FontBase.tex
% Comment to ward off arxiv errors
%auto-ignore 

\def\CurrentFontFile{}
\newdimen\DefaultFontSize
\DefaultFontSize=10pt
\newdimen\CurrentFontSize
\def\SetFont#1{\SetFontSpec{#1}{\CurrentFontSize}}
\def\SetFontSize#1{\SetFontSpec{\CurrentFontFile}{#1}}
\def\SetFontSpec#1#2{\edef\CurrentFontFile{#1}\CurrentFontSize=#2\font\ActiveFont#1 at\CurrentFontSize\ActiveFont}
\SetFontSpec{ptmr}{\DefaultFontSize}

\def\rm{\fam0\SetFont{\BaseFont}}
\def\it{\fam\itfam\SetFont{\ItalicFont}}
\def\sl{\fam\slfam\SetFont{\SlantedFont}}
\def\bf{\fam\bffam\SetFont{\BoldFont}}
\def\tt{\fam\ttfam\SetFont{\TypewriterFont}}

\def\SetTextFontGroup#1#2{\def\InstallProc{\csname InstallFontGroup#1\endcsname}\expandafter
\ifx\InstallProc\relax\errmessage{Unable to install font group #1}\else
\InstallProc{#2}\fi}

\def\SetMathFontGroup#1#2{\def\InstallProc{\csname InstallMathFontGroup#1\endcsname}\expandafter
\ifx\InstallProc\relax\errmessage{Unable to install math font group #1}\else
\InstallProc{#2}\fi}

\def\InstallFontGroupTimes#1{\gdef\BaseFont{ptmr}\gdef\ItalicFont{ptmri}\gdef\SlantedFont{ptmro}\gdef\BoldFont{ptmb}\gdef\TypewriterFont{pcrr}\SetFontSpec{\BaseFont}{#1}}
\def\InstallFontGroupPalatino#1{\gdef\BaseFont{pplr}\gdef\ItalicFont{pplri}\gdef\SlantedFont{pplro}\gdef\BoldFont{pplb}\gdef\TypewriterFont{pcrr}\SetFontSpec{\BaseFont}{#1}}

\def\InstallMathFontGroupTimesOld#1{%
%\font\tf=ztmcmr at 12pt
%\font\sf=ztmcmr at 9pt
%\font\ssf=ztmcmr at 7pt
\font\tf=cmr12
\font\sf=cmr9
\font\ssf=cmr7
\textfont0=\tf 
\scriptfont0=\sf
\scriptscriptfont0=\ssf
%\font\tf=ztmcmrm at 12pt
%\font\sf=ztmcmrm at 9pt
%\font\ssf=ztmcmrm at 7pt
\font\tf=cmmi12
\font\sf=cmmi9
\font\ssf=cmmi7
\skewchar\tf='177
\skewchar\sf='177
\skewchar\ssf='177
\textfont1=\tf 
\scriptfont1=\sf
\scriptscriptfont1=\ssf
\font\tfs=cmsy10 at 12pt
\font\sfs=cmsy9
\font\ssfs=cmsy7
\skewchar\tfs='60
\skewchar\sfs='60
\skewchar\ssfs='60
\textfont2=\tfs 
\scriptfont2=\sfs
\scriptscriptfont2=\ssfs
\textfont3=\tenex
\scriptfont3=\tenex
\scriptscriptfont3=\tenex
\font\tfsa=txsya at 12pt
\font\sfsa=txsya at 9pt
\font\ssfsa=txsya at 7pt
\textfont\msafam=\tfsa
\scriptfont\msafam=\sfsa
\scriptscriptfont\msafam=\ssfsa
\font\tfsb=txsyb at 12pt
\font\sfsb=txsyb at 9pt
\font\ssfsb=txsyb at 7pt
\textfont\msbfam=\tfsb
\scriptfont\msbfam=\sfsb
\scriptscriptfont\msbfam=\ssfsb}

