}+[\bar{A}^{\mbox{space}},\bar{A}^{\mbox{space}}]$,
$\bar{\nabla}$ and $\bar{R}$ are the Levi-Civita connection and the scalar
curvature of $(\mathbb{R}^{3},\bar{g})$ respectively.

######  Remark 2.5.

Unless we give (2.8) and (2.9), one can not determine the values on the right
hand side of (2.5), (2.6) and (2.9). In the next, we are going to write their
specific expressions via
$(\bar{g},\bar{N},\bar{F},\bar{E},\bar{\beta},\bar{\Phi},\bar{\Psi})$:

$\displaystyle 2\bar{N}^{-2}\cdot
T(\partial_{t}-\beta,\partial_{t}-\beta)|_{t=0}=\bar{g}^{ik}\bar{g}^{jl}\langle\bar{F}_{ij},\bar{F}_{kl}\rangle/2+2\bar{g}_{ij}\langle\bar{E}^{i},\bar{E}^{j}\rangle+2|\bar{\Psi}-\hat{\hbox
to0.0pt{/\hss}\nabla}_{\bar{\beta}}\bar{\Phi}|^{2}/\bar{N}^{2}$
$\displaystyle+\frac{\bar{g}^{ij}}{\bar{N}^{2}}\langle\bar{\beta}^{k}\bar{F}_{ki}-\bar{N}\bar{E}^{l}\bar{g}_{li},\bar{\beta}^{k}\bar{F}_{kj}-\bar{N}\bar{E}^{l}\bar{g}_{lj}\rangle+2\\{\bar{g}^{ij}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi})^{{\dagger}}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{j}\bar{\Phi})+V(|\bar{\Phi}|^{2})\\}$

and

$\displaystyle-\bar{N}^{-1}\cdot T(\partial_{t}-\beta,\partial_{i})|_{t=0}$
$\displaystyle=$
$\displaystyle\langle\bar{E}^{l},\bar{F}_{li}\rangle+\bar{N}^{-1}\bar{\beta}^{k}\bar{g}^{pq}\langle\bar{F}_{pk},\bar{F}_{qi}\rangle-\bar{N}^{-1}(\bar{\Psi}-\bar{\beta}^{k}\hat{\hbox
to0.0pt{/\hss}\nabla}_{k}\bar{\Phi})^{{\dagger}}\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi}$ $\displaystyle-\bar{N}^{-1}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi})^{{\dagger}}(\bar{\Psi}-\bar{\beta}^{k}\hat{\hbox
to0.0pt{/\hss}\nabla}_{k}\bar{\Phi}),$

where $\hat{\hbox to0.0pt{/\hss}\nabla}$ is given by

$\displaystyle\hat{\hbox
to0.0pt{/\hss}\nabla}_{X}\bar{\Phi}:=X(\bar{\Phi})-\frac{\sqrt{-1}}{2}(\bar{A}^{\mbox{space}}(X))^{I}\sigma_{I}\bar{\Phi}$

and $X$ is a tangent vector field on $\mathbb{R}^{3}$. Moreover, we also have

$\displaystyle\tilde{J}(\beta-\partial_{t})|_{t=0}=\bar{\Phi}^{{\dagger}}S(\bar{\beta}^{k}\hat{\hbox
to0.0pt{/\hss}\nabla}_{k}\bar{\Phi}-\bar{\Psi})-(\bar{\beta}^{k}\hat{\hbox
to0.0pt{/\hss}\nabla}_{k}\bar{\Phi}-\bar{\Psi})^{{\dagger}}S\bar{\Phi},$

where for any $\mathbb{C}^{2}-$valued function $\psi$, we define
$S\psi:=(S^{I}\psi)\cdot T_{I}$. Recall that $S^{I}:=\sqrt{-1}\sigma_{I}/2$
and $(T_{I})_{I=1,2,3}$ is the basis of $\mathfrak{su}(2)$.

###  2.10. The null frame

At each point $x\in\mathbb{R}^{4}$, we introduce a pair of null vectors
$(L,\underline{L})$ with

$L:=\partial_{t}+\partial_{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{L}:=\partial_{t}-\partial_{r},$

where $x:=(x^{0},x^{1},x^{2},x^{3})$, $t:=x^{0}$ and
$r:=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$. Sometimes, we also denote
$x^{i}$ by $x_{i}$($i=1,2,3$) and $L$ by $\bar{\partial}_{0}$. Let
$\tilde{S}_{1}$ and $\tilde{S}_{2}$ be two orthonormal smooth tangent vector
fields to the unit sphere $\mathbb{S}^{2}$, where the orthogonality is in the
sense of the standard metric of $\mathbb{S}^{2}$. For convenience
$A,B,C,D,\cdots$ means any of the vectors $\tilde{S}_{1}$ and $\tilde{S}_{2}$
at times. Given a 1-tensor $\pi:=\pi_{\beta}dx^{\beta}$ and a 2-tensor
$p:=p_{\alpha\beta}dx^{\alpha}\otimes dx^{\beta}$, we define $\pi_{X}:=\pi(X)$
and $p_{XY}:=p(X,Y)$, provided $X,Y$ are two vector fields.

Suppose that

$eu:=dt\otimes dt+\sum\limits_{i=1}^{3}dx^{i}\otimes dx^{i}$

is the standard Euclidean metric of $\mathbb{R}^{4}$. Then we have

$\displaystyle
eu_{\underline{L}L}=eu_{LA}=eu_{\underline{L}A}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eu_{LL}=eu_{\underline{L}\underline{L}}=2$

and

$\displaystyle eu_{AB}=\delta_{AB}:=\left\\{\begin{array}[]{rcl}&0&A\not=B\\\
&1&A=B.\end{array}\right.$

The inverse of $eu$ is

$\displaystyle
eu^{\underline{L}L}=eu^{LA}=eu^{\underline{L}A}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eu^{LL}=eu^{\underline{L}\underline{L}}=1/2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eu^{AB}=\delta_{AB}.$

Noting that $\tilde{S}_{1}$ and $\tilde{S}_{2}$ are defined only locally on
$\mathbb{S}^{2}$, we replace them with the projections

$\displaystyle\bar{\partial}_{i}:=\partial_{i}-\omega_{i}\cdot\partial_{r},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\omega_{i}:=x_{i}/r\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,i=1,2,3.$

It is nor hard to see that
$\\{\bar{\partial}_{1},\bar{\partial}_{2},\bar{\partial}_{3}\\}$ gives a set
of global and linear dependent vector fields on $\mathbb{S}^{2}$. Moreover,
one can also represent $\\{\bar{\partial}_{i}|i=1,2,3\\}$ by $\tilde{S}_{1}$
and $\tilde{S}_{2}$, i.e.

(2.11)
$\displaystyle\bar{\partial}_{i}=\tilde{S}_{1}^{i}\cdot\tilde{S}_{1}+\tilde{S}_{2}^{i}\cdot\tilde{S}_{2},$

where $\tilde{S}_{j}:=\tilde{S}_{j}^{i}\cdot\partial_{i}$ and $j=1,2$.

We call $\\{L,\underline{L},\tilde{S}_{1},\tilde{S}_{2}\\}$ the null frame and
introduce the coming notation. Let
$\mathcal{T}:=\\{L,\tilde{S}_{1},\tilde{S}_{2}\\}$,
$\mathcal{U}:=\\{\underline{L},L,\tilde{S}_{1},\tilde{S}_{2}\\}$,
$\mathcal{L}:=\\{L\\}$ and $\mathcal{S}:=\\{\tilde{S}_{1},\tilde{S}_{2}\\}$.
For any $l$ of these families $\mathcal{V}_{1},\cdots,\mathcal{V}_{l}$(namely,
$\mathcal{V}_{1},\cdots,\mathcal{V}_{l}\in\\{\mathcal{T},\mathcal{U},\mathcal{L},\mathcal{S}\\}$)
and an arbitrary $k$-tensor
$p:=p_{\alpha_{1}\cdots\alpha_{k}}dx^{\alpha_{1}}\otimes\cdots\otimes
dx^{\alpha_{k}}$ with $k\geqslant l$, we define two norms $|p|$ and
$|p|_{\mathcal{V}_{1}\cdots\mathcal{V}_{l}}$ as

$\displaystyle|p|^{2}:=\sum\limits_{\alpha_{1},\cdots,\alpha_{k}=0}^{3}(p_{\alpha_{1}\cdots\alpha_{k}})^{2}$

and

$\displaystyle|p|^{2}_{\mathcal{V}_{1}\cdots\mathcal{V}_{l}}:$
$\displaystyle=$
$\displaystyle\sum\limits_{V_{1},V^{\prime}_{1}\in\mathcal{V}_{1}}\cdots\sum\limits_{V_{l},V^{\prime}_{l}\in\mathcal{V}_{l}}\sum\limits_{\alpha_{l+1},\cdots,\alpha_{k}=0}^{3}eu^{V_{1}V_{1}^{\prime}}\cdots
eu^{V_{l}V^{\prime}_{l}}$ $\displaystyle\times
p(V_{1},\cdots,V_{l},\partial_{\alpha_{l+1}},\cdots,\partial_{\alpha_{k}})\cdot
p(V^{\prime}_{1},\cdots,V^{\prime}_{l},\partial_{\alpha_{l+1}},\cdots,\partial_{\alpha_{k}})$

It is not difficult to check that $|p|_{\mathcal{V}_{1}\cdots\mathcal{V}_{l}}$
is independent of the choice of $\\{\tilde{S}_{1},\tilde{S}_{2}\\}$ on
$\mathbb{S}^{2}$.

###  2.11. The Minkowski metric

Recall that the Minkowski metric $m$ of $\mathbb{R}^{4}$ is given by

$\displaystyle m:=-dt\otimes dt+\sum\limits_{i=1}^{3}dx^{i}\otimes dx^{i}.$

From Section 4 of [12] it follows that

$\displaystyle
m_{LL}=m_{\underline{L}\underline{L}}=m_{LA}=m_{\underline{L}A}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m_{L\underline{L}}=m_{\underline{L}L}=-2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m_{AB}=\delta_{AB}$

The inverse of the metric has the form

$\displaystyle
m^{LL}=m^{\underline{L}\underline{L}}=m^{LA}=m^{\underline{L}A}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m^{L\underline{L}}=m^{\underline{L}L}=-1/2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m^{AB}=\delta^{AB}.$

Recall that $\partial$ is the Levi-Civita connection of $(\mathbb{R}^{4},m)$.
We shall use it to define a new differential operator $\bar{\partial}$ as
follows. Provided $p$ is a $k$-tensor and $q:=r-t$, $\bar{\partial}p$ is given
by

$\displaystyle\bar{\partial}p:=\partial p-\partial_{\partial r}p\otimes dq,$

where we recall $\partial_{r}:=\frac{\partial}{\partial r}$. Easily, the
readers, reviewing the definition of $\bar{\partial}_{\beta}$ in Subsection
2.10, can check that

$\displaystyle\bar{\partial}p=\bar{\partial}_{\beta}\left(p_{\alpha_{1}\cdots\alpha_{k}}\right)\cdot
dx^{\alpha_{1}}\otimes\cdots\otimes dx^{\alpha_{k}}\otimes dx^{\beta}.$

######  Remark 2.6.

From Lemma 2.6 of [10] it follows that for any 2-tensor $p$ and
$\mathcal{V},\mathcal{W}\in\\{\mathcal{T},\mathcal{U},\mathcal{L},\mathcal{S}\\}$,
the quantity $|\bar{\partial}p|_{\mathcal{V}\mathcal{W}}$ is equivalent to
that of (4.5) in [12]. Moreover, $|p|_{\mathcal{V}\mathcal{W}}$ and $|\partial
p|_{\mathcal{V}\mathcal{W}}$ are all equivalent to those of (4.3) and (4.4) in
[12].

##  3\. Local well-posedness

Recall that $A$ is the Yang-Mills potential. For simplicity, we decompose $A$
as

$A:=A^{\mbox{time}}+A^{\mbox{space}},$

where

$A^{\mbox{time}}(x,t):=A_{0}(x,t)dt\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A^{\mbox{space}}(x,t):=A_{i}(x,t)dx^{i}.$

From now on, we always assume that $(\mathbb{R}^{4},g)$ satisfies the wave
coordinates condition.

