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\controldates{17-JUL-2000,17-JUL-2000,17-JUL-2000,17-JUL-2000}
 
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\dateposted{July 19, 2000}
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\PII{S 1079-6762(00)00081-0}
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\copyrightinfo{2000}{American Mathematical Society}

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%\numberwithin{equation}{section}

\begin{document}

\title[Initial data for the Einstein equations]{On the 
connectedness of the space of initial data for the
Einstein equations}

% author one information
\author{Brian Smith}
\address{University of Alabama at Birmingham, Birmingham, AL 35205}
\email{smith@math.uab.edu}
\thanks{This research was supported in part by NSF grant DMS~9704760.}

% author two information
\author{Gilbert Weinstein}
\address{University of Alabama at Birmingham, Birmingham, AL 35205}
\email{weinstei@math.uab.edu}


\subjclass[2000]{Primary 83C05; Secondary 58G11}
\date{May 27, 1999}
\commby{Richard Schoen}

\begin{abstract}
Is the space of initial data for the Einstein vacuum equations
connected? As a partial answer to this question, we prove the
following result: Let ${\EuScript M}$ be the space of asymptotically flat
metrics of non-negative scalar curvature on ${\mathbb R}^3$ which admit a
global foliation outside a point by $2$-spheres of positive mean
and Gauss curvatures. Then ${\EuScript M}$ is connected.
\end{abstract}

\maketitle

\section*{Introduction}

The Einstein vacuum equations of general relativity read:
\begin{equation}    \label{eq:einstein}
    \Rfour_{\mu\nu} - \frac12 \Rfour \gb_{\mu\nu} = 0,
\end{equation}
where $\Rfour_{\mu\nu}$ is the Ricci curvature tensor of a
Lorentzian $4$-manifold, and $\Rfour$ the scalar curvature. The
basic problem for these equations is the Cauchy problem:
\emph{given data on a time-slice $M$, consisting of a Riemannian
metric $g$ and a second fundamental form $k$ on $M$, find the
evolution of space-time according to~\eqref{eq:einstein}}. Not all
the equations in~\eqref{eq:einstein} are evolution equations.
Using the twice-contracted Gauss equation and the Codazzi
equations of the Riemannian submanifold $M$, one finds that the
normal-normal and normal-tangential components
of~\eqref{eq:einstein} are:
\begin{align}
    \label{eq:vc-gauss}
    R - \abs{k}^2 + (\tr k)^2 &= 0, \\
    \label{eq:vc-codazzi}
    \nabla^j k_{ij} - \nabla_i \tr k &= 0,
\end{align}
where $R$ is the scalar curvature of $M$, and $k$ its second
fundamental form.  These equations, called the \emph{Vacuum
Constraint Equations}, involve no time derivatives and hence are
to be considered as restrictions on the data $g$ and $k$;
see~\cite{wald}. We will only consider \emph{asymptotically flat}
(AF) solutions of these equations, i.e., solutions satisfying the
decay:
\begin{align*}
   g_{ij} - \delta_{ij} &= O(r^{-1}), \\
   k_{ij} &= O(r^{-2}), \\
   R &\in L^1(M).
\end{align*}


It is standard to choose the \emph{maximal gauge} $\tr k=0$
in~\eqref{eq:vc-gauss}--\eqref{eq:vc-codazzi}, which, as shown by
Bartnik~\cite{bartnik84}, involves no loss of generality. In this
case, we get the \emph{Maximal Gauge Vacuum Constraint Equations}:
\begin{align}
    \label{eq:mgvc-gauss}
    R &= \abs{k}^2, \\
    \label{eq:mgvc-trace}
    \tr k &= 0, \\
    \label{eq:mgvc-codazzi}
    \Div k &= 0.
\end{align}
These form an underdetermined system of elliptic equations on $M$
for $g$ and $k$.

Much work has been devoted to finding solutions
of~\eqref{eq:mgvc-gauss}--\eqref{eq:mgvc-codazzi}; see for
example~\cite{cantor79,cantor81,
choquetbruhat74,choquet00,christodoulou81,
fischer80,
york71} and the references therein. However, certain
fundamental questions remain unanswered. For example, it is not
known whether the space of AF initial data on a given $3$-manifold
$M$ is connected, not even in the case $M=\R^3$. Since the
evolution equations trace a continuous path in the phase space of
initial data, either answer to this question would be of
considerable significance for the dynamics of the Einstein
equations.