\def\InstallMathFontGroupTimes#1{%
\mtsize = #1
\mssize = #1
\msssize = #1
\multiply\mssize by 900
\divide\mssize by 1200
\multiply\msssize by 700
\divide\msssize by 1200
\font\tf=cmr12 at \mtsize
\font\sf=cmr9 at \mssize
\font\ssf=cmr7 at \msssize
\textfont0=\tf 
\scriptfont0=\sf
\scriptscriptfont0=\ssf
\font\tf=cmmi12 at \mtsize
\font\sf=cmmi9 at \mssize
\font\ssf=cmmi7 at \msssize
\skewchar\tf='177
\skewchar\sf='177
\skewchar\ssf='177
\textfont1=\tf 
\scriptfont1=\sf
\scriptscriptfont1=\ssf
\font\tfs=cmsy10 at \mtsize
\font\sfs=cmsy9 at \mssize
\font\ssfs=cmsy7 at \msssize
\skewchar\tfs='60
\skewchar\sfs='60
\skewchar\ssfs='60
\textfont2=\tfs 
\scriptfont2=\sfs
\scriptscriptfont2=\ssfs
\font\txex=txex
\textfont3=\txex
\scriptfont3=\txex
\scriptscriptfont3=\txex
\font\tfsa=txsya at \mtsize
\font\sfsa=txsya at \mssize
\font\ssfsa=txsya at \msssize
\textfont\msafam=\tfsa
\scriptfont\msafam=\sfsa
\scriptscriptfont\msafam=\ssfsa
\font\tfsb=txsyb at \mtsize
\font\sfsb=txsyb at \mssize
\font\ssfsb=txsyb at \msssize
\textfont\msbfam=\tfsb
\scriptfont\msbfam=\sfsb
\scriptscriptfont\msbfam=\ssfsb


\font\trsfs=rsfs10 at \mtsize
\font\srsfs=rsfs7 at \mssize
\font\ssrsfs=rsfs5 at \msssize
\textfont\rsfsfam=\trsfs
\scriptfont\rsfsfam=\srsfs
\scriptscriptfont\rsfsfam=\ssrsfs

\font\tzc=pzcmi at \mtsize
\font\szc=pzcmi at \mssize
\font\sszc=pzcmi at \msssize
\textfont\zcfam=\tzc
\scriptfont\zcfam=\szc
\scriptscriptfont\zcfam=\sszc

\font\teu=eurm10 at \mtsize
\font\seu=eurm7 at \mssize
\font\sseu=eurm5 at \msssize
\textfont\eufam=\teu
\scriptfont\eufam=\seu
\scriptscriptfont\eufam=\sseu
}

\newdimen\mtsize
\newdimen\mssize
\newdimen\msssize
\newfam\msafam
\newfam\msbfam
\newfam\rsfsfam
\newfam\zcfam
\newfam\eufam
\def\InstallTxMath#1{
\mtsize = #1
\mssize = #1
\msssize = #1
\multiply\mssize by 900
\divide\mssize by 1200
\multiply\msssize by 700
\divide\msssize by 1200

\font\tf=txr at \mtsize
\font\sf=txr at \mssize
\font\ssf=txr at \msssize
\textfont0=\tf 
\scriptfont0=\sf
\scriptscriptfont0=\ssf

\font\tf=txmi1 at \mtsize
\font\sf=txmi1 at \mssize
\font\ssf=txmi1 at \msssize
\skewchar\tf='177
\skewchar\sf='177
\skewchar\ssf='177
\textfont1=\tf 
\scriptfont1=\sf
\scriptscriptfont1=\ssf

\font\tfs=txsy at \mtsize
\font\sfs=txsy at \mssize
\font\ssfs=txsy at \msssize
\skewchar\tfs='60
\skewchar\sfs='60
\skewchar\ssfs='60
\textfont2=\tfs 
\scriptfont2=\sfs
\scriptscriptfont2=\ssfs

\font\txex=txex
\textfont3=\txex
\scriptfont3=\txex
\scriptscriptfont3=\txex

\font\tfsa=txsya at \mtsize
\font\sfsa=txsya at \mssize
\font\ssfsa=txsya at \msssize
\textfont\msafam=\tfsa
\scriptfont\msafam=\sfsa
\scriptscriptfont\msafam=\ssfsa