###  3.1. The Yang-Mills equations in wave coordinates and Lorentzian gauges

The following computation is obvious

$\displaystyle\nabla^{\alpha}\tilde{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle
g^{\alpha\lambda}(\partial_{\lambda}\partial_{\alpha}A_{\beta}-\partial_{\lambda}\partial_{\beta}A_{\alpha}+[\partial_{\lambda}A_{\alpha},A_{\beta}]+[A_{\alpha},\partial_{\lambda}A_{\beta}])$
$\displaystyle-g^{\alpha\lambda}\Gamma^{\theta}_{\lambda\beta}\cdot(\partial_{\alpha}A_{\theta}-\partial_{\theta}A_{\alpha}),$

where $\Gamma$ is the Christoffel symbols of $g$ and we have used the wave
coordinates condition to deduce (3.1).

By elementary manipulations (3.1) becomes

(3.2) $\displaystyle\nabla^{\alpha}\tilde{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle
g^{\alpha\lambda}\partial_{\lambda}\partial_{\alpha}A_{\beta}-\partial_{\beta}(g^{\alpha\lambda}\partial_{\lambda}A_{\alpha})+\partial_{\beta}g^{\alpha\lambda}\cdot\partial_{\lambda}A_{\alpha}+g^{\alpha\lambda}[\partial_{\lambda}A_{\alpha},A_{\beta}]+g^{\alpha\lambda}[A_{\alpha},\partial_{\lambda}A_{\beta}]$
$\displaystyle-g^{\alpha\lambda}\Gamma^{\theta}_{\lambda\beta}\cdot(\partial_{\alpha}A_{\theta}-\partial_{\theta}A_{\alpha}).$

We can transform (3.2) into

$\displaystyle\nabla^{\alpha}\tilde{F}_{\alpha\beta}=g^{\alpha\lambda}\partial_{\lambda}\partial_{\alpha}A_{\beta}+\partial_{\beta}g^{\alpha\lambda}\cdot\partial_{\lambda}A_{\alpha}+g^{\alpha\lambda}[A_{\alpha},\partial_{\lambda}A_{\beta}]-g^{\alpha\lambda}\Gamma^{\theta}_{\lambda\beta}\cdot(\partial_{\alpha}A_{\theta}-\partial_{\theta}A_{\alpha}),$

if we assume that $g^{\alpha\lambda}\partial_{\lambda}A_{\alpha}=0$, which is
equivalent to $\mbox{div}_{g}A\equiv
g^{\alpha\lambda}\nabla_{\lambda}A_{\alpha}=0$ called the Lorentz gauge
condition(the equivalence follows from the wave gauge condition).

It is easy to see

$\displaystyle\partial_{\beta}g^{\alpha\lambda}=-g^{\alpha\theta}\cdot\partial_{\beta}g_{\theta\mu}\cdot
g^{\mu\lambda}.$

Therefore, the Yang-Mills equation in wave coordinates and Lorentzian gauges
can be written as

(3.3) $\displaystyle
g^{\alpha\lambda}\partial_{\lambda}\partial_{\alpha}A_{\beta}+f_{\beta}(g,A,\partial
g,\partial A,\Phi,\partial\Phi)=0,$

where

$\displaystyle f_{\beta}(g,A,\partial g,\partial
A,\Phi,\partial\Phi):=-g^{\alpha\theta}\cdot\partial_{\beta}g_{\theta\mu}\cdot
g^{\mu\lambda}\cdot\partial_{\lambda}A_{\alpha}+g^{\alpha\lambda}[A_{\alpha},\partial_{\lambda}A_{\beta}]+g^{\alpha\lambda}[A_{\lambda},\partial_{\alpha}A_{\beta}]$
$\displaystyle-\frac{1}{2}g^{\alpha\lambda}g^{\theta\mu}\cdot(\partial_{\lambda}g_{\mu\beta}+\partial_{\beta}g_{\lambda\mu}-\partial_{\mu}g_{\lambda\beta})\cdot(\partial_{\alpha}A_{\theta}-\partial_{\theta}A_{\alpha})-g^{\alpha\lambda}[A_{\lambda},\partial_{\beta}A_{\alpha}]+g^{\alpha\lambda}[A_{\lambda},[A_{\alpha},A_{\beta}]]$
$\displaystyle-\Phi^{{\dagger}}S(\partial_{\beta}\Phi-\sqrt{-1}A^{I}_{\beta}\sigma_{I}\Phi/2)+(\partial_{\beta}\Phi-\sqrt{-1}A^{I}_{\beta}\sigma_{I}\Phi/2)^{{\dagger}}S\Phi.$

###  3.2. Einstein equation in wave coordinates

Referring to Section 7.4 of Chapter 6 in [1] we get the coming formula

$\displaystyle
Ric_{\alpha\beta}=-\frac{1}{2}g^{\lambda\mu}\partial_{\lambda}\partial_{\mu}g_{\alpha\beta}+h_{\alpha\beta}(g,\partial
g)+\frac{1}{2}(g_{\alpha\lambda}\partial_{\beta}\hat{F}^{\lambda}+g_{\beta\lambda}\partial_{\alpha}\hat{F}^{\lambda})$

with

$\displaystyle h_{\alpha\beta}(g,\partial
g):=P_{\alpha\beta}^{\rho\sigma\gamma\delta\lambda\mu}(g,g^{-1})\cdot\partial_{\rho}g_{\gamma\delta}\cdot\partial_{\sigma}g_{\lambda\mu},$

where $Ric$ is the Ricci tensor of $g$ and the tensor $P$ is a polynomial in
$g$ and $g^{-1}$.

Hence, (2.1) can be reduced to

$\displaystyle
g^{\lambda\mu}\partial_{\lambda}\partial_{\mu}g_{\alpha\beta}+\tilde{f}_{\alpha\beta}(g,A,\Phi,\partial
g,\partial A,\partial\Phi)=0,$

where

$\displaystyle\tilde{f}_{\alpha\beta}(g,A,\Phi,\partial g,\partial
A,\partial\Phi):=2g^{\lambda\mu}\langle\partial_{\lambda}A_{\alpha}-\partial_{\alpha}A_{\lambda}+[A_{\lambda},A_{\alpha}],\partial_{\mu}A_{\beta}-\partial_{\beta}A_{\mu}+[A_{\mu},A_{\beta}]\rangle$
$\displaystyle-\frac{1}{2}g_{\alpha\beta}g^{\lambda\rho}g^{\mu\sigma}\langle\partial_{\lambda}A_{\mu}-\partial_{\mu}A_{\lambda}+[A_{\lambda},A_{\mu}],\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}+[A_{\rho},A_{\sigma}]\rangle-2h_{\alpha\beta}(g,\partial
g)$ $\displaystyle-
g_{\alpha\lambda}\partial_{\beta}\hat{F}^{\lambda}-g_{\beta\lambda}\partial_{\alpha}\hat{F}^{\lambda}+2(\partial_{\alpha}\Phi-\sqrt{-1}A^{I_{1}}_{\alpha}\sigma_{I_{1}}\Phi/2)^{{\dagger}}(\partial_{\beta}\Phi-\sqrt{-1}A^{I_{2}}_{\beta}\sigma_{I_{2}}\Phi/2)$
$\displaystyle+2(\partial_{\beta}\Phi-\sqrt{-1}A^{I_{2}}_{\beta}\sigma_{I_{2}}\Phi/2)^{{\dagger}}(\partial_{\alpha}\Phi-\sqrt{-1}A^{I_{1}}_{\alpha}\sigma_{I_{1}}\Phi/2)+2g_{\alpha\beta}\cdot
V(|\Phi|^{2}).$

The wave coordinates conditions tell us that $\hat{F}^{\lambda}=0$. Hence we
have

$\displaystyle\tilde{f}_{\alpha\beta}(g,A,\Phi,\partial g,\partial
A,\partial\Phi):=2g^{\lambda\mu}\langle\partial_{\lambda}A_{\alpha}-\partial_{\alpha}A_{\lambda}+[A_{\lambda},A_{\alpha}],\partial_{\mu}A_{\beta}-\partial_{\beta}A_{\mu}+[A_{\mu},A_{\beta}]\rangle$
$\displaystyle-\frac{1}{2}g_{\alpha\beta}g^{\lambda\rho}g^{\mu\sigma}\langle\partial_{\lambda}A_{\mu}-\partial_{\mu}A_{\lambda}+[A_{\lambda},A_{\mu}],\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}+[A_{\rho},A_{\sigma}]\rangle-2h_{\alpha\beta}(g,\partial
g)$
$\displaystyle+2(\partial_{\alpha}\Phi-\sqrt{-1}A^{I_{1}}_{\alpha}\sigma_{I_{1}}\Phi/2)^{{\dagger}}(\partial_{\beta}\Phi-\sqrt{-1}A^{I_{2}}_{\beta}\sigma_{I_{2}}\Phi/2)+2g_{\alpha\beta}\cdot
V(|\Phi|^{2})$
$\displaystyle+2(\partial_{\beta}\Phi-\sqrt{-1}A^{I_{2}}_{\beta}\sigma_{I_{2}}\Phi/2)^{{\dagger}}(\partial_{\alpha}\Phi-\sqrt{-1}A^{I_{1}}_{\alpha}\sigma_{I_{1}}\Phi/2).$

###  3.3. Higgs equations in wave coordinates and Lorentzian gauges

It is easy to check that Higgs equations in wave coordinates and Lorentzian
gauges are equivalent to

$\displaystyle
g^{\lambda\mu}\partial_{\lambda}\partial_{\mu}\Phi+U(\Phi,\partial\Phi,A)=0,$

where

$\displaystyle U(\Phi,\partial\Phi,A):$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2}g^{\lambda\mu}A_{\mu}^{I_{1}}\sigma_{I_{1}}(\partial_{\lambda}\Phi)-\frac{\sqrt{-1}}{2}g^{\lambda\mu}A_{\lambda}^{I_{2}}\sigma_{I_{2}}(\partial_{\mu}\Phi)$
$\displaystyle-\frac{1}{4}g^{\lambda\mu}A^{I_{2}}_{\lambda}A^{I_{1}}_{\mu}\sigma_{I_{2}}\sigma_{I_{1}}(\Phi)-V^{\prime}(|\Phi|^{2})\Phi$

and the equivalence follows from wave coordinates conditions and Lorentzian
gauges.

###  3.4. Reducing EYMH equations to quasi-linear systems on a new bundle over
$(\mathbb{R}^{4},eu)$

Firstly, we want to construct a new vector bundle $BU$ over
$(\mathbb{R}^{4},eu)$. It is given by

$\displaystyle BU:=(T^{*}\mathbb{R}^{4}\otimes
T^{*}\mathbb{R}^{4})\times(T^{*}\mathbb{R}^{4}\otimes\mathfrak{su}(2))\times\mathbb{C}^{2}$

endowed with a metric $\lceil\cdot,\cdot\rceil$, where the symbol “$\times$”
means the Cartesian product of vector bundles, and $T^{*}\mathbb{R}^{4}$ is
the cotangent bundle of $\mathbb{R}^{4}$. More precisely, for any
$(g_{i},A_{i},\Phi_{i})\in BU$($i=1,2$), we define

$\displaystyle\lceil(g_{1},A_{1},\Phi_{1}),(g_{2},A_{2},\Phi_{2})\rceil:=((g_{1},g_{2}))+\langle\langle
A_{1},A_{2}\rangle\rangle+\Phi_{1}^{{\dagger}}\Phi_{2},$

where

$\displaystyle((g_{1},g_{2})):=\sum\limits_{\alpha,\theta}(g_{1})_{\alpha\theta}\cdot(g_{2})_{\alpha\theta},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\langle\langle
A_{1},A_{2}\rangle\rangle:=\sum\limits_{\alpha}\langle(A_{1})_{\alpha},(A_{2})_{\alpha}\rangle.$

Furthermore, we define a connection D on $BU$ by the following identity

$\displaystyle\textbf{\mbox{D}}(g,A,\Phi):=(\partial g,\partial
A,\partial\Phi)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for any $(g,A,\Phi)\in
BU$.}$

It is not difficult to check that D is compatible to the metric
$\lceil\cdot,\cdot\rceil$.