The standard method for solving
\eqref{eq:mgvc-gauss}--\eqref{eq:mgvc-codazzi} has been the
\emph{conformal method}. In this method the free data is the
conformal class of an asymptotically flat Riemannian metric $g$,
and a trace-free divergence-free symmetric $2$-tensor $k$ on $M$.
Since the trace-free and divergence-free conditions on $k$ are
invariant under the transformation $g\mapsto\phi^4 g$,
$k\mapsto\phi^{-2} k$, it suffices to find $\phi$ so
that~\eqref{eq:mgvc-gauss} is satisfied.  This will be so provided
that the Lichnerowicz equation is satisfied:
\[
   \Delta \phi - \frac18 R\phi + \abs{k}^2 \phi^{-7}=0.
\]
A solution of this equation can be found if the negative part of
the scalar curvature is small enough in the $L^{3/2}$ norm;
see~\cite{christodoulou81}. In particular, the question above can
be reduced to the following purely geometric problem: is the space
of AF metrics of non-negative scalar curvature on a
$3$-manifold $M$ connected?

In this paper, we announce, and sketch the proof of a result which
gives a partial answer in the affirmative to the question posed
above; details will appear in~\cite{smithweinstein}. We say that
a topological $2$-sphere $S$ in $M$ is \emph{quasiconvex} if both
the Gauss and the mean curvature of $S$ are
positive~\cite{christodoulou93}. Let $\M$ be the space of
$C^{2,\alpha}_{-1}$ metrics $g$ on $\R^3$ with non-negative scalar
curvature $R\in L^1$ which admit a global coordinate system whose
coordinate spheres are quasiconvex, and which satisfy in this
coordinate system:
\begin{align*}
   g_{ij}-\delta_{ij} &=O(r^{-1}) \qquad\text{as $r\to\infty$}, \\
   g_{ij}-\delta_{ij} &=0\qquad\qquad\quad \text{at $r=0$}.
\end{align*}
The $C^{2,\alpha}_{-1}$ topology on $\M$ is generated by the
following system of neighborhoods of any metric $g\in\M$:
\[
   \left\{g'\in\M \st \sum_{\alpha=0}^2 \sup
   \abs{(1+r)^{m+1}\D^m(g_{ij}-g'_{ij})} +
   [\D^2g_{ij}-\D^2g'_{ij}]_{\alpha,-3}
   < \epsilon \right\},
\]
where
\[
    [f]_{\alpha,-k} = \sup_r \left( (1+r)^{k+\alpha}
    \sup_{x,y\in B_r}\frac{\abs{f(x)-f(y)}}{\abs{x-y}^\alpha}
    \right)
\]
is the weighted H\"older norm with exponent $\alpha$ of $f$ on
$\R^3$. In fact, in view of the general covariance of the Einstein
Equations, we are only interested in the quotient of $\M$ by the
group $\G$ of diffeomorphisms of $\R^3$.

\begin{maintheorem}
The quotient of $\M/\G$ is path connected in the quotient topology
induced by $C^{2,\alpha}_{-1}$ on $\M$.
\end{maintheorem}

Of course, this raises the following question: when does an AF
metric $g$ of non-negative scalar curvature belong to $\M$?
Clearly, a necessary condition is that $g$ possesses no compact
minimal surfaces.  However we do not even know whether the absence
of compact minimal surfaces suffices to guarantee the existence of
a global foliation with positive mean curvature.

To prove our Main Theorem, we generalize a method introduced by
Bartnik~\cite{bartnik93} to construct quasispherical metrics of
prescribed scalar curvature.  A metric is \emph{quasispherical} if
it can be foliated by \emph{round spheres}, spheres of constant
curvature.  Bartnik observed that prescribing scalar curvature for
this type of metric could be viewed as a parabolic equation on the
sphere for one of the metric coefficients, $u=\abs{\nabla
r}^{-1}$, where $r$ is the foliating function, provided that the
mean curvature was also positive.  We combine this with the
Poincar\'{e} Uniformization as in \cite{christodoulou93} to get a
general method to prescribe scalar curvature for metrics in $\M$.
As an application, we prove the Main Theorem.