\font\tfsb=txsyb at \mtsize
\font\sfsb=txsyb at \mssize
\font\ssfsb=txsyb at \msssize
\textfont\msbfam=\tfsb
\scriptfont\msbfam=\sfsb
\scriptscriptfont\msbfam=\ssfsb
}
\def\Bbb#1{{\fam\msbfam\relax#1}}
\def\Cal#1{{\fam\rsfsfam\relax#1}}
\SetTextFontGroup{Times}{\DefaultFontSize}
\SetMathFontGroup{Times}{\DefaultFontSize}

common.tex
% Comment to ward off arxiv errors
%auto-ignore 

\input amstex.tex
\documentstyle{amsppt}
\def\cal{\Cal}
\input eplain
\input FontBase.tex
\input amssym.tex
\DefaultFontSize=12pt
\SetTextFontGroup{Times}{12pt}
\SetMathFontGroup{Times}{12pt}

\def\InstallTimesMath#1{\SetMathFontGroup{Times}{#1}}
%\def\InstallTimesMath#1{\InstallTxMath{#1}}

\vsize=8.6in
\hsize=6in
\hoffset 0.25 in
\voffset 0.25 in
\parskip = 8 pt
\parindent = 0pt
%\advance \baselineskip by 8pt

% High level formatting
\newcount\sectioncount
\newcount\subsectioncount
\def\secword{Section}
\def\seclabel#1{\definexref{#1}{\number\sectioncount}{sec}}
\def\subsecword{Section}
\def\subseclabel#1{\defineref{#1}{\number\sectioncount.\number\subsectioncount}{subsec}}

\def\bl{}
\def\el{}
\def\li#1{\par\penalty500\hangindent=4em
\hangafter=0\indent\llap{\SetFontSpec{ptmr}{12pt}#1\hskip 0.5em}}
%\def\define#1{{\SetFontSpec{ptmb}{12pt}#1}}
\def\define#1{#1}%

\newif\ifdraft
\drafttrue
\def\Version#1{\draftfalse\gdef\versiondate{#1}}

\newif\iflatexing\latexingfalse
\newif\ifjournal\journalfalse

\def\section#1{\advance\sectioncount by 1 \subsectioncount = 0 \proccount=0
\vskip 10pt plus 6pt minus 2pt \goodbreak
{\InstallTimesMath{16pt}\SetFontSpec{pplb}{16pt}\number\sectioncount. #1\par}}

\def\subsection#1{\advance\subsectioncount by 1
\vskip 5pt plus 3pt \goodbreak
{\InstallTimesMath{13pt}\SetFontSpec{pplb}{13pt}
\number\sectioncount.\number\subsectioncount\ #1\par}}

\def\raggedcenter{\leftskip=0pt plus4em\rightskip=\leftskip
\parfillskip=0pt\spaceskip=.3333em\xspaceskip=.5em\pretolerance=9999
\tolerance=9999\parindent=0pt\hyphenpenalty=9999\exhyphenpenalty=9999}

\def\startmatter{\bgroup\noindent\vskip 2cm plus 0.5cm 
\baselineskip=12pt
\ifx\titletext{}\else{\raggedright\SetFontSpec{pplb}{20pt}\hbox to \hsize{\vbox{\titletext}}}\bigskip\fi
\SetFontSpec{ptmr}{12pt}%
\InstallTimesMath{12pt}%
\ifx\authortext{}\else{\bf \authortext}\hfill
\vskip 0pt{\authorhome\hfill\parskip=0pt\vskip 0pt\email\authoremail}
\smallskip\fi
\ifdraft Draft: \today\else\versiondate\fi\vskip 1cm
\ifx\abstracttext{}\else{{\SetFontSpec{pplb}{10pt}Abstract. 
}\SetFontSpec{ptmr}{10pt}\abstracttext\smallskip}\fi
\egroup\advance \baselineskip by 4pt}
\let\endmatter\bye

\def\title#1{\global\def\titletext{#1}}
\def\titletext{}
\def\Author#1#2#3{\global\def\authortext{#1}\gdef\authorhome{#2}\gdef\authoremail{#3}}
\def\authortext{}
\def\authorhome{}
\def\Abstract#1{\global\def\abstracttext{#1}}
\def\abstracttext{}
\def\headtitle{{\SetFontSpec{ptmri}{10pt}\titletext\ifdraft\ (Draft: \today)\fi}} 
\def\headauthor{{\SetFontSpec{ptmri}{10pt}\authortext}} 
\def\email#1{{E-mail: #1}}%