From the discussion in Subsection 3.1, 3.2 and 3.3 we infer that if
$u:=(g,A,\Phi)\in BU$ solves the EYMH equations in wave coordinates and
Lorentzian gauges, then it is also a solution of the following quasi-linear
system

(3.4) $\displaystyle
h^{\lambda\mu}(u)\cdot\textbf{\mbox{D}}_{\lambda}\textbf{\mbox{D}}_{\mu}u+l(u,\textbf{\mbox{D}}u)=0,$

where

$\displaystyle
h^{\lambda\mu}(u):=g^{\lambda\mu}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,l(u,\textbf{\mbox{D}}u):=\left(\tilde{f}_{\alpha\beta}(u,\textbf{\mbox{D}}u)dx^{\alpha}\otimes
dx^{\beta},f_{\theta}(u,\textbf{\mbox{D}}u)dx^{\theta},\hat{U}(u,\textbf{\mbox{D}}u)\right),$

where $\hat{U}(u,\textbf{\mbox{D}}u):=U(\Phi,\partial\Phi,A)$.

###  3.5. Determining the initial value of $\textbf{\mbox{D}}_{0}u$ on
$\mathbb{R}^{3}$

In order to determining $\textbf{\mbox{D}}_{0}u|_{t=0}$, we must give the
values of
$(\partial_{t}g_{ij},\partial_{t}\beta^{i},\partial_{t}N,\partial_{t}A_{\alpha})|_{t=0}$.
From Chapter 6 of [1] it follows that they can not be chosen arbitrarily; they
should satisfy some restrictions.

By (6.1) of Section 6.1 in Chapter 6 of [1] we know

(3.5)
$\displaystyle\partial_{t}g_{ij}|_{t=0}=-2\bar{N}K_{ij}+\bar{g}_{jh}\bar{\nabla}_{i}\bar{\beta}^{h}+\bar{g}_{ih}\bar{\nabla}_{j}\bar{\beta}^{h}.$

From Lorentzian gauge condition $\mbox{div}_{g}A=0$ and wave coordinates
condition we infer that

$\displaystyle
g^{0\alpha}\partial_{t}A_{\alpha}=-g^{i\alpha}\partial_{i}A_{\alpha},$

implying

$\displaystyle-N^{-2}\partial_{t}A_{0}+N^{-2}\beta^{i}\partial_{t}A_{i}=-N^{-2}\beta^{i}\partial_{i}A_{0}-(g_{*}^{ij}-N^{-2}\beta^{i}\beta^{j})\partial_{i}A_{j}.$

Restricting the above identity to $\mathbb{R}^{3}$ yields

(3.6)
$\displaystyle(\partial_{t}A_{0})|_{t=0}=\bar{\beta}^{i}(\partial_{t}A_{i})|_{t=0}+\bar{\beta}^{i}\partial_{i}\bar{A}_{0}+(\bar{N}^{2}\bar{g}^{ij}-\bar{\beta}^{i}\bar{\beta}^{j})\partial_{i}\bar{A}_{j},$

where $\bar{A}:=A^{\mbox{space}}|_{t=0}$ and
$\bar{A}_{0}:=A^{\mbox{time}}(\partial_{t})|_{t=0}$. Now the problem turns to
be how to determine $(\partial_{t}A_{i})|_{t=0}$. Easily, from (2.8) it
follows that

(3.7)
$\displaystyle(\partial_{t}A_{i})|_{t=0}=\partial_{i}\bar{A}_{0}-[\bar{A}_{0},\bar{A}_{i}]+\bar{N}\cdot\bar{E}^{j}\cdot\bar{g}_{ji}.$

Substituting (3.7) into (3.6) yields

(3.8)
$\displaystyle(\partial_{t}A_{0})|_{t=0}=2\bar{\beta}^{i}\partial_{i}\bar{A}_{0}-\bar{\beta}^{i}[\bar{A}_{0},\bar{A}_{i}]+\bar{N}\cdot\bar{g}_{ij}\bar{E}^{i}\bar{\beta}^{j}+(\bar{N}^{2}\bar{g}^{ij}-\bar{\beta}^{i}\bar{\beta}^{j})\partial_{i}\bar{A}_{j}.$

In other words, if $\bar{A}_{0}$ and $\bar{A}$ are given, then
$(\partial_{t}A_{0})|_{t=0}$ and $(\partial_{t}A_{i})|_{t=0}$ can be specified
via (3.6) and (3.7).

The wave coordinates condition tells us

$\displaystyle
g^{\alpha\beta}\Gamma^{0}_{\alpha\beta}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g^{\alpha\beta}\Gamma^{k}_{\alpha\beta}=0,$

which are equivalent to

(3.9)
$\displaystyle\partial_{t}N=\frac{1}{2}Ng_{*}^{ij}\partial_{t}g_{ij}-N\mbox{div}_{g_{t}}\beta+\beta(N)$

and

$\displaystyle\partial_{t}\beta^{k}$ $\displaystyle=$
$\displaystyle(N^{2}g_{*}^{ij}-\beta^{i}\beta^{j}){}^{t}\Gamma^{k}_{ij}-\frac{1}{2}\beta^{k}\cdot\partial_{t}g_{ij}\cdot
g_{*}^{ij}+\frac{1}{2}\beta^{k}\cdot(L_{\beta}g)_{ij}\cdot g_{*}^{ij}$
$\displaystyle+N^{-1}\partial_{t}N\beta^{k}+\frac{1}{2}g_{*}^{kh}\partial_{h}\\{g(\beta,\beta)-N^{2}\\}-N^{-1}\beta^{k}\beta(N)$
$\displaystyle+\beta^{i}\cdot\left({}^{t}\nabla_{i}\beta^{k}\right)-\beta^{i}g_{*}^{hk}g_{ip}\cdot\left({}^{t}\nabla_{h}\beta^{p}\right).$

Substituting (3.9) into (3.5) yields

(3.11) $\displaystyle\partial_{t}\beta^{k}$ $\displaystyle=$
$\displaystyle(N^{2}g_{*}^{ij}-\beta^{i}\beta^{j}){}^{t}\Gamma^{k}_{ij}+\frac{1}{2}\beta^{k}\cdot(L_{\beta}g)_{ij}\cdot
g_{*}^{ij}-\beta^{k}\mbox{div}_{g_{t}}\beta$
$\displaystyle+\frac{1}{2}g_{*}^{kh}\partial_{h}\\{g(\beta,\beta)-N^{2}\\}+\beta^{i}\cdot\left({}^{t}\nabla_{i}\beta^{k}\right)-\beta^{i}g_{*}^{hk}g_{ip}\cdot\left({}^{t}\nabla_{h}\beta^{p}\right).$

Restricting (3.9) and (3.5) to $\mathbb{R}^{3}$ and then substituting (3.5)
into them lead to

(3.12)
$\displaystyle\partial_{t}N|_{t=0}=-\bar{N}^{2}\cdot\mbox{tr}_{\bar{g}}K+\bar{\beta}(\bar{N})$

and

$\displaystyle\partial_{t}\beta^{k}|_{t=0}$ $\displaystyle=$
$\displaystyle(\bar{N}^{2}\bar{g}^{ij}-\bar{\beta}^{i}\bar{\beta}^{j})\bar{\Gamma}^{k}_{ij}+\frac{1}{2}\bar{\beta}^{k}\cdot\mbox{tr}_{\bar{g}}(L_{\bar{\beta}}\bar{g})-\bar{\beta}^{k}\mbox{div}_{\bar{g}}\bar{\beta}$
$\displaystyle+\frac{1}{2}\bar{g}^{kh}\partial_{h}\\{\bar{g}(\bar{\beta},\bar{\beta})-\bar{N}^{2}\\}+\bar{\beta}^{i}\cdot\bar{\nabla}_{i}\bar{\beta}^{k}-\bar{\beta}^{i}\bar{g}^{hk}\bar{g}_{ip}\cdot\bar{\nabla}_{h}\bar{\beta}^{p}.$

###  3.6. Local existence and uniqueness in the wave coordinates and
Lorentzian gauge

Thanks to Subsection 3.4 and 3.5, we have formulated the intrinsic Cauchy
problem for EYMH equations in the form of standard PDE analyses. Hence, one
can now use the results in Appendix 3 of [1] to obtain a local in time, global
in space, existence and uniqueness theorem in the wave coordinates and
Lorentzian gauges. The methods we rely on are almost the same as those of
Section 7 and 8 in Chapter 6 of [1]. Before getting the local existence and
uniqueness theorem, we need two lemmas.

######  Lemma 3.1.

If $(g,A,\Phi)$ satisfies the EYMH equations in the wave coordinates and
Lorentzian gauge, then the wave functions $\hat{F}^{\lambda}$ and the function
$\mbox{div}_{g}A$ satisfy a system of second order and linear homogeneous
differential equations with principal terms the wave equation in the metric
$g$.

Proof. It is easy if the readers apply Bianchi identities. The process of
proof is almost the same as that of Lemma 10.1 in Chapter 6 of [1]. Hence we
omit it. $\hfill\Box$

######  Lemma 3.2.

Given a solution of the EYMH equations in wave coordinates and Lorentzian
gauge, whose initial datum satisfy $\hat{F}^{\lambda}|_{t=0}=0$ and
$\mbox{div}_{g}A|_{t=0}=0$, the conditions
$\partial_{t}\hat{F}^{\lambda}|_{t=0}=0$ and
$\partial_{t}(\mbox{div}_{g}A)|_{t=0}=0$ are satisfied if and only if the
initial datum satisfy the EYMH constraints.

Proof. The result follows from straightforward computation. $\hfill\Box$

######  Theorem 3.3.

Let $e$ be the standard Euclidean metric of $\mathbb{R}^{3}$ and $\tilde{D}$
is the Levi-Civita connection of $(\mathbb{R}^{3},e)$.

Hypotheses on the initial datum sets
$(\mathbb{R}^{3},\bar{g},K,\bar{A}_{0},\bar{A}^{\mbox{space}},\bar{E},\bar{\beta},\bar{N},\bar{\Phi},\bar{\Psi})$
and
$(\partial_{t}g_{ij}|_{t=0},\partial_{t}A_{0}|_{t=0},\partial_{t}A_{i}|_{t=0},\partial_{t}\beta^{k}|_{t=0},\partial_{t}N|_{t=0})$:

1\. $\bar{g}$ is a Riemannian metric on $\mathbb{R}^{3}$ uniformly equivalent
to $e$ and such that

$\displaystyle\tilde{D}\bar{g}\in
H_{s-1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar{g}\in\bar{C}^{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,s\in\mathbb{Z}\cap\left[3,\infty\right),$

where $\mathbb{Z}$ is the set of all the integers. Furthermore,
$(\partial_{t}g_{ij})|_{t=0}$ is given by (3.5).

2\. $K$ is a symmetric 2-tensor on $\mathbb{R}^{3}$ such that $K\in H_{s-1}.$

3\. $\bar{A}_{0}$ belongs to $\bar{C}^{0}$ and $\partial\bar{A}_{0}\in
H_{s-1}$. And $(\partial_{t}A_{0})|_{t=0}$ is given by (3.8).

4\. $\bar{A}^{\mbox{space}}\in\bar{C}^{0}$ and
$\tilde{D}(\bar{A}^{\mbox{space}})\in H_{s-1}$. Moreover,
$(\partial_{t}A_{i})|_{t=0}$ is given by (3.7).

5\. $\bar{\beta}\in\bar{C}^{0}$ and $\tilde{D}\bar{\beta}\in H_{s-1}$.
Moreover, there exists a positive constant $b$ such that
$e(\bar{\beta},\bar{\beta})\leqslant b$. And $\partial_{t}\beta^{k}|_{t=0}$ is
given by (3.5).

6\. $\bar{N}\in\bar{C}^{0}$ and $\partial\bar{N}\in H_{s-1}$. Moreover, there
exist two positive constants $C_{1}$ and $C_{2}$ such that
$C_{1}\leqslant\bar{N}\leqslant C_{2}$. Besides, $\partial_{t}N|_{t=0}$ is
given by (3.12).