Denote by $r$ the foliating function normalized so that the area
of the spheres is $4\pi r^2$, and by $\gamma$ the induced metric
on the spheres. Any smooth enough metric $g\in\M$ can be written
as:
\begin{equation}    \label{eq:metric}
    g = u^2 dr^2 + e^{2v} \gammab_{AB}(\betah^A dr + r d\theta^A)
    (\betah^B dr + r d\theta^B),
\end{equation}
where $(\theta^1,\theta^2)$ are local coordinates on $\Sphere^2$,
$\gammab_{AB}$ is a fixed (independent of $r$) round metric of
area $4\pi$, and $\betah=\betah^A\D_A$ is the \emph{shift vector}.
Here, and throughout, we use the summation convention: repeated
indices are summed over their range, $0,1,2,3$ for Greek indices,
$1,2,3$ for lower case Latin indices, and $1,2$ for upper case
Latin indices. Let $\chi$ be the second fundamental form,
$H=\tr_\gamma\chi$ be the mean curvature of the spheres, and
$\Pi=\Lie_\betah \gamma$ be the deformation tensor of $\betah$ on
the spheres; then it can be checked that
\begin{align}
    \label{eq:chib}
    \chib = r u\chi &= \left( (1+r v_r)\gamma-\Pi/2 \right),\\
    \label{eq:Hb}
    \Hb = r u H &= \left(2 +
    2 r v_r-e^{-2v}\Div_{\gammab}\beta\right),
\end{align}
where $\beta=e^{2v}\betah$. It is important to note that both
$\abs{\chib}_\gamma^2$ and $\Hb$ can be calculated in terms of
only $\beta$, $v$, $r$, and the round metric $\gammab$ on
$\Sphere^2$. Let $N$ be the outer unit normal to the foliation
spheres, let $\Nb=ruN=r\D_r-\betah$, let $\Deltash_\gamma$ be the
Laplacian on the spheres with respect to $\gamma$, and let
\begin{equation}    \label{eq:kappa}
   \kappa=r^{-2}e^{-2v}(1-\Deltash v)
\end{equation}
be the Gauss curvature of the spheres. Then the equation for the
scalar curvature $R$ of $g$ can be written as
\begin{equation}    \label{eq:tilde}
   \Hb \D_{\Nb} u
   = r^2 u^2 \Deltash_\gamma u + \Ab u - \Bb u^3,
\end{equation}
where
\begin{gather*}
   \Ab=\D_{\Nb}\Hb - \Hb +\frac12 \abs{\chib}_\gamma^2 + \frac12 \Hb^2, \\
   \Bb=r^2(\kappa-\frac12 R)=e^{-2v}(1-\Deltash v)-\frac12 r^2 R.
\end{gather*}
Noting that the Laplacian with respect to $\gammab$ is
$\Deltash=r^2 e^{2v} \Deltash_\gamma$, we obtain, provided that
$H>0$, the following \emph{Bernoulli-type} parabolic equation for
$u$ on the unit sphere:
\begin{equation} \label{eq:main}
    r\D_r u - \beta\cdot\nablash u = \Gamma u^2 \Deltash u + A u
    - B u^3,
\end{equation}
where $\nablash u$ is the tangential component of the gradient of
$u$ , $\Gamma=e^{-2v}/\Hb$, $A=\Ab/\Hb$ and $B=\Bb/\Hb$. It
follows from the comment following
equations~\eqref{eq:chib}--\eqref{eq:Hb} that the coefficients
$\Gamma$, $A$ and $B$ can be calculated in terms of only $\beta$,
$v$, $r$, the round metric $\gammab$ on $\Sphere^2$, and $R$.  The
quasispherical case can be recovered by setting $v=0$, and
$\kappa=1$, see~\cite{bartnik93}.

The proof of the Main Theorem is based on the study of
equation~\eqref{eq:main}.  The deformation to a flat metric is
accomplished in several steps.  First, the metric is smoothed out
with the scalar curvature $R$ truncated to be compactly supported.
Next, we deform the metric to one satisfying $2\kappa>R$. Then, we
deform to a metric with compactly supported $\beta$ and $v$.
Finally, we deform to a flat metric. The last three steps are all
based on the following strategy.  The deformation $g_\lambda$ is
defined explicitly on a ball $B_{r_{0}}$.  In the exterior of
$B_{r_0}$ we consider $\beta_\lambda$, $v_\lambda$, and
$R_\lambda$ as free data, and solve equation~\eqref{eq:main} on
$[r_0,\infty)\times\Sphere^2$ for $u_\lambda$ with initial
conditions $u_\lambda|_{S_{r_0}}$. In order for this to be
feasible, and for the resulting metric $g_\lambda$ to yield a
continuous path, we must ensure that $\beta_{\lambda}$,
$v_{\lambda}$, and $R_{\lambda}$ are continuous in the appropriate
spaces, that $\Hb_\lambda$ is positive, and that $R_{\lambda}$ is
non-negative. In addition, one must verify conditions that
guarantee the global existence of the solution $u_\lambda$, its
appropriate decay as $r\to\infty$, and continuity with respect to
$\lambda$.  The regularity of $u_{\lambda}$ across $S_{r_{0}}$ is
obtained by solving equation~\eqref{eq:main} on
$[r',r_0+\epsilon)\times\Sphere^2$, $00$. Furthermore,
it is well known that, for some choices of coefficients and
initial data, a classical solution can blow up in finite time.
Thus, our main objective here is to derive conditions which
guarantee the existence of a global positive solution on the time
interval $[r_0,\infty]$.

