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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{17-JUL-2000,17-JUL-2000,17-JUL-2000,17-JUL-2000} \documentclass{era-l} % \renewcommand{\subjclassname}{% % \textup{2000} Mathematics Subject Classification} %\usepackage[active]{srcltx} %\usepackage{amsmath} %\usepackage{amssymb} %\usepackage[active]{srcltx} \usepackage{euscript} \issueinfo{6}{08}{}{2000} \dateposted{July 19, 2000} \pagespan{52}{63} \PII{S 1079-6762(00)00081-0} %\def\copyrightyear{2000} \copyrightinfo{2000}{American Mathematical Society} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{maintheorem}{Main Theorem} \renewcommand{\themaintheorem}{\hspace{-1ex}} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\R}{\mathbb R} \newcommand{\M}{\EuScript M} \newcommand{\Lie}{\EuScript L} \newcommand{\G}{\EuScript G} \newcommand{\nablash}{\nabla{\kern -.75 em \raise 1.5 true pt\hbox{{\bf/}}}\kern +.1 em} \newcommand{\Deltash}{\Delta{\kern -.69 em \raise .2 true pt\hbox{{\bf/}}}\kern +.1 em} \newcommand{\Rslash}{R{\kern -.60 em \raise 1.5 true pt\hbox{{\bf/}}}\kern +.1 em} \newcommand{\tr}{\operatorname{tr}} \newcommand{\Div}{\operatorname{div}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\Rfour}{\bar R} \newcommand{\gammab}{\bar\gamma} \newcommand{\chib}{\bar\chi} \newcommand{\gb}{\bar g} \newcommand{\Nb}{\bar N} \newcommand{\Hb}{\bar H} \newcommand{\Ab}{\bar A} \newcommand{\Bb}{\bar B} \newcommand{\betah}{{\hat\beta}} \newcommand{\chit}{\tilde\chi} \newcommand{\Nt}{\tilde N} \newcommand{\Ht}{\tilde H} \newcommand{\Sphere}{\mathbb S} \newcommand{\D}{\partial} \newcommand{\Ric}{\operatorname{Ric}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\bA}{\mathbf{A}} \newcommand{\st}{\colon\>} %\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_0 0. \end{equation} If the initial data $u_0 0$, 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 $0 0$ 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 $0 0$ for $r 0$. 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 $r_0 0$, 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. 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