7\. $\bar{E}\in H_{s-1}$.

8\. $\bar{\Phi}\in\bar{C}^{0}$ and $\partial\bar{\Phi}\in H_{s-1}$.

9\. $\bar{\Psi}\in H_{s-1}$.

10\.
$(\mathbb{R}^{3},\bar{g},K,\bar{A}_{0},\bar{A}^{\mbox{space}},\bar{E},\bar{\beta},\bar{N},\bar{\Phi},\bar{\Psi})$
satisfies the EYMH constraints.

Conclusions:

The initial datum sets admit a development $(V_{T},g,A,\Phi)$ for some $T>0$,
such that $A\in\tilde{E}_{s}(T)$, the spacetime metric $g$ is a regular sliced
Lorentzian metric in $\tilde{E}_{s}(T)$, $\Phi\in\tilde{E}_{s}(T)$ and
$(g,A,\Phi)$ satisfies on $V_{T}$ the EYMH equations. Furthermore,
$(g,A,\Phi)$ meets the wave coordinates and Lorentzian gauge conditions.

Two such developments in the wave coordinates and Lorentzian gauge
$(V_{T},g_{1},A_{1},\Phi_{1})$ and $(V_{T},g_{2},A_{2},\Phi_{2})$, which are
in $\tilde{E}_{s}(T)$, and which take the same initial values
$(\bar{g},K,\bar{A}_{0},\bar{A}^{\mbox{space}},\bar{E},\bar{\beta},\bar{N},\bar{\Phi},\bar{\Psi})$
on $\mathbb{R}^{3}$, coincide on $V_{T}$.

Sketch of the proof. Note that (3.4) are quasi-diagonal, hyperquasi-
linear(i.e. $h^{\lambda\mu}$ depends on $u$ but not on $\textbf{\mbox{D}}u$),
second order systems of the type treated in Appendix 3 of [1]. They satisfies
the hypotheses enunciated in that appendix. So the existence and uniqueness
theorem for (3.4) then follows.

By Lemma 3.2 we know that, since the initial datum satisfy the EYMH
constraints and

(3.14)
$\displaystyle\hat{F}^{\lambda}|_{t=0}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\mbox{div}_{g}A)|_{t=0}=0,$

the following identities hold true

$\displaystyle\partial_{t}\hat{F}^{\lambda}|_{t=0}=0\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\partial_{t}(\mbox{div}_{g}A)|_{t=0}=0,$

where it is obvious that the conditions (3.7), (3.8), (3.5), (3.12) and (3.5)
lead to (3.14). Furthermore, Lemma 3.1 tells us that, if $(g,A,\Phi)$
satisfies (3.4), then $\hat{F}^{\lambda}$ and $\mbox{div}_{g}A$ satisfy a
system of second order linear homogeneous differential equations with
principal terms the wave equation in the metric $g$. Combining the above two
lemmas we arrive at that

$\hat{F}^{\lambda}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{div}_{g}A=0,$

provided the initial datum satisfy the EYMH constraints and (3.14). Hence, a
solution for (3.4), with initial datum satisfying the EYMH constraints and
(3.14), is a solution for the full EYMH system. $\hfill\Box$

##  4\. The equations of $(h:=g-m,A,\Phi)$

Given the initial datum set
$(\mathbb{R}^{3};\bar{g},K,\bar{A}^{\mbox{space}},\bar{A}_{0},\bar{E},\bar{\beta}\equiv
0,\bar{N},\bar{\Phi},\bar{\Psi})$ satisfying the EYMH constraints:

$\displaystyle\bar{R}-|K|^{2}_{\bar{g}}+(\mbox{tr}_{\bar{g}}K)^{2}$
$\displaystyle=$
$\displaystyle\bar{g}^{ik}\bar{g}^{jl}\langle\bar{F}_{ij},\bar{F}_{kl}\rangle/2+3\bar{g}_{ij}\langle\bar{E}^{i},\bar{E}^{j}\rangle+2|\bar{\Psi}|^{2}/\bar{N}^{2}$
$\displaystyle+2\\{\bar{g}^{ij}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi})^{{\dagger}}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{j}\bar{\Phi})+V(|\bar{\Phi}|^{2})\\}$

with
$\bar{F}:=d\bar{A}^{\mbox{space}}+[\bar{A}^{\mbox{space}},\bar{A}^{\mbox{space}}]$,

$\displaystyle(\mbox{div}_{\bar{g}}K)_{i}-\partial_{i}(\mbox{tr}_{\bar{g}}K)=\langle\bar{E}^{l},\bar{F}_{li}\rangle-\bar{N}^{-1}(\bar{\Psi})^{{\dagger}}\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi}-\bar{N}^{-1}(\hat{\hbox
to0.0pt{/\hss}\nabla}_{i}\bar{\Phi})^{{\dagger}}\bar{\Psi}$

and

$\displaystyle\bar{\Psi}^{{\dagger}}S\bar{\Phi}-\bar{\Phi}^{{\dagger}}S\bar{\Psi}=\bar{N}\cdot[\bar{A}^{\mbox{space}}_{i},\bar{E}^{i}]+\bar{E}(\bar{N})+\bar{N}\mbox{div}_{\bar{g}}\bar{E},$

we are going to get a solution for the EYMH equations. Suppose

$\displaystyle\bar{g}\in H_{N+1},\,\,\,\,\,\,\,\,K\in
H_{N+1},\,\,\,\,\,\,\,\,\bar{E}\in H_{N+1},\,\,\,\,\,\,\,\,\bar{N}\in
H_{N+1},$ $\displaystyle\bar{A}^{\mbox{space}}\in
H_{N+1},\,\,\,\,\,\,\,\,\bar{A}_{0}\in H_{N+1},\,\,\,\,\,\,\,\,\bar{\Phi}\in
H_{N+1},\,\,\,\,\,\,\,\,\bar{\Psi}\in H_{N},$

where the “$N$” in “$H_{N+1}$” is the same as that in “$\mathcal{E}_{N}$” and
is an integer not smaller than 4. Furthermore, we have to assume that
$\bar{g}$ is uniformly equivalent to $e$ and $\bar{N}$ is bounded above and
below by some positive constants.

In order to satisfy the wave coordinates and Lorentzian gauge conditions, we
define the initial datum $\Phi|_{t=0}$, $\partial_{t}\Phi|_{t=0}$,
$g_{\mu\nu}|_{t=0}$, $\partial_{t}g_{\mu\nu}|_{t=0}$, $A_{\alpha}|_{t=0}$, and
$\partial_{t}A_{\alpha}|_{t=0}$ as follows:

(4.1) $\displaystyle
g_{ij}|_{t=0}:=\bar{g}_{ij},\,\,\,\,\,\,\,\,g_{00}|_{t=0}:=-\bar{N}^{2},\,\,\,\,\,\,\,\,g_{0i}|_{t=0}:=0,$
(4.2) $\displaystyle
A^{\mbox{space}}|_{t=0}:=\bar{A}^{\mbox{space}},\,\,\,\,\,\,\,\,A^{\mbox{time}}|_{t=0}:=\bar{A}_{0}dt|_{t=0},\,\,\,\,\,\,\,\,\Phi|_{t=0}:=\bar{\Phi},$
(4.3)
$\displaystyle\partial_{t}g_{ij}|_{t=0}:=-2\bar{N}K_{ij},\,\,\,\,\,\,\,\,\partial_{t}g_{00}|_{t=0}:=2\bar{N}^{3}\cdot\mbox{tr}_{\bar{g}}K,$
(4.4)
$\displaystyle\partial_{t}g_{0l}|_{t=0}:=\bar{N}^{2}\bar{g}^{ij}\partial_{j}\bar{g}_{il}-\frac{1}{2}\bar{N}^{2}\bar{g}^{ij}\partial_{l}\bar{g}_{ij}-\bar{N}\partial_{l}\bar{N},$
(4.5)
$\displaystyle\partial_{t}A_{0}|_{t=0}:=\bar{N}^{2}\bar{g}^{ij}\partial_{i}\bar{A}^{\mbox{space}}_{j},\,\,\,\,\partial_{t}A_{i}|_{t=0}:=\partial_{i}\bar{A}_{0}-[\bar{A}_{0},\bar{A}^{\mbox{space}}_{i}]+\bar{N}\cdot\bar{E}^{j}\cdot\bar{g}_{ji},$
(4.6)
$\displaystyle\partial_{t}\Phi|_{t=0}:=\bar{\Psi}+\sqrt{-1}(\bar{A}_{0})^{I}\sigma_{I}\bar{\Phi}/2.$

From (2.4) it follows that giving $\partial_{t}g_{0l}|_{t=0}$ and
$\partial_{t}g_{00}|_{t=0}$ is equivalent to giving
$\partial_{t}\beta^{k}|_{t=0}$ and $\partial_{t}N|_{t=0}$.

Now we obtain a solution $(g,A,\Phi)\in\tilde{E}_{N+1}(T)$ to the EYMH
equations for some $T>0$, which also satisfies the wave coordinates and
Lorentzian gauge conditions.

On the other hand, from (3.17) of [11] it follows that

$\displaystyle
Ric_{\mu\nu}=-\frac{1}{2}\overset{\sim}{\Box}_{g}g_{\mu\nu}+\frac{1}{2}\tilde{P}(\partial_{\mu}g,\partial_{\nu}g)+\frac{1}{2}\tilde{Q}_{\mu\nu}(\partial
g,\partial g),$

where

$\displaystyle\tilde{P}(\partial_{\mu}g,\partial_{\nu}g):=g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}\cdot\left(\frac{1}{4}\partial_{\mu}g_{\beta\beta^{\prime}}\partial_{\nu}g_{\alpha\alpha^{\prime}}-\frac{1}{2}\partial_{\nu}g_{\alpha\beta}\partial_{\mu}g_{\alpha^{\prime}\beta^{\prime}}\right)$

and

$\displaystyle\tilde{Q}_{\mu\nu}(\partial g,\partial
g):=g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}\partial_{\alpha}g_{\beta\mu}\partial_{\alpha^{\prime}}g_{\beta^{\prime}\nu}-g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}\left(\partial_{\alpha}g_{\beta\mu}\partial_{\beta^{\prime}}g_{\alpha^{\prime}\nu}-\partial_{\beta^{\prime}}g_{\beta\mu}\partial_{\alpha}g_{\alpha^{\prime}\nu}\right)$
$\displaystyle+g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}(\partial_{\mu}g_{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha}g_{\beta\nu}-\partial_{\alpha}g_{\alpha^{\prime}\beta^{\prime}}\partial_{\mu}g_{\beta\nu})+g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}(\partial_{\nu}g_{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha}g_{\beta\mu}-\partial_{\alpha}g_{\alpha^{\prime}\beta^{\prime}}\partial_{\nu}g_{\beta\mu})$
$\displaystyle+\frac{1}{2}g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}(\partial_{\beta^{\prime}}g_{\alpha\alpha^{\prime}}\partial_{\mu}g_{\beta\nu}-\partial_{\mu}g_{\alpha\alpha^{\prime}}\partial_{\beta^{\prime}}g_{\beta\nu})+\frac{1}{2}g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}(\partial_{\beta^{\prime}}g_{\alpha\alpha^{\prime}}\partial_{\nu}g_{\beta\mu}-\partial_{\nu}g_{\alpha\alpha^{\prime}}\partial_{\beta^{\prime}}g_{\beta\mu}).$

Hence, (2.1) is equivalent to

$\displaystyle\overset{\sim}{\Box}_{g}g_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-2g^{\mu\lambda}\langle\partial_{\alpha}A_{\mu}-\partial_{\mu}A_{\alpha}+[A_{\alpha},A_{\mu}],\partial_{\beta}A_{\lambda}-\partial_{\lambda}A_{\beta}+[A_{\beta},A_{\lambda}]\rangle$
$\displaystyle+\frac{1}{2}g_{\alpha\beta}g^{\rho\sigma}g^{\mu\lambda}\langle\partial_{\rho}A_{\mu}-\partial_{\mu}A_{\rho}+[A_{\rho},A_{\mu}],\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}+[A_{\sigma},A_{\lambda}]\rangle$
$\displaystyle+\tilde{P}(\partial_{\alpha}g,\partial_{\beta}g)+\tilde{Q}_{\alpha\beta}(\partial
g,\partial
g)-(\hat{\nabla}_{\alpha}\Phi)^{{\dagger}}\hat{\nabla}_{\beta}\Phi-(\hat{\nabla}_{\beta}\Phi)^{{\dagger}}\hat{\nabla}_{\alpha}\Phi-
g_{\alpha\beta}V(|\Phi|^{2}).$