The principal ingredient in this and future subsections is a
simple a priori bound on solutions of~\eqref{eq:main}. To derive
this bound, we use the familiar substitution $w=u^{-2}$ well-known
from the elementary method used to solve the corresponding
Bernoulli ordinary differential equation. If $u>0$
satisfies~\eqref{eq:main} on $[r_0,r_1]$, then $w$ satisfies
\begin{equation}    \label{eq:w}
     r\D_{r} w-{\beta} \cdot\nablash u =
     2(-\Gamma u^{-1}\Deltash u-Aw+B).
\end{equation}
Since this equation is only used to derive pointwise a priori
bounds, and since $u$ has a maximum where $w$ has a minimum and
vice versa, there is no need to transform the gradient and
Laplacian terms. For example, it follows from~\eqref{eq:w} that
\begin{equation} \label{eq:est}
     r\D_{r}w_*+2A^*w_* \geq 2B_*,
\end{equation}
which upon integration yields the lower bound:
\begin{equation}      \label{eq:lower}
     rw_{*} \geq
     \left[ r_{0}w_{*}(r_{0})+
     \int_{r_{0}}^{r} 2B_*
     \exp\left(\int_{r_{0}}^{r'}(2A^{*}-1)\frac{dt}{t} \right)
     dr'\right]
     \exp\left(\int_{r_{0}}^{r}(1-2A^{*})\frac{dt}{t}\right).
\end{equation}
Similarly, one obtains the upper bound:
\begin{equation}     \label{eq:upper}
     rw^{*} \leq
     \left[ r_{0}w^{*}(r_{0})+
     \int_{r_{0}}^{r} 2B^* \hspace{-1pt}
     \exp\left(\int_{r_{0}}^{r'}(2A_{*}-1)\frac{dt}{t} \right)
     dr'\right]
     \exp\left(\int_{r_{0}}^{r}(1-2A_{*})\frac{dt}{t}\right).
\end{equation}
In particular it follows immediately that $w$ is bounded above,
and hence $u\geq c>0$, where $c$ depends on $r_1$. Suppose
furthermore that
\begin{equation}     \label{eq:K}
           K = \frac{1}{r_0}
          \left(\sup_{r_00.
\end{equation}
If the initial data $u_00$, and that $u$ is a solution which is bounded above
and below, $C^{-1}\leq u\leq C$ on $A_I$, where $I\subset I'$. Then
standard parabolic Schauder theory gives
\begin{equation}                 \label{eq:ubound}
     \norm{u}_{4,\alpha;I} \leq C',
\end{equation}
where $C'$ depends on $C$ and the length of $I$. Using the scaling
properties of the $H^{k,\alpha}_I$-norms and of
equation~\eqref{eq:main} we can also derive~\eqref{eq:ubound} for
$I_\lambda=[\lambda r_0, \lambda r_1]$ with $C'$ independent of
$\lambda$ provided $I_\lambda\subset I'$. Furthermore, if in
addition $|2A-1|0$
such that $\beta,\Gamma,A,B\in H^{2,\alpha}_I$ satisfy
\begin{gather*}
   \norm{\beta}_{2,\alpha;I},\norm{\Gamma}_{2,\alpha;I},
   \norm{B}_{2,\alpha;I},
   \norm{r(2A-1)}_{2,\alpha;I} \leq C, \\
   C^{-1}\leq\Gamma\leq C,
\end{gather*}
and $u_0\in C^{4,\alpha}(\Sphere^{2})$ satisfies $00$ is a classical
solution of equation~\eqref{eq:main} on $I\times\Sphere^2$ with
coefficients $\beta,\Gamma,A,B\in H^{2,\alpha}_I$ and with initial
data $u_0\in C^{4,\alpha}(\Sphere^2)$.  Let $\tilde B\in
H^{2,\alpha}_I$ satisfy $\tilde B\geq B$. Then the equation
\[
    r\D_r u-\beta\cdot\nablash u=\Gamma u^2\Deltash u+Au-\tilde Bu^3
\]
has a unique solution $\tilde u \in H^{4,\alpha}_I$ with the same
initial data $\tilde u(r_0,\theta)=u_0(\theta)$. Furthermore, we
have $0<\tilde u\leq u$.
\end{theorem}

\begin{proof}
It suffices to prove a supremum a priori bound for $\tilde u$ on
any interval $[r_0,r_1)$ where the solution $\tilde u>0$ exists.
Subtracting the equation for $u$ from the equation for $\tilde u$,
we get an equation for $v=\tilde u-u$:
\begin{equation}
    r\D_{r}v -\beta\cdot\nablash v =\Gamma
    \tilde u^2\Deltash v + \tilde A v - (\tilde B-B)\tilde u^3,
\end{equation}
where $\tilde A=A + \Gamma(\tilde u+ u)\Deltash u - B(\tilde u^2+
\tilde u u + \tilde u^2)$. Since $(\tilde B-B)\tilde u^3>0$, the
maximum principle applies to give $v\leq0$.  It follows that
$\tilde u\leq u$ on $[r_0,r_1)$.
\end{proof}

For the remainder of Section~\ref{sec:bernoulli} we set
$I=[r_0,\infty)$.