\headline={\ifnum\pageno>1\SetFontSpec{ptmr}{10pt}%
\ifx\titletext{}\folio.\hfill\else
\ifodd\pageno\headtitle\hfill\folio\else
\folio\hfill\headauthor\fi\fi\else\hfil\fi}
\nopagenumbers

\catcode`@=11 % at signs are  letters
\def\makeheadline{\vbox to\z@{\vskip-35\p@
  \line{\vbox to8.5\p@{}\the\headline}\vss}\nointerlineskip}
\catcode`@=12 % at signs are no longer letters

% ams bib stuff
\catcode`@=11 % at signs are  letters
\def\providecommand#1{
   \def\@commandname{#1}%
   \@getoptionalarg\@postprovidecommand
}
\def\@postprovidecommand#1{}
\catcode`@=12 % at signs are no longer letters
\def\textbf#1{{\bf#1}}

\newcount\proccount
\def\myproclaim#1{\advance\proccount by 1\medbreak\noindent{\bf #1.%
\enspace}\bgroup\sl}
\def\procnumber{\number\sectioncount.\number\proccount}

\def\endmyproclaim{\egroup\par
\ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi}

\def\Proof{\endmyproclaim{\it Proof:\hskip 0.5em}}
\def\NoProof{\endmyproclaim}
\def\EOProof{\unskip\quad$\square$\goodbreak\par}

\def\propword{Proposition}
\def\proplabel#1{\definexref{#1}{\procnumber}{prop}}
\def\prop{\myproclaim{\propword\ \procnumber}\def\label{\proplabel}}

\def\theoremword{Theorem}
\def\theoremlabel#1{\definexref{#1}{\procnumber}{theorem}}
\def\thm{\myproclaim{\theoremword\ \procnumber}\def\label{\theoremlabel}}

\def\lemmaword{Lemma}
\def\lemmalabel#1{\definexref{#1}{\procnumber}{lemma}}
\def\lem{\myproclaim{\lemmaword\ \procnumber}\def\label{\lemmalabel}}

\def\corword{Corollary}
\def\corlabel#1{\definexref{#1}{\procnumber}{cor}}
\def\cor{\myproclaim{\corword\ \procnumber}\def\label{\corlabel}}

\def\Definition{\advance\proccount by 1\medbreak\noindent{\bf 
Definition \procnumber}\enspace\bgroup\sl}
\def\EODefinition{\egroup\par\bigskip}


\def\defnumber{\number\sectioncount.\number\defcount}





\def\Remark{{\bf Remark.}\enspace}
\def\EORemark{\medskip\par}

% multiline math
\def\({$$\eqalign\begingroup}
\def\){\endgroup$$}

\def\Acknowledge{\bigskip{\bf Acknowledgements.}\enspace}
\def\EOAcknowledge{\par}

\catcode`@=11 % at signs are  letters

\def\eqaligntd#1{\null\,\vcenter\bgroup\openup\jot\m@th
  \ialign\bgroup\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$&$\displaystyle{{}##}$\hfil
\crcr}
\def\endeqaligntd{\crcr\egroup\egroup\,}

\def\eqalignd{\null\,\vcenter\bgroup\openup\jot\m@th
  \ialign\bgroup\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr}      
\def\endeqalignd{\crcr\egroup\egroup\,}

\def\eqalignnod{\displ@y \tabskip\centering
  \halign to\displaywidth\bgroup\hfil$\@lign\displaystyle{##}$\tabskip\z@skip
    &$\@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\@lign##$}\tabskip\z@skip\crcr}
\def\endeqalignnod{\crcr\egroup}

%\def\eqalignnod{\displ@y \tabskip\centering
%  \halign to\displaywidth\bgroup\hfil$\@lign\displaystyle{##}$\tabskip\z@skip
%    &$\@lign\displaystyle{{}##}$\hfil\tabskip\centering
%    &\llap{$\@lign##$}\tabskip\z@skip\crcr}