Define two 2-tensors

$\displaystyle
h_{\mu\nu}:=g_{\mu\nu}-m_{\mu\nu}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,H^{\mu\nu}:=g^{\mu\nu}-m^{\mu\nu},$

where $m^{\mu\nu}$ and $g^{\mu\nu}$ are the inverses of $m_{\mu\nu}$ and
$g_{\mu\nu}$ respectively(Recall that $m_{\mu\nu}$ is the Minkowski metric of
$\mathbb{R}^{4}$). We want to obtain the equation of $h_{\mu\nu}$. From Lemma
4.1 of [10] it follows that

(4.8) $\displaystyle H^{\mu\nu}=-h^{\mu\nu}+O^{\mu\nu}(h^{2}),$

where $O^{\mu\nu}(h^{2})$ is a two-tensor vanishing to the second order at
$h=0$. Besides, by Lemma 3.2 of [11], we know that if $h$ is small, (4) is
equivalent to

$\displaystyle\overset{\sim}{\Box}_{g}h_{\mu\nu}$ $\displaystyle=$
$\displaystyle P(\partial_{\mu}h,\partial_{\nu}h)+Q_{\mu\nu}(\partial
h,\partial h)+G_{\mu\nu}(h)(\partial h,\partial
h)-(h_{\mu\nu}+m_{\mu\nu})V(|\Phi|^{2})$
$\displaystyle+2(h^{\alpha\beta}-m^{\alpha\beta})\langle\partial_{\alpha}A_{\mu}-\partial_{\mu}A_{\alpha}+[A_{\alpha},A_{\mu}],\partial_{\beta}A_{\nu}-\partial_{\nu}A_{\beta}+[A_{\beta},A_{\nu}]\rangle$
$\displaystyle-2O^{\alpha\beta}(h^{2})\langle\partial_{\alpha}A_{\mu}-\partial_{\mu}A_{\alpha}+[A_{\alpha},A_{\mu}],\partial_{\beta}A_{\nu}-\partial_{\nu}A_{\beta}+[A_{\beta},A_{\nu}]\rangle$
$\displaystyle+\frac{1}{2}\\{m^{\alpha\rho}m^{\beta\sigma}h_{\mu\nu}+m^{\alpha\rho}m^{\beta\sigma}m_{\mu\nu}-h^{\alpha\rho}m^{\beta\sigma}m_{\mu\nu}-m^{\alpha\rho}h^{\beta\sigma}m_{\mu\nu}$
$\displaystyle+O^{\alpha\rho\beta\sigma}_{\mu\nu}(h^{2})\\}\langle\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}+[A_{\alpha},A_{\beta}],\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}+[A_{\rho},A_{\sigma}]\rangle$
$\displaystyle-(\partial_{\mu}\Phi-\sqrt{-1}A^{I_{1}}_{\mu}\sigma_{I_{1}}\Phi/2)^{{\dagger}}(\partial_{\nu}\Phi-\sqrt{-1}A^{I_{2}}_{\nu}\sigma_{I_{2}}\Phi/2)$
$\displaystyle-(\partial_{\nu}\Phi-\sqrt{-1}A^{I_{2}}_{\nu}\sigma_{I_{2}}\Phi/2)^{{\dagger}}(\partial_{\mu}\Phi-\sqrt{-1}A^{I_{1}}_{\mu}\sigma_{I_{1}}\Phi/2)$
$\displaystyle:=$ $\displaystyle F_{\mu\nu}$

where $O^{\alpha\rho\beta\sigma}_{\mu\nu}(h^{2})$ vanishes to the second order
at $h=0$,

$\displaystyle
P(\partial_{\mu}h,\partial_{\nu}h):=m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}\cdot\left(\frac{1}{4}\partial_{\mu}h_{\beta\beta^{\prime}}\partial_{\nu}h_{\alpha\alpha^{\prime}}-\frac{1}{2}\partial_{\nu}h_{\alpha\beta}\partial_{\mu}h_{\alpha^{\prime}\beta^{\prime}}\right)$

and

$\displaystyle Q_{\mu\nu}(\partial h,\partial
h):=m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}\partial_{\alpha}h_{\beta\mu}\partial_{\alpha^{\prime}}h_{\beta^{\prime}\nu}-m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}\left(\partial_{\alpha}h_{\beta\mu}\partial_{\beta^{\prime}}h_{\alpha^{\prime}\nu}-\partial_{\beta^{\prime}}h_{\beta\mu}\partial_{\alpha}h_{\alpha^{\prime}\nu}\right)$
$\displaystyle+m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}(\partial_{\mu}h_{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha}h_{\beta\nu}-\partial_{\alpha}h_{\alpha^{\prime}\beta^{\prime}}\partial_{\mu}h_{\beta\nu})+m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}(\partial_{\nu}h_{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha}h_{\beta\mu}-\partial_{\alpha}h_{\alpha^{\prime}\beta^{\prime}}\partial_{\nu}h_{\beta\mu})$
$\displaystyle+\frac{1}{2}m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}(\partial_{\beta^{\prime}}h_{\alpha\alpha^{\prime}}\partial_{\mu}h_{\beta\nu}-\partial_{\mu}h_{\alpha\alpha^{\prime}}\partial_{\beta^{\prime}}h_{\beta\nu})+\frac{1}{2}m^{\alpha\alpha^{\prime}}m^{\beta\beta^{\prime}}(\partial_{\beta^{\prime}}h_{\alpha\alpha^{\prime}}\partial_{\nu}h_{\beta\mu}-\partial_{\nu}h_{\alpha\alpha^{\prime}}\partial_{\beta^{\prime}}h_{\beta\mu}),$

is a null form and $G_{\mu\nu}(h)(\partial h,\partial h)$ is a quadratic form
in $\partial h$ with coefficients smoothly dependent on $h$ and vanishing when
$h$ vanishes, i.e. $G_{\mu\nu}(0)(\partial h,\partial h)=0$.

Using (4.8) again we get

$\displaystyle\overset{\sim}{\Box}_{g}A_{\beta}=\\{-m^{\alpha\theta}h^{\mu\lambda}-h^{\alpha\theta}m^{\mu\lambda}+m^{\alpha\theta}m^{\mu\lambda}+O^{\alpha\theta\mu\lambda}(h^{2})\\}\partial_{\beta}h_{\theta\mu}\partial_{\lambda}A_{\alpha}$
$\displaystyle+\frac{1}{2}\\{-m^{\alpha\lambda}h^{\theta\mu}-h^{\alpha\lambda}m^{\theta\mu}+m^{\alpha\lambda}m^{\theta\mu}+O^{\alpha\lambda\theta\mu}(h^{2})\\}$
$\displaystyle\times(\partial_{\lambda}h_{\mu\beta}+\partial_{\beta}h_{\mu\lambda}-\partial_{\mu}h_{\lambda\beta})(\partial_{\alpha}A_{\theta}-\partial_{\theta}A_{\alpha})-(m^{\alpha\lambda}-h^{\alpha\lambda}+O^{\alpha\lambda}(h^{2}))[A_{\alpha},\partial_{\lambda}A_{\beta}]$
$\displaystyle+(m^{\alpha\lambda}-h^{\alpha\lambda}+O^{\alpha\lambda}(h^{2}))\\{[A_{\lambda},\partial_{\beta}A_{\alpha}]-[A_{\lambda},\partial_{\alpha}A_{\beta}]-[A_{\lambda},[A_{\alpha},A_{\beta}]]\\}$
$\displaystyle+\Phi^{{\dagger}}S(\partial_{\beta}\Phi-\sqrt{-1}A^{I}_{\beta}\sigma_{I}\Phi/2)-(\partial_{\beta}\Phi-\sqrt{-1}A^{I}_{\beta}\sigma_{I}\Phi/2)^{{\dagger}}S\Phi$
$\displaystyle:=J_{\beta},$

provided $h$ is sufficiently small. Easily, the coming identity follows again
from (4.8)

(4.11) $\displaystyle\overset{\sim}{\Box}_{g}\Phi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2}(m^{\lambda\mu}-h^{\lambda\mu}+O^{\lambda\mu}(h^{2}))A_{\mu}^{I_{1}}\sigma_{I_{1}}(\partial_{\lambda}\Phi)+\frac{\sqrt{-1}}{2}(m^{\lambda\mu}-h^{\lambda\mu}+O^{\lambda\mu}(h^{2}))A_{\lambda}^{I_{2}}\sigma_{I_{2}}(\partial_{\mu}\Phi)$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{4}(m^{\lambda\mu}-h^{\lambda\mu}+O^{\lambda\mu}(h^{2}))A^{I_{2}}_{\lambda}A^{I_{1}}_{\mu}\sigma_{I_{2}}\sigma_{I_{1}}(\Phi)+V^{\prime}(|\Phi|^{2})\Phi:=W.$

##  5\. Beginning of the proof of Theorem 1.1

As described in the introduction, $T$ is the maximal existence time of the
solution $(g,A,\Phi)$ and assumed to be finite. We have defined the time
$T^{*}$ to be

(5.1) $\displaystyle T^{*}:=\sup\\{T_{0}:\exists C=C(T),\,s.t.\,\,\,\,\forall
t\in[0,T_{0}),\,\,\,\,\mathcal{E}_{N}(t)\leqslant 2C\varepsilon\\},$

where $\mathcal{E}_{N}$ is given by (1.4). Our goal is to show that if
$\varepsilon>0$ is small enough, then the inequality in (5.1) implies the same
inequality with $2C$ replaced by $C$ for all $t<T^{*}$.

The first step is to derive the preliminary decay estimates for $h$, $A$ and
$\Phi$ under the assumption (5.1). However, our method is the same as that of
Theorem 5.2 of [10]. Hence we only list the result and omit the proof.

######  Theorem 5.1.

Let $h$, $A$ and $\Phi$ verify the inequality in (5.1). Then we have

(5.2) $|\partial Z^{I}h(x,t)|+|\partial Z^{I}A(x,t)|+|\partial
Z^{I}\Phi(x,t)|\lesssim\left\\{\begin{array}[]{rcl}\varepsilon(1+|q|)^{-2-\gamma}&\text{$q>0$,}&\\\
\varepsilon(1+|q|)^{-3/2}&\text{$q<0$,}&\end{array}\right.|I|\leqslant N-3.$

Furthermore

(5.3)
$|Z^{I}h(x,t)|+|Z^{I}A(x,t)|+|Z^{I}\Phi(x,t)|\lesssim\left\\{\begin{array}[]{rcl}\varepsilon(1+|q|)^{-1-\gamma}&\text{$q>0$,}&\\\
\varepsilon(1+|q|)^{-1/2}&\text{$q<0$,}&\end{array}\right.|I|\leqslant N-3.$

And

(5.4)
$|\bar{\partial}Z^{I}h(x,t)|+|\bar{\partial}Z^{I}A(x,t)|+|\bar{\partial}Z^{I}\Phi(x,t)|\lesssim\left\\{\begin{array}[]{rcl}\varepsilon(1+|q|)^{-2-\gamma}&\text{$q>0$,}&\\\
\varepsilon(1+|q|)^{-3/2}&\text{$q<0$,}&\end{array}\right.|I|\leqslant N-4.$

###  5.1. Estimates for the inhomogeneous terms $F_{\mu\nu}$, $J_{\beta}$ and
$W$

(5.3) tells us that $|Z^{I}h|+|Z^{I}A|+|Z^{I}\Phi|\leqslant 1/2$, provided
$\varepsilon$ is small enough and $|I|\leqslant N-3$. The upper bound “1/2”
plays a key role in the sequel.