\subsection{Asymptotic behavior}

To study the asymptotic behavior of solutions of
equation~\eqref{eq:main} we define:
\[
    m = \frac{r}{2}\,(1-u^{-2}).
\]
If $u$ is a global solution of equation~\eqref{eq:main} on
$I\times\Sphere^2$, then $m$ satisfies
\begin{equation}     \label{eq:meq}
     r\D_{r}m-\beta\cdot\nablash m=
     r\Gamma  \frac{\Deltash u}{u}
     -(2A-1)m+{r}(A-B).
\end{equation}
We note that if $r\abs{2A-1}\leq C$, and
\begin{equation}
    \label{L1}
    |A-B|^* \in L^{1}(I),
\end{equation}
then using the maximum principle, it follows that $\abs{m}$ is
bounded. Applying Schauder theory to equation~\eqref{eq:meq} we
then obtain:

\begin{theorem} \label{thm:L1}
Let $I=[r_0,\infty)$, let $\beta, \Gamma, A, B \in
H^{2,\alpha}_I$, suppose that~\eqref{L1} is satisfied, and let $u
\in H^{2,\alpha}_I$ be a positive solution of
equation~\eqref{eq:main}.  Suppose that there is a constant $C>0$
such that
\begin{gather*}
   \norm{u}_{2,\alpha;I},
   \norm{r(2A-1)}_{2,\alpha;I},\Vert{B}\Vert_{2,\alpha;I},
   \Vert{|A-B|^*}\Vert_{L^1(I)}
   \leq C, \\
   C^{-1}\leq\Gamma\leq C.
\end{gather*}
Then $m=r(1-u^{-2})/2$ satisfies
\begin{equation} \label{eq:mnrmbdd}
    \norm{m}_{4,\alpha;I}0$, $i=1,2$, are bounded
classical solutions of equation~\eqref{eq:2eqs:u} on
$I\times\Sphere^2$.  Let $B=B_1-B_2$, and suppose also that
$|B|^*\in L^1(I)$, that $m_1$ is bounded, and that there is a
constant $C>0$ such that
\begin{gather*}
    \norm{u_1}_{2,\alpha;I},\norm{u_2}_{2,\alpha;I},
    \norm{m_1}_{2,\alpha;I},\norm{r(2A-1)}_{2,\alpha;I},
    \Vert{B}\Vert_{2,\alpha;I},
    \Vert{|B|^*}\Vert_{L^1(I)}
    \leq C, \\
    C^{-1}\leq\Gamma\leq C.
\end{gather*}
Then $\tilde m=r(1-\tilde u^{-2})/2$ satisfies
\[
    \norm{\tilde m}_{4,\alpha;I} \leq C',
\]
where $C'$ depends only on $C$.
\end{theorem}

\subsection{Continuous dependence on parameters}

\begin{theorem}   \label{thm:continuous}
Let $I=[r_0,\infty)$, and suppose that $u_\lambda\in
H_I^{k+2,\alpha}$, $a\leq\lambda\leq b$, is a family of solutions
of~\eqref{eq:main} with
$\beta_\lambda,\Gamma_\lambda,A_\lambda,B_\lambda$ satisfying
$r(2A_\lambda-1),r(2B_\lambda-1),\beta_\lambda,\Gamma_\lambda\in
C^0([a,b],H^{k,\alpha}_I)$ and with the  initial data
$u_\lambda(r_0)\in
C^0\bigl([a,b],C^{k+2,\alpha}(\Sphere^2)\bigr)$. Suppose also that
one of the following conditions is satisfied:
\begin{enumerate}
\item $|A_\lambda-B_\lambda|^*\in C^0\bigl([a,b],L^1(I)\bigr)$;
\item A continuous family of solutions
$m'_\lambda\in C^0\bigl([a,b],H^{k+2,\alpha}_I\bigr)$
of~\eqref{eq:meq} exist.
\end{enumerate}
Then $u_\lambda,m_\lambda\in C^0\bigl([a,b],H^{k,\alpha}_I\bigr)$.
\end{theorem}