\catcode`@=12 % at signs are no longer letters

\def\be{$$}
\def\ee{$$}
\def\bel#1{\gdef\eqlabel{#1}$$}
\def\eel{\eqdef{\eqlabel}$$}
\def\bea{\bgroup\let\\\cr$$\eqalignd}
\def\eea{\endeqalignd$$\egroup}
\def\beal#1{\bgroup\let\\\cr\gdef\eqlabel{#1}$$\eqalignd}
\def\eeal{\endeqalignd\eqdef{\eqlabel}$$\egroup}
\def\beat{\bgroup\let\\\cr$$\eqaligntd}
\def\eeat{\endeqaligntd$$\egroup}
\def\beatl#1{\bgroup\let\\\cr\gdef\eqlabel{#1}$$\eqaligntd}
\def\eeatl{\endeqaligntd\eqdef{\eqlabel}$$\egroup}
\def\bean{\bgroup\let\\\crcr\let\label\ealabel
$$\eqalignnod}
\def\eean{\endeqalignnod$$\egroup}
\def\ealabel#1{&\eqdef{#1}\crcr}

\let\ref\refn
\def\nonumber{}

% Some style types
%\def\emph{\SetFontSpec{phvb}{10pt}}
%\def\Emph{\SetFontSpec{phvb}{14pt}}
\def\tt{\fam\ttfam\SetFontSpec{\TypewriterFont}{8pt}}
\def\sf{\SetFontSpec{phvr}{10pt}}

% Extra definitions
\def\ds{\displaystyle}
\def\newpage{\vfill\eject}
\def\til{\lower 0.75ex\hbox{\char126}}

% Holder spaces
\def\hCf#1#2{C^{#1,#2}}
\def\hC#1{\hCf{#1}{\alpha}}
\def\Cza{\hC{0}}
\def\Czad{\Cza_\delta}
\def\Czaloc{\Cza_{\rm loc}}
\def\Cka{\hC k}

% Other math defs
\def\urlf{\SetFontSpec{pcrr}{11pt}}
\def\emph#1{{\it#1}}
\def\bysame{\leavevmode\hbox to3em{\hrulefill}\thinspace}

\def\references{\section{References}}

\input epsf
\def\epsbox#1{\epsfbox{#1}}
\def\epsboxw#1#2{{\epsfxsize=#1\epsfbox{#2}}}
\newcount\figurecount
\def\figlabel#1{\definexref{#1}{\number\figurecount}{fig}}
\def\fig#1#2{\advance\figurecount by 1\noindent\vskip\parskip\bigskip
{\def\label{\figlabel}\vbox{\hbox to \hsize{\hfil#1\hfil}\par\hfil
Figure \number\figurecount: #2\hfil\bigskip}}}