######  Proposition 5.2.

Assume that $h=g-m$, $A$ and $\Phi$ satisfy the inequality in (5.1). Let
$F_{\mu\nu}$, $J_{\beta}$ and $W$ be as in (4), (4) and (4.11) respectively.
Then we have

$\displaystyle|Z^{I}F|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J|+|K|\leqslant|I|}(|\partial
Z^{J}h|_{\mathcal{T}\mathcal{U}}\cdot|\partial
Z^{K}h|_{\mathcal{T}\mathcal{U}}+|\bar{\partial}Z^{J}h|\cdot|\partial
Z^{K}h|)$ $\displaystyle+\sum\limits_{|J|+|K|\leqslant|I|-1}|\partial
Z^{J}h|_{\mathcal{L}\mathcal{T}}\cdot|\partial
Z^{K}h|+\sum\limits_{|J|+|K|\leqslant|I|-2}|\partial Z^{J}h|\cdot|\partial
Z^{K}h|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{3}}h|\cdot|\partial
Z^{J_{2}}h|\cdot|\partial
Z^{J_{1}}h|+\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial Z^{J_{2}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|+\sum\limits_{|I_{1}|+\cdots+|I_{2s}|=|I|}|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|I_{1}|+\cdots+|I_{2s}|=|I|}|Z^{J_{1}}h|\cdot|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}\Phi|\cdot|\partial
Z^{J_{2}}\Phi|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}\Phi|\cdot|\partial
Z^{J_{3}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|=|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}\Phi|\cdot|Z^{J_{4}}\Phi|,$
$\displaystyle|Z^{I}J|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|I_{1}|+|I_{2}|\leqslant|I|}|\partial
Z^{I_{1}}h|\cdot|\partial
Z^{I_{2}}A|+\sum\limits_{|I_{1}|+|I_{2}|\leqslant|I|}|Z^{I_{1}}A|\cdot|\partial
Z^{I_{2}}A|$
$\displaystyle+\sum\limits_{|I_{1}|+|I_{2}|+|I_{3}|\leqslant|I|}|Z^{I_{1}}A|\cdot|Z^{I_{2}}A|\cdot|Z^{I_{3}}A|+\sum\limits_{|I_{1}|+|I_{2}|\leqslant|I|}|Z^{I_{1}}\Phi|\cdot|\partial
Z^{I_{2}}\Phi|$
$\displaystyle+\sum\limits_{|I_{1}|+|I_{2}|+|I_{3}|=|I|}|Z^{I_{1}}\Phi|\cdot|Z^{I_{2}}\Phi|\cdot|Z^{I_{3}}A|$

and

(5.7) $\displaystyle|Z^{I}W|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|I_{1}|+|I_{2}|\leqslant|I|}|Z^{I_{1}}A|\cdot|\partial
Z^{I_{2}}\Phi|+\sum\limits_{|I_{1}|+|I_{2}|+|I_{3}|\leqslant|I|}|Z^{I_{1}}A|\cdot|Z^{I_{2}}A|\cdot|Z^{I_{3}}\Phi|$
$\displaystyle+\sum\limits_{|I_{1}|+\cdots+|I_{2s+1}|=|I|}|V^{(s+1)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s+1}|Z^{I_{i}}\Phi|,$

provided $|Z^{J}h|\leqslant\tilde{C}<1$ for all multi-indices
$|J|\leqslant|I|$ and vector fields $Z\in\mathfrak{L}$. Here the “$J$” in
(5.2) is the same as that in (4).

Proof. For simplicity, we only show (5.2). The other estimates can be deduced
by the same approach.

Reviewing the definition of $F$ gives

$\displaystyle Z^{I}F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
Z^{I}\\{P(\partial_{\mu}h,\partial_{\nu}h)+Q_{\mu\nu}(\partial h,\partial
h)+G_{\mu\nu}(h)(\partial h,\partial h)\\}$
$\displaystyle+\mbox{Term}_{1}+\mbox{Term}_{2}+\mbox{Term}_{3}+\mbox{Term}_{4}+\mbox{Term}_{5}+\mbox{Term}_{6}+\mbox{Term}_{7},$

where

$\displaystyle\mbox{Term}_{1}:=2Z^{I}\\{(h^{\alpha\beta}-m^{\alpha\beta})\langle\partial_{\alpha}A_{\mu}-\partial_{\mu}A_{\alpha}+[A_{\alpha},A_{\mu}],\partial_{\beta}A_{\nu}-\partial_{\nu}A_{\beta}+[A_{\beta},A_{\nu}]\rangle\\},$
$\displaystyle\mbox{Term}_{2}:=-2Z^{I}\\{O^{\alpha\beta}(h^{2})\langle\partial_{\alpha}A_{\mu}-\partial_{\mu}A_{\alpha}+[A_{\alpha},A_{\mu}],\partial_{\beta}A_{\nu}-\partial_{\nu}A_{\beta}+[A_{\beta},A_{\nu}]\rangle\\},$
$\displaystyle\mbox{Term}_{3}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}Z^{I}\big{\\{}(m^{\alpha\rho}m^{\beta\sigma}h_{\mu\nu}+m^{\alpha\rho}m^{\beta\sigma}m_{\mu\nu}-h^{\alpha\rho}m^{\beta\sigma}m_{\mu\nu}-m^{\alpha\rho}h^{\beta\sigma}m_{\mu\nu})$
$\displaystyle\times\langle\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}+[A_{\alpha},A_{\beta}],\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}+[A_{\rho},A_{\sigma}]\rangle\big{\\}},$
$\displaystyle\mbox{Term}_{4}:=Z^{I}\left\\{O^{\alpha\rho\beta\sigma}_{\mu\nu}(h^{2})\langle\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}+[A_{\alpha},A_{\beta}],\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}+[A_{\rho},A_{\sigma}]\rangle\right\\},$
$\displaystyle\mbox{Term}_{5}:=-Z^{I}\\{(h_{\mu\nu}+m_{\mu\nu})V(|\Phi|^{2})\\},$
$\displaystyle\mbox{Term}_{6}:=-Z^{I}\\{(\partial_{\mu}\Phi-\sqrt{-1}A^{I_{1}}_{\mu}\sigma_{I_{1}}\Phi/2)^{{\dagger}}(\partial_{\nu}\Phi-\sqrt{-1}A^{I_{2}}_{\nu}\sigma_{I_{2}}\Phi/2)\\}$

and

$\displaystyle\mbox{Term}_{7}:=-Z^{I}\\{(\partial_{\nu}\Phi-\sqrt{-1}A^{I_{2}}_{\nu}\sigma_{I_{2}}\Phi/2)^{{\dagger}}(\partial_{\mu}\Phi-\sqrt{-1}A^{I_{1}}_{\mu}\sigma_{I_{1}}\Phi/2)\\}.$

(9.28) of [12] tells us that, if $|Z^{J}h|\leqslant\tilde{C}<1$ for all multi-
indices $|J|\leqslant|I|$ and vector fields $Z\in\mathfrak{L}$, one will
obtain

$\displaystyle|Z^{I}(P+Q+G)|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J|+|K|\leqslant|I|}(|\partial
Z^{J}h|_{\mathcal{T}\mathcal{U}}\cdot|\partial
Z^{K}h|_{\mathcal{T}\mathcal{U}}+|\bar{\partial}Z^{J}h|\cdot|\partial
Z^{K}h|)$ $\displaystyle+\sum\limits_{|J|+|K|\leqslant|I|-1}|\partial
Z^{J}h|_{\mathcal{L}\mathcal{T}}\cdot|\partial
Z^{K}h|+\sum\limits_{|J|+|K|\leqslant|I|-2}|\partial Z^{J}h|\cdot|\partial
Z^{K}h|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{3}}h|\cdot|\partial
Z^{J_{2}}h|\cdot|\partial Z^{J_{1}}h|.$

Moreover, it is easy to get

$\displaystyle|\mbox{Term}_{1}|\lesssim\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}\Bigg{|}Z^{J_{1}}\partial
A+\sum\limits_{|L_{1}|+|L_{2}|=|J_{1}|}|Z^{L_{1}}A|\cdot|Z^{L_{2}}A|\Bigg{|}\cdot\Bigg{|}Z^{J_{2}}\partial
A+\sum\limits_{|L_{3}|+|L_{4}|=|J_{2}|}|Z^{L_{3}}A|\cdot|Z^{L_{4}}A|\Bigg{|}.$

From induction argument it follows that for any multi-index $I$, there exist a
set of universal constants $\\{C_{J}:0\leqslant|J|\leqslant|I|\\}$ such that

(5.9) $\displaystyle
Z^{I}\partial_{\alpha}=\sum\limits_{0\leqslant|J|\leqslant|I|}C_{J}\cdot\partial_{\alpha}Z^{J},$

implying

$\displaystyle|\mbox{Term}_{1}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial
Z^{J_{2}}A|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|.$

The same method leads to

$\displaystyle|\mbox{Term}_{2}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial
Z^{J_{2}}A|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|,$
$\displaystyle|\mbox{Term}_{3}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial
Z^{J_{2}}A|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|$

and

$\displaystyle|\mbox{Term}_{4}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial
Z^{J_{2}}A|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|.$

On the other hand, by elementary computation we obtain

$\displaystyle|\mbox{Term}_{5}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|I_{1}|+\cdots+|I_{2s}|=|I|}|Z^{J_{1}}h|\cdot|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|$
$\displaystyle+\sum\limits_{|I_{1}|+\cdots+|I_{2s}|=|I|}|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|.$

Furthermore, the following estimates are not difficult to get:

$\displaystyle|\mbox{Term}_{6}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}\Phi|\cdot|\partial
Z^{J_{2}}\Phi|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}\Phi|\cdot|\partial
Z^{J_{3}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|=|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}\Phi|\cdot|Z^{J_{4}}\Phi|$

and

$\displaystyle|\mbox{Term}_{7}|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}\Phi|\cdot|\partial
Z^{J_{2}}\Phi|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}\Phi|\cdot|\partial
Z^{J_{3}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|=|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}\Phi|\cdot|Z^{J_{4}}\Phi|.$

Then the result of this proposition follows. $\hfill\Box$

##  6\. Energy estimates for the EYMH equations

In this section we prove the following result.

######  Theorem 6.1.

Let $h_{\mu\nu}=g_{\mu\nu}-m_{\mu\nu}$, $A_{\beta}$ and $\Phi$ be a local in
time solution to (4), (4) and $(\ref{99})$ respectively satisfying the wave
coordinates and Lorentzian gauge conditions on the interval $[0,T^{*})$.
Suppose also $\gamma\in(0,1/2)$ and $\mu\in(0,1/2)$. Assume that we have the
following estimates for $t\in[0,T^{*})$ and all multi-indices $|I|\leqslant
N-4$:

(6.1) $\displaystyle|\partial
H|_{\mathcal{T}\mathcal{U}}+\frac{|H|_{\mathcal{L}\mathcal{T}}}{1+|q|}+\frac{|ZH|_{\mathcal{L}\mathcal{L}}}{1+|q|}\lesssim\varepsilon,$
(6.4) $\displaystyle|\partial
Z^{I}h|+\frac{|Z^{I}h|}{1+|q|}+|\bar{\partial}Z^{I}h|\leqslant\left\\{\begin{array}[]{rcl}&C\varepsilon(1+|q|)^{-2-\gamma}&\quad\text{when
$q>0$,}\\\ &C\varepsilon(1+|q|)^{-3/2}&\quad\text{when
$q<0$,}\end{array}\right.$ (6.7) $\displaystyle|\partial
Z^{I}A|+\frac{|Z^{I}A|}{1+|q|}+|\bar{\partial}Z^{I}A|\leqslant\left\\{\begin{array}[]{rcl}&C\varepsilon(1+|q|)^{-2-\gamma}&\quad\text{when
$q>0$,}\\\ &C\varepsilon(1+|q|)^{-3/2}&\quad\text{when
$q<0$,}\end{array}\right.$ (6.10) $\displaystyle|\partial
Z^{I}\Phi|+\frac{|Z^{I}\Phi|}{1+|q|}+|\bar{\partial}Z^{I}\Phi|\leqslant\left\\{\begin{array}[]{rcl}&C\varepsilon(1+|q|)^{-2-\gamma}&\quad\text{when
$q>0$,}\\\ &C\varepsilon(1+|q|)^{-3/2}&\quad\text{when
$q<0$,}\end{array}\right.$

and

(6.11) $\displaystyle\mathcal{E}_{N}(0)\leqslant\varepsilon.$

Then there is a positive constant $C^{\prime}$ dependent of $T$ such that we
have the energy estimate

$\displaystyle\mathcal{E}_{N}(t)\leqslant C^{\prime}\varepsilon^{2},$

for all $t\in[0,T^{*})$.