\section{Deformation of metrics in ${\EuScript M}$}
\label{sec:deform}

We are now in a position to sketch the proof of the Main Theorem.
Recall that any metric $g\in\M$ can be written as
in~\eqref{eq:metric}. We define a nested sequence of subsets
$\M=\M_{0}\supset\dots\supset\M_{4}$:
\begin{align*}
    \M_1 &= \{g\in\M_0\st r(1-u)\in H^{4,\alpha}_{[r_0,\infty)};\>
    r\beta,rv \in H^{8,\alpha}_{[r_0,\infty)}, \forall r_0>0,
        \supp R \text{\ is compact}\}, \\
    \M_2 &= \{g\in\M_1\st 2\kappa-R>0\}, \\
    \M_3 &= \{g\in\M_2\st \beta,v\text{ are compactly
        supported} \}, \\
    \M_4 &= \{g\in\M_3\st \text{$g$ is flat}\}.
\end{align*}
Let us say that \emph{$\M_i$ is connected to $\M_{i+1}$} if for
each $g\in\M_i$ there is a path $\Gamma$ in $\M_i$, continuous in
the topology of $C^{2,\alpha}_{-1}$, with $\Gamma(0)=g$ and
$\Gamma(1)\in\M_{i+1}$. We will show that $\M_{i}$ is connected to
$\M_{i+1}$ for each $i=0,\dots,3$. The Main Theorem follows by
joining these paths.

\begin{lemma}   \label{lemma:smoothing}
$\M_{0}$ is connected to $\M_{1}$.
\end{lemma}
\begin{proof}
Let $g=g_0\in\M_0$.  It is not difficult, using a truncation
followed by a standard smoothing, to construct a deformation
$g_\lambda$, continuous in $C^{2,\alpha}_{-1}$, from $g_0$ to a
smooth metric $g_1$ which is flat outside a large enough ball,
with scalar curvature $R_\lambda\in L^1$, and such that
$g_\lambda-g_0$ is small in ${C^{2,\alpha}_{-1/2}}$ for all
$\lambda$.  Since $g_\lambda$ is close to $g$, the coordinate
spheres are still quasiconvex in $g_\lambda$, and the negative
part $R_\lambda^-$ of the scalar curvature of $g_\lambda$ is small
in $L^{3/2}$. It follows that the operator
$-8\Delta_{g_\lambda}+R_\lambda$ is injective~\cite{cantor81}, and
hence also a bijection; see~\cite{chaljubchoquet79,
parkerlee87}. We can now choose a smooth positive function
of compact support $S_\lambda$ which is close to $R_\lambda$ in
$C^{\alpha}_{-5/2}$, and solve the equation
\[
    (-8\Delta + R_\lambda)\psi_\lambda=R_\lambda-S_\lambda.
\]
It follows from the above that $\psi_\lambda$ is small in
$C^{2,\alpha}_{-1/2}$. Taking $\phi_\lambda=1+\psi_\lambda$, we
see that the metrics $\phi_\lambda^4 g_\lambda$ have positive
scalar curvature, quasiconvex coordinate spheres, and form a
continuous path from $g_0$ to a smooth metric $\tilde g_1=\phi_1^4
g_1$. Since $R_1-S_1$ is of compact support, and $g_1$ is flat
outside a compact set, it follows that $\tilde g_1\in
C^{k,\alpha}_{-1}$ for all $k$.  On each coordinate sphere
equipped with the metric $\gamma$ induced by $\tilde g_1$, it is
possible, using the techniques of~~
\cite[Chapter 2]{christodoulou93}, to find a uniformization factor $r^2e^{2v}$,
with bounds as required in $\M_1$, so that
$\gammab=r^{-2}e^{-2v}\gamma$ is a round metric with surface area
$4\pi r^2$. We conclude that the continuous path
$\phi_\lambda^4g_\lambda$ joins $g\in\M_0$ to a metric $\tilde
g_1\in\M_1$.
\end{proof}

It is important to note that since the round metric $\gammab$ on
the coordinate spheres will in general vary with $r$, it is most
likely necessary to change the background flat metric when writing
$\tilde g_1$ as in~\eqref{eq:metric}. Nevertheless, these two flat
metrics are asymptotic as $r\to\infty$; see~\cite{smithweinstein}
for details.

As outlined in the Introduction, the deformation is obtained in
the next three steps by deforming $g_\lambda$ explicitly inside a
ball $B_{r_0}$, while solving~\eqref{eq:main} outside $B_{r_0}$
with the deformation of $\beta_\lambda$, $v_\lambda$ and
$R_\lambda$ defined so that $\kappa_\lambda,\Hb_\lambda>0$,
$R_\lambda\geq0$, and so that theorems from
Section~\ref{sec:bernoulli} guarantee global existence, asymptotic
behavior as $r\to\infty$, and continuity of $u_\lambda$ in
$H^{4,\alpha}_{[r_0,\infty)}$.  Note that in order to ensure
continuity at the end point of the deformation, it is necessary to
have a $\Hb_\lambda$ uniformly bounded below by a positive
constant. Now, if $g_\lambda$ is a path in $\M_i$, $i=1,2,3$, such
that for some $00$ for
$r0$. Let $\varphi(r)$ be a smooth
cut-off function on $[0,\infty)$, satisfying $0\leq\varphi\leq1$,
$\varphi=1$ on $[0,r_0]$, and $\varphi=0$ on $[r_1,\infty)$.
Define $\varphi_\lambda(r)=(1-\lambda) + \lambda\varphi(r)$ and
define $R_{\lambda}=\varphi_\lambda R$. Then $R_\lambda$ is
monotonically decreasing in $\lambda$, $R_{\lambda}=R$ on
$B_{r_{0}}$, and $\supp(R_{1})\subset B_{r_{1}}$.  Thus, Theorems
\ref{thm:monotone} and \ref{thm:m:monotone} can be used to solve
equation~\eqref{eq:main} on $[r_0,\infty)\times\Sphere^2$ for
$u_{\lambda}\in H^{4,\alpha}$. The continuity of $u_\lambda$ and
$m_\lambda$ with respect to $\lambda$ is obtained from
Theorem~\ref{thm:continuous}. Clearly, $g_1\in\M_2$ and the lemma
follows.
\end{proof}

\begin{lemma}
$\M_{2}$ is connected to $\M_{3}$.
\end{lemma}
\begin{proof}
Let $g\in\M_{2}$, and put $R_\lambda=R$. For
$\lambda\in[1,\infty)$ define
$\tilde\beta_\lambda=\bigl(\phi_\lambda\bigr)^* \beta$, $\tilde
v_\lambda=\bigl(\phi_\lambda\bigr)^* v$, where
$\phi_\lambda(r,\theta)=(\lambda r,\theta)$.  Note that
$r\tilde\beta_\lambda$ and $r\tilde v_\lambda$ are continuous in
$H^{6,\alpha}_I$ since $r\tilde\beta_\lambda$ and $r\tilde
v_\lambda$ are uniformly bounded in $H^{8,\alpha}_I$.  Now, let
$\beta_\lambda=\varphi\beta+(1-\varphi)\tilde\beta_\lambda$, and
define $v_\lambda$ by $e^{2v_\lambda}=\varphi
e^{2v}+(1-\varphi)e^{2\tilde v_\lambda}$, where $\varphi(r)$ is a
cut-off function as in the proof of the previous lemma.  It
follows from~\eqref{eq:Hb} that
\[
   e^{2v_\lambda} \Hb_\lambda = \varphi e^{2v}\Hb + (1-\varphi)
   e^{2\tilde v_\lambda} \tilde H_\lambda + (e^{2v}-e^{2\tilde
   v_\lambda}) r \varphi',
\]
where $\tilde H_\lambda=\bigl(\phi_\lambda\bigr)^* \Hb$.  Thus,
since $v$ and $\tilde v_\lambda$ tend to zero as $r\to\infty$, it
follows that if $r_0$ and $r_1/r_0$ are large enough, then
$\Hb_\lambda>0$ for $r>r_0$. Furthermore, in view
of~\eqref{eq:kappa}, the Gauss curvature $\kappa_\lambda$ is given
by
\begin{multline*}
   r^2 e^{2v_\lambda} \kappa_\lambda = 1 - \Deltash v_\lambda
   = r^2 (\varphi e^{2v}\kappa + (1-\varphi) e^{2\tilde v_\lambda}
   \tilde\kappa_\lambda) + \abs{\nablash v}^2 \\
   - 2 e^{-2v_\lambda}
   (\varphi e^{2v}\abs{\nablash v}^2 + (1-\varphi)e^{2\tilde
   v_\lambda} \abs{\nablash \tilde v_\lambda}^2),
\end{multline*}
where $\tilde\kappa_\lambda=\bigl(\phi_\lambda\bigr)^*\kappa$.
Hence, since also $\abs{\nablash v}$ and $\abs{\nablash\tilde
v_\lambda}$ tend to zero as $r\to\infty$, we see that if $r_0$ is
large enough, then $\kappa_\lambda>0$ for $r>r_0$. By choosing
$r_0$ large enough, we can also ensure that $R=0$ outside
$B_{r_0}$. As in the proof of the previous lemma, we define
$g_{\lambda} = g$ inside $B_{r_{0}}$, and solve
equation~\eqref{eq:main} for $u_\lambda$ outside $B_{r_0}$.  The
existence of $u_\lambda$ for all $r\geq r_0 $ is now guaranteed by
Theorem~\ref{thm:K}. Note that outside $B_{r_1}$,
$\beta_\lambda=\tilde\beta_\lambda$, $v_\lambda=\tilde v_\lambda$,
hence we have a uniformly bounded solution
${\lambda}^{-1}\bigl(\phi_\lambda\bigr)^* m$ of
equation~\eqref{eq:meq}, and therefore
Theorem~\ref{thm:m:monotone} applies to give the asymptotic
behavior of $u_{\lambda}$ for $r\to\infty$. It is easy to see that
the path $g_\lambda$ can be extended continuously to $[1,\infty]$,
and since $\beta_\lambda$ and $v_\lambda$ tend to zero as
$\lambda\to\infty$ for $r>r_1$, it follows that $g_\infty\in\M_3$.
\end{proof}