Assuming the conclusions of Theorem 6.1 for a moment we finish the proof of
the main Theorem 1.1.

###  6.1. End of the proof of Theorem 1.1

Recall that $T^{*}$ was defined as the maximal time with the property that the
bound

$\mathcal{E}_{N}(t)\leqslant 2C\varepsilon$

holds for all $t\in[0,T^{*})$. Direct check shows that the estimates of
Theorem 5.1 imply the assumption (6.1)-(6.10). The conclusion of Theorem 6.1
states that the energy

$\mathcal{E}_{N}(t)\leqslant
C^{\prime}\varepsilon^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,T^{*}).$

Thus choosing a sufficiently small $\varepsilon>0$ we can show that
$\mathcal{E}_{N}(t)\leqslant C\varepsilon$ thus contracting the maximality of
$T^{*}$ and consequently proving that $(g,A,\Phi)$ is a global solution.
Therefore, it remains to prove Theorem 6.1.

###  6.2. Proof of Theorem 6.1

Recall that $h_{\mu\nu}$, $A_{\beta}$ and $\Phi$ satisfy the wave equations
$\overset{\sim}{\Box}_{g}h_{\mu\nu}=F_{\mu\nu}$,
$\overset{\sim}{\Box}_{g}A_{\beta}=J_{\beta}$ and
$\overset{\sim}{\Box}_{g}\Phi=W$ respectively. Our goal is to compute the
energy norms of $Z^{I}h$, $Z^{I}A$ and $Z^{I}\Phi$, where $Z\in\mathfrak{L}$.

From (11.10) of [12] it follows that

$\overset{\sim}{\Box}_{g}Z^{I}h_{\mu\nu}=F^{I}_{\mu\nu}$

with

$F^{I}:=\hat{Z}^{I}F-D^{I},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,D^{I}:=\hat{Z}^{I}\overset{\sim}{\Box}_{g}h^{1}-\overset{\sim}{\Box}_{g}Z^{I}h^{1},\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\hat{Z}:=Z+c_{Z}.$

Similarly, we also have

$\overset{\sim}{\Box}_{g}Z^{I}A=\hat{Z}^{I}J-W^{I}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\overset{\sim}{\Box}_{g}Z^{I}\Phi=\hat{Z}^{I}W-L^{I},$

where
$W^{I}:=\hat{Z}^{I}\overset{\sim}{\Box}_{g}A-\overset{\sim}{\Box}_{g}Z^{I}A$
and
$L^{I}:=\hat{Z}^{I}\overset{\sim}{\Box}_{g}\Phi-\overset{\sim}{\Box}_{g}Z^{I}\Phi$.
(11.13) of [12] tells us

$\displaystyle\int_{\Sigma_{t}}|\partial
Z^{I}h|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}|\bar{\partial}Z^{I}h|^{2}w^{\prime}$
$\displaystyle\lesssim$ $\displaystyle\int_{\Sigma_{0}}|\partial
Z^{I}h|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}\Big{\\{}\frac{\varepsilon|\partial
Z^{I}h|^{2}}{1+\tau}w+\frac{w\cdot(1+\tau)}{\varepsilon}(|\hat{Z}^{I}F|^{2}+|D^{I}|^{2})\Big{\\}},$
$\displaystyle\lesssim$ $\displaystyle\int_{\Sigma_{0}}|\partial
Z^{I}h|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}\Big{\\{}\varepsilon|\partial
Z^{I}h|^{2}w+\frac{w}{\varepsilon}(|\hat{Z}^{I}F|^{2}+|D^{I}|^{2})\Big{\\}},$

where $\Sigma_{t}:=\mathbb{R}^{3}\times\\{t\\}$. Applying the same methods
yields

$\displaystyle\int_{\Sigma_{t}}|\partial
Z^{I}A|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}|\bar{\partial}Z^{I}A|^{2}w^{\prime}$
$\displaystyle\lesssim$ $\displaystyle\int_{\Sigma_{0}}|\partial
Z^{I}A|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}\left\\{\varepsilon|\partial
Z^{I}A|^{2}w+\frac{w}{\varepsilon}(|\hat{Z}^{I}J|^{2}+|W^{I}|^{2})\right\\}.$

and

$\displaystyle\int_{\Sigma_{t}}|\partial
Z^{I}\Phi|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}|\bar{\partial}Z^{I}\Phi|^{2}w^{\prime}$
$\displaystyle\lesssim$ $\displaystyle\int_{\Sigma_{0}}|\partial
Z^{I}\Phi|^{2}w+\int_{0}^{t}\int_{\Sigma_{\tau}}\left\\{\varepsilon|\partial
Z^{I}\Phi|^{2}w+\frac{w}{\varepsilon}(|\hat{Z}^{I}W|^{2}+|L^{I}|^{2})\right\\}.$

We begin with the following estimates on the inhomogeneous terms $F$, $J$ and
$W$.

######  Lemma 6.2.

Under the assumptions of Theorem 6.1, we have

$\displaystyle|Z^{I}F|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|K|\leqslant|I|}\left\\{\varepsilon|\partial
Z^{K}h|+\varepsilon(1+|q|)^{-3/2}|\bar{\partial}Z^{K}h|+\varepsilon^{2}\frac{|Z^{K}h|}{(1+|q|)^{3}}\right\\}$
$\displaystyle+\varepsilon(1+|q|)^{-1}\sum\limits_{|K|\leqslant|I|}(|\partial
Z^{K}A|+|\partial
Z^{K}\Phi|)+\varepsilon^{2\tilde{\gamma}}(1+|q|)^{-\tilde{\gamma}}$
$\displaystyle+\varepsilon^{3}(1+|q|)^{-3/2}\sum\limits_{|K|\leqslant|I|}(|Z^{K}A|+|Z^{K}\Phi|+|Z^{K}h|)$
(6.16) $\displaystyle|Z^{I}J|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|K|\leqslant|I|}|\partial
Z^{K}h|\cdot\varepsilon(1+|q|)^{-1}+\varepsilon^{2}(1+|q|)^{-1}\sum\limits_{|K|\leqslant|I|}|Z^{K}A|$
$\displaystyle+\varepsilon\sum\limits_{|K|\leqslant|I|}(|\partial
Z^{K}A|+|\partial Z^{K}\Phi|)(1+|q|)^{-1/2}$

and

(6.17) $\displaystyle|Z^{I}W|$ $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|K|\leqslant|I|}|\partial
Z^{K}\Phi|\cdot\varepsilon(1+|q|)^{-1/2}+\varepsilon^{2}(1+|q|)^{-1}\sum\limits_{|K|\leqslant|I|}|Z^{K}\Phi|$

Proof. We only prove (6.2), since the other cases are easy(Note that
throughout the process we have to use the assumption
$\tilde{\gamma}>3/2+\gamma$). According to Proposition 5.2 we have

$\displaystyle|Z^{I}F|$ $\displaystyle\lesssim$
$\displaystyle\mbox{Term}+\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|\partial
Z^{J_{2}}A|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|$
$\displaystyle+\sum\limits_{|I_{1}|+\cdots+|I_{2s}|=|I|}|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|I_{1}|+\cdots+|I_{2s}|=|I|}|Z^{J_{1}}h|\cdot|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}\Phi|\cdot|\partial
Z^{J_{2}}\Phi|+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}\Phi|\cdot|\partial
Z^{J_{3}}\Phi|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|=|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}\Phi|\cdot|Z^{J_{4}}\Phi|,$

where

Term $\displaystyle:=$
$\displaystyle\sum\limits_{|J|+|K|\leqslant|I|}(|\partial
Z^{J}h|_{\mathcal{T}\mathcal{U}}\cdot|\partial
Z^{K}h|_{\mathcal{T}\mathcal{U}}+|\bar{\partial}Z^{J}h|\cdot|\partial
Z^{K}h|)$ $\displaystyle+\sum\limits_{|J|+|K|\leqslant|I|-1}|\partial
Z^{J}h|_{\mathcal{L}\mathcal{T}}\cdot|\partial
Z^{K}h|+\sum\limits_{|J|+|K|\leqslant|I|-2}|\partial Z^{J}h|\cdot|\partial
Z^{K}h|$
$\displaystyle+\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{3}}h|\cdot|\partial
Z^{J_{2}}h|\cdot|\partial Z^{J_{1}}h|.$

From Theorem 5.1 it follows that

(6.18) Term $\displaystyle\lesssim$
$\displaystyle\sum\limits_{|K|\leqslant|I|}\left\\{\varepsilon|\partial
Z^{K}h|+\varepsilon(1+|q|)^{-3/2}|\bar{\partial}Z^{K}h|+\varepsilon^{2}\frac{|Z^{K}h|}{(1+|q|)^{3}}\right\\}.$

In addition, (6.7) implies

(6.19) $\displaystyle\sum\limits_{|I_{1}|+|I_{2}|\leqslant|I|}|\partial
Z^{I_{1}}A|\cdot|\partial
Z^{I_{2}}A|\lesssim\varepsilon(1+|q|)^{-3/2}\sum\limits_{|K|\leqslant|I|}|\partial
Z^{K}A|.$

Furthermore, from (6.7) we get

$\displaystyle|Z^{I}A|\leqslant C\varepsilon(1+|q|)^{-1/2}.$

Hence, one can obtain

$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|\partial
Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\leqslant
C^{2}\varepsilon^{2}(1+|q|)^{-1}\sum\limits_{|K|\leqslant|I|}|\partial
Z^{K}A|.$

The same method yields the following inequalities:

$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}A|\cdot|Z^{J_{4}}A|\leqslant
C^{3}\varepsilon^{3}(1+|q|)^{-3/2}\sum\limits_{|K|\leqslant|I|}|Z^{K}A|,$
$\displaystyle\sum\limits_{|I_{1}|+\cdots+|I_{2s}|=|I|}|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|\leqslant
C^{2\tilde{\gamma}}\varepsilon^{2\tilde{\gamma}}(1+|q|)^{-\tilde{\gamma}},$
$\displaystyle\sum\limits_{|J_{1}|+|I_{1}|+\cdots+|I_{2s}|=|I|}|Z^{J_{1}}h|\cdot|V^{(s)}(|\Phi|^{2})|\cdot\prod\limits_{i=1}^{2s}|Z^{I_{i}}\Phi|\leqslant
C^{2\tilde{\gamma}}\varepsilon^{2\tilde{\gamma}}(1+|q|)^{-\tilde{\gamma}}\sum\limits_{|K|\leqslant|I|}|Z^{K}h|,$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|\leqslant|I|}|\partial
Z^{J_{1}}\Phi|\cdot|\partial Z^{J_{2}}\Phi|\leqslant
C\varepsilon(1+|q|)^{-3/2}\sum\limits_{|K|\leqslant|I|}|\partial Z^{K}\Phi|,$
$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|\leqslant|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}\Phi|\cdot|\partial
Z^{J_{3}}\Phi|\leqslant
C^{2}\varepsilon^{2}(1+|q|)^{-1}\sum\limits_{|K|\leqslant|I|}|\partial
Z^{K}\Phi|,$

and

$\displaystyle\sum\limits_{|J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|=|I|}|Z^{J_{1}}A|\cdot|Z^{J_{2}}A|\cdot|Z^{J_{3}}\Phi|\cdot|Z^{J_{4}}\Phi|\leqslant
C^{3}\varepsilon^{3}(1+|q|)^{-3/2}\sum\limits_{|K|\leqslant|I|}|Z^{K}\Phi|.$

Combining the above estimates gives (6.2). $\hfill\Box$

######  Lemma 6.3.

Under the assumptions of Theorem 6.1, one can get

(6.20)
$\displaystyle\varepsilon^{-1}\int_{0}^{t}\int|Z^{I}F|^{2}wdxd\tau\lesssim\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int(|\partial
Z^{K}h|^{2}+|\partial Z^{K}A|^{2}+|\partial Z^{K}\Phi|^{2})wdxd\tau$
$\displaystyle+\varepsilon^{4\tilde{\gamma}-1}+\sum\limits_{|K|\leqslant|I|}\varepsilon\int_{0}^{t}\int|\bar{\partial}Z^{K}h|^{2}w^{\prime}dxd\tau,$
(6.21)
$\varepsilon^{-1}\int_{0}^{t}\int|Z^{I}J|^{2}wdxd\tau\lesssim\sum\limits_{|K|\leqslant|I|}\varepsilon\int_{0}^{t}\int(|\partial
Z^{K}h|^{2}+|\partial Z^{K}A|^{2}+|\partial Z^{K}\Phi|^{2})wdxd\tau$

and

(6.22)
$\displaystyle\varepsilon^{-1}\int_{0}^{t}\int|Z^{I}W|^{2}wdxd\tau\lesssim\varepsilon\cdot\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int|\partial
Z^{K}\Phi|^{2}wdxd\tau.$

Proof. Throughout the process we have to apply Corollary 13.3 of [12](to
transform $Z^{K}h$, $Z^{K}A$ and $Z^{K}\Phi$ into $\partial Z^{K}h$, $\partial
Z^{K}A$ and $\partial Z^{K}\Phi$ respectively), the inequality $1\leqslant
1+\tau\leqslant 1+T$(Note that the constant $C$ relies on the maximal
existence time $T$) and the assumption $\tilde{\gamma}>3/2+\gamma$(to ensure
that some integral is finite). Indeed, if the integral is denoted by $In$, we
can give its specific expression

$\displaystyle
In:=\varepsilon^{4\tilde{\gamma}-1}\int_{\mathbb{R}^{3}}(1+|q|)^{-2\tilde{\gamma}}w\,dx^{1}dx^{2}dx^{3}.$

Since $w\lesssim(1+|q|)^{1+2\gamma}$, under polar coordinates system it is
easy to get

$\displaystyle
In\lesssim\varepsilon^{4\tilde{\gamma}-1}\int_{\mathbb{S}^{2}}d\mathbb{S}^{2}\int_{0}^{\infty}(1+|q|)^{1+2\gamma-2\tilde{\gamma}}r^{2}\,dr.$

Hence, the assumption $\tilde{\gamma}>3/2+\gamma$ implies $In<\infty$.

The other part of the proof is similar to that of Lemma 11.3 of [12]. So we
omit it. $\hfill\Box$

Now we deal with $D^{I}$, $W^{I}$ and $L^{I}$.

######  Lemma 6.4.

Under the assumptions of Theorem 6.1, we have

(6.23)
$\varepsilon^{-1}\int_{0}^{t}\int|D^{I}|^{2}wdxd\tau\lesssim\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int\left(|\partial
Z^{K}h|^{2}w+|\bar{\partial}Z^{K}h|^{2}w^{\prime}\right)dxd\tau+\varepsilon^{3},$
$\displaystyle\varepsilon^{-1}\int_{0}^{t}\int|W^{I}|^{2}wdxd\tau$
$\displaystyle\lesssim$
$\displaystyle\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int\left(|\partial
Z^{K}h|^{2}w+|\bar{\partial}Z^{K}h|^{2}w^{\prime}\right)dxd\tau$
$\displaystyle+\varepsilon^{3}+\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int|\partial
Z^{K}A|^{2}w\,dxd\tau$

and

$\displaystyle\varepsilon^{-1}\int_{0}^{t}\int|L^{I}|^{2}wdxd\tau$
$\displaystyle\lesssim$
$\displaystyle\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int\left(|\partial
Z^{K}h|^{2}w+|\bar{\partial}Z^{K}h|^{2}w^{\prime}\right)dxd\tau$
$\displaystyle+\varepsilon^{3}+\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int|\partial
Z^{K}\Phi|^{2}w\,dxd\tau$

Proof. We only prove (6.4) since the other estimates follow from the same
methods. According to Proposition 5.3 of [12] we arrive at

$\displaystyle|\overset{\sim}{\Box}_{g}Z^{I}\Phi-\hat{Z}^{I}\overset{\sim}{\Box}_{g}\Phi|$
$\displaystyle\lesssim$
$\displaystyle\sum\limits_{|K|\leqslant|I|}\sum\limits_{|J|+(|K|-1)_{+}\leqslant|I|}\left(\frac{|Z^{J}H|}{1+|q|}+\frac{|Z^{J}H|_{\mathcal{L}\mathcal{L}}}{1+|q|}\right)|\partial
Z^{K}\Phi|$
$\displaystyle+\sum\limits_{|K|\leqslant|I|}\left(\sum\limits_{|J|+(|K|-1)_{+}\leqslant|I|-1}\frac{|Z^{J}H|_{\mathcal{L}\mathcal{T}}}{1+|q|}+\sum\limits_{|J|+(|K|-1)_{+}\leqslant|I|-2}\frac{|Z^{J}H|}{1+|q|}\right)|\partial
Z^{K}\Phi|.$

Our goal is to obtain the estimate for the quantity

$\sum\limits_{|I|\leqslant
N}\int_{0}^{t}\int|\overset{\sim}{\Box}_{g}Z^{I}\Phi-\hat{Z}^{I}\overset{\sim}{\Box}_{g}\Phi|^{2}w\,dxd\tau.$

Let us first deal with the terms in (6.2) with $|K|\leqslant N-4$. In this
case we use the decay estimate (6.10). It is clear that now we only have to
consider the expression

(6.27) $\displaystyle\sum\limits_{\begin{subarray}{c}|J|\leqslant|I|\\\
|J^{\prime}|\leqslant|I|-1\\\
|K|\leqslant|I|-2\end{subarray}}\int_{0}^{t}\int\left\\{\frac{|Z^{J}H|^{2}}{(1+|q|)^{2}}+\frac{|Z^{J}H|^{2}_{\mathcal{L}\mathcal{L}}+|Z^{J^{\prime}}H|^{2}_{\mathcal{L}\mathcal{T}}+|Z^{K}H|^{2}}{(1+|q|)^{2}}\right\\}\varepsilon^{2}(1+|q|)^{-3}w\,dxd\tau$
$\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}|J|\leqslant|I|\\\
|J^{\prime}|\leqslant|I|-1\\\
|K|\leqslant|I|-2\end{subarray}}\int_{0}^{t}\int\left\\{|Z^{J}H|^{2}+|Z^{J}H|^{2}_{\mathcal{L}\mathcal{L}}+|Z^{J^{\prime}}H|^{2}_{\mathcal{L}\mathcal{T}}+|Z^{K}H|^{2}\right\\}\varepsilon^{2}(1+|q|)^{-1}w\,dxd\tau.$

From the proof of Lemma 11.5 in [12] it follows that (6.27) is bounded by

$\displaystyle
C\varepsilon^{2}\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int\left(\frac{|\partial
Z^{K}h|^{2}}{1+\tau}w+|\bar{\partial}Z^{K}h|^{2}w^{\prime}\right)dxd\tau$
$\displaystyle+C\varepsilon^{2}\sum\limits_{|K|\leqslant|I|-1}\int_{0}^{t}\int\frac{|\partial
Z^{K}h|^{2}}{(1+\tau)^{1-2C\varepsilon}}wdxd\tau+C\varepsilon^{4}$

which is equivalent to

$\displaystyle
C\varepsilon^{2}\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int\left(|\partial
Z^{K}h|^{2}w+|\bar{\partial}Z^{K}h|^{2}w^{\prime}\right)dxd\tau+C\varepsilon^{4},$

where we let the parameter $M$ in the expression
$H_{0}^{\mu\nu}:=-\chi(r/t)\chi(r)M\delta^{\mu\nu}/r$ equal to 0(the
expression can be found at the beginning of the proof of Lemma 11.5 in [12]).
For more details we refer to the last inequality on page 1460 of [12].

Returning to (6.2) we now deal with the case $|K|\geqslant N-3$, which implies
$|J|\leqslant 4$. By the same way of proving Lemma 11.5 of [12] we arrive at
that the contribution of the terms with $|K|\geqslant N-3$ to
$|\overset{\sim}{\Box}_{g}Z^{I}\Phi-\hat{Z}^{I}\overset{\sim}{\Box}_{g}\Phi|$
can be bounded by

$\varepsilon\sum\limits_{|K|=|I|}\frac{|\partial
Z^{K}\Phi|}{1+\tau}+\varepsilon\sum\limits_{|K|<|I|}\frac{|\partial
Z^{K}\Phi|}{(1+\tau)^{1-C\varepsilon}},$

which is equivalent to

$\varepsilon\sum\limits_{|K|\leqslant|I|}|\partial Z^{K}\Phi|.$

$\hfill\Box$

Now let us finish the proof of Theorem 6.1. Applying (6.2), (6.2) and (6.2)
together with Lemma 6.3 and Lemma 6.4 yields

(6.28) $\displaystyle\int_{\Sigma_{t}}(|\partial Z^{I}h|^{2}+|\partial
Z^{I}A|^{2}+|\partial
Z^{I}\Phi|^{2})w+\int_{0}^{t}\int_{\Sigma_{\tau}}(|\bar{\partial}Z^{I}h|^{2}+|\bar{\partial}Z^{I}A|^{2}+|\bar{\partial}Z^{I}\Phi|^{2})w^{\prime}$
$\displaystyle\lesssim$
$\displaystyle\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int(|\partial
Z^{K}h|^{2}+|\partial Z^{K}A|^{2}+|\partial
Z^{K}\Phi|^{2})w+\int_{\Sigma_{0}}(|\partial Z^{I}h|^{2}+|\partial
Z^{I}A|^{2}+|\partial Z^{I}\Phi|^{2})w$
$\displaystyle+\varepsilon\sum\limits_{|K|\leqslant|I|}\int_{0}^{t}\int(|\bar{\partial}Z^{K}h|^{2}+|\bar{\partial}Z^{K}A|^{2}+|\bar{\partial}Z^{K}\Phi|^{2})w^{\prime}+\varepsilon^{3}$

Denote

$\tilde{\mathcal{E}}_{k}(t):=\sup\limits_{0\leqslant\tau\leqslant
t}\sum\limits_{\begin{subarray}{c}Z\in\mathfrak{L}\\\ |I|\leqslant
k\end{subarray}}\int_{\Sigma_{\tau}}(|\partial Z^{I}h|^{2}+|\partial
Z^{I}A|^{2}+|\partial Z^{I}\Phi|^{2})w\,dx$

and

$\mathcal{S}_{k}(t):=\sum\limits_{\begin{subarray}{c}Z\in\mathfrak{L}\\\
|I|\leqslant
k\end{subarray}}\int_{0}^{t}\int_{\Sigma_{\tau}}(|\bar{\partial}Z^{I}h|^{2}+|\bar{\partial}Z^{I}A|^{2}++|\bar{\partial}Z^{I}\Phi|^{2})w^{\prime}\,dx.$

Then we get

(6.29) $\displaystyle\tilde{\mathcal{E}}_{k}(t)+\mathcal{S}_{k}(t)$
$\displaystyle\lesssim$
$\displaystyle\tilde{\mathcal{E}}_{k}(0)+\varepsilon\int_{0}^{t}\tilde{\mathcal{E}}_{k}(\tau)d\tau+\varepsilon\mathcal{S}_{k}(t)+\varepsilon^{3},$

which implies

$\displaystyle\tilde{\mathcal{E}}_{k}(t)\leqslant C(T)\varepsilon^{2}.$

$\hfill\Box$

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Zonglin Jia

Institute of Applied Physics and Computational Mathematics, China Academy of
Engineering Physics, Beijing, 100088, P. R. China

Email<EMAIL_ADDRESS>
Boling Guo

Institute of Applied Physics and Computational Mathematics, China Academy of
Engineering Physics, Beijing, 100088, P. R. China

Email<EMAIL_ADDRESS>