\begin{lemma}   \label{lemma:flat}
$\M_{3}$ is connected to $\M_{4}$.
\end{lemma}
\begin{proof}
Let $g\in\M_3$, choose $r_0>0$ so that $R$, $\beta$, and $v$ are
supported in $B_{r_0}$, and let $\varphi(r)$ be a cut-off function
as above. For $\lambda\in[1,\infty)$, define
$\beta_\lambda=\varphi\, \bigl(\phi_{1/\lambda}\bigr)^*\beta$, and
$\tilde v_\lambda=\bigl(\phi_{1/\lambda}\bigr)^* v$.  Let
$r_00$, hence $h=\inf\tilde H_\lambda$
is independent of $\lambda$.  Let $f(r)$ be a smooth non-negative
function supported on $[r_0,r_2]$, satisfying on $[r_0,r_1]$ the
inequality:
\[
   f > - r^{-a-1} (\varphi h + r \varphi'),
\]
where $a=\max\{-(2+2r\D_r \tilde v_\lambda),0\}$.  Let
$\xi(r)=r^a\int_{r_0}^r f(s)\, ds$, $\psi=\xi+\varphi$, and
$v_\lambda=\zeta(\tilde v_\lambda + \frac12\log \psi)$.  Since
$\xi\geq0$, it now follows from~\eqref{eq:Hb} that we have for
$r_0 -a\xi + r\xi' - r^{a+1} f =
   0.
\]
Furthermore, since $\beta_\lambda=0$ in $B_{r_2}\setminus
B_{r_1}$, we can also choose $\zeta$ so as to ensure that
$\Hb_\lambda>0$ there, provided that $r_2/r_1$ is large enough.
Since the deformation of $v$ is radial, it is clear that
$\kappa_\lambda>0$.  Define $g_\lambda=\lambda^2
\bigl(\phi_{1/\lambda}\bigr)^* g$ in $B_{r_0}$, and as before,
solve for $u_\lambda$ in~\eqref{eq:main} on
$[r_0,\infty)\times\Sphere^2$ with initial data
$u_\lambda|_{S_{r_0}}$.  Global existence and asymptotic behavior
as $r\to\infty$ is obtained from Theorems~\ref{thm:K}
and~\ref{thm:L1}.  The path $g_\lambda$ can be extended
continuously to $[1,\infty]$, and since $\beta_\lambda$,
$v_\lambda$, and $R_\lambda$ tend to zero as $\lambda\to\infty$,
it follows that $u_\lambda$ tends to $1$. Consequently $g_\infty$
is flat, and the continuous path $g_\lambda$ joins $g_1$ to a flat
metric $g_\infty\in\M_4$.  However, note that $g_1\ne g$, since
clearly $v_1\ne v$.  In order to complete the proof of the lemma,
we now define a continuous path $g_\lambda$, $\lambda\in[0,1]$,
between $g$ and $g_1$.  Define $g_\lambda=g$ in $B_{r_0}$,
$\beta_\lambda=\beta$, $R_\lambda=R$, and
$v_\lambda=\zeta\bigl(v+(1/2)\log(1-\lambda + \lambda\psi)\bigr)$.
Then from~\eqref{eq:Hb} we get
$e^{2v_\lambda}\Hb_\lambda=(1-\lambda)e^{2v_0}\Hb_0+\lambda
e^{2v_1}\Hb_1>0$ in $[r_0,r_1]$, and as before $\Hb_\lambda>0$
also in $[r_1,r_2]$ provided $r_2/r_1$ is large enough. Clearly,
as before, we have $\kappa_\lambda>0$. Thus, we can solve for
$u_\lambda$ in~\eqref{eq:main} as above.
\end{proof}

Note that due to the uniformization step in
Lemma~\ref{lemma:smoothing}, the final flat metric $g_\infty$ in
Lemma~\ref{lemma:flat} may be different from the background flat
metric originally given in $\M_0$.

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\end{document}