macros.tex
% Comment to ward off arxiv errors
%auto-ignore 
\def\bnorm#1{\left|\left|#1\right|\right|}
\def\Eflat{{\overline E} }
\def\gflat{{\overline g} }
\def\Reals{{\Bbb R}}
\def\Ints{{\Bbb Z}}
\def\Rn{\Reals^n}
\def\abs#1{\left|#1\right|}
\def\partialbeta{\partial^{\,\beta}}
\def\loc{_{\rm loc}}
\def\scl{_{\rm scl}}
\def\cpct{_{\rm c}}
\def\spOmega{;\;\Omega}
\def\Holder{H\"older }
\def\gchi{\raise 0.4 ex\hbox{$\chi$}}
\def\evat#1#2{\left.#1\;\right|_{#2}}
\def\evdM#1{\evat{#1}{\partial M}}
\def\calA{{\cal A}}
\def\calB{{\cal B}}
\def\calD{{\cal D}}
\def\calL{{\cal L}}
\def\calV{{\cal V}}
\def\calP{{\cal P}}
\def\calM{{\cal M}}
\def\calS{{\cal S}}
\def\calF{{\cal F}}
\def\calG{{\cal G}}
\def\diff{{\rm d}\,}
\def\Id{{\rm Id}}
\def\graph{{\rm graph}\;}
\def\del{_\delta}
\def\mdel{_{-\delta}}
\def\srho{_{\rho}}
\def\Lop{\mathop L\nolimits}
\def\Bop{\mathop B\nolimits}
\def\tM{\tilde{M}\nolimits}
\def\tK{\tilde{K}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
\def\div{\mathop{\rm div}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\ondm{\quad\hbox{on $\partial M$}}
\def\tg{{\tilde{g}}}
\def\tR{{\tilde{R}}}
\def\th{{\tilde{h}}}
\def\tu{\tilde{u}}
\def\hatg{{\hat g}}
\def\hatK{{\hat K}}
\def\tsigma{\tilde{\sigma}}
\def\lie{{\mathop{\bf L}}\nolimits}
\def\ck{{\Bbb L}}
\def\vl{\Delta_{\Bbb L}}
\def\rl{\Delta_{\rm R}}
\def\Leps{\vl}
\def\Beps{B\,}
\def\Lflat{\overline{\Delta}_{\Bbb L}}
\def\Bflat{\overline{B}}
\def\vlflat{\overline{\Delta}_{\Bbb L}}
\def\ckflat{\overline{\Bbb L}}
\def\jia{_j^{\,i,\alpha}}
\def\ija{_{ij}^{\;\alpha}}
\def\jiapa{\jia\partial_\alpha}
\def\ijapa{\ija\partial_\alpha}
\def\hBr{B_r^+}
\def\dV{\;dV}
\def\dA{\;dA}
\def\tdV{\;\tilde{dV}}
\def\tdA{\;\tilde{dA}}
\def\converges#1{\mathrel{\ds\mathop{\longrightarrow}_{#1}}}
\def\bull{$\bullet$}
\def\nuin{{\nu_{\rm int}}}
\def\nuout{{\nu_{\rm ext}}}
\def\PL{{\cal P}_{\Bbb L}}
\def\Lie#1{{\cal L}_{#1}}
\def\dimc{\kappa}
\def\expfrac#1#2{{#1\over#2}}
\def\ip<#1,#2>{\left<#1,#2\right>}
\def\P#1#2#3{{\calP}^{#1,#2}_{#3}}
\def\cz{\check}
\def\low{_{\rm low}}
\def\H#1{H^{#1}}
\def\id{\mathop{\rm id}}
\def\cpctsubset{\subset\subset}
\def\Htt{{H^{s-1}_{\delta-1}}}
\def\Hmt{{H^{s}_{\delta}}}
\def\Hout{{H^{s-2}_{\delta-2}}}
\def\ra{\rightarrow}
\def\vlk{\Delta_{\Bbb L_k}}
\def\divk{\mathop{ {\rm div}_{k}}}
\def\Vol{\mathop{ {\rm Vol}}\nolimits}
\def\im{\mathop{{\rm im}}\nolimits}
\def\Lap{\Delta\,}

% Sobolov spaces
\def\W#1#2{W^{{#1},{#2}}}
\def\Woq{\W{1}{q}}
\def\Wkq{\W{k}{q}}
\def\Woqloc{\W{1}{q}_{\rm loc}}
\def\Woqd{\W{1}{q}_{\delta}}
\def\Woqn{\W{1}{q}_{0}}

rough-compact.tex
\input common.tex
\input macros.tex

%%%%% START of original document

\Version{June 15, 2004}
\draftfalse
\title{\raggedright Rough Solutions of the Einstein Constraint Equations on Compact Manifolds}
\ifjournal
\Author{David Maxwell}{Department of Mathematical Sciences, University of Alaska Fairbanks, Fairbanks, Alaska\ \ 99775}{}
\else
\Author{David Maxwell}{University of Alaska Fairbanks}{david.maxwell@uaf.edu}
\fi
\Abstract{We construct low regularity solutions of the vacuum
Einstein constraint equations on compact manifolds.  On $3$-manifolds
we obtain solutions with metrics in $H^s$ where $s>3/2$.  The constant
mean curvature (CMC) conformal method leads to a construction of all CMC
initial data with this level of regularity.  These results extend
a construction from \cite{maxwell-rough} that treated the
asymptotically Euclidean case.}
\startmatter
\section{Introduction}
Initial data for the vacuum Cauchy problem in general relativity consists of
a Riemannian manifold $(M^3,g)$ and a symmetric
$(0,2)$-tensor $K$. One seeks an isometric embedding of $(M,g)$ 
into a Ricci-flat Lorenzian manifold
such that $K$ is the extrinsic curvature of this embedding.  Since
the ambient manifold is Ricci flat, the initial data $(M,g,K)$ must
satisfy the compatibility conditions
\beal{constraints}
R_g-\abs{K}_g^2+ \tr_g K ^2 &= 0\\
\div_g K -d\,\tr_g K &= 0,
\eeal
where $R_g$ is the scalar curvature of the metric $g$.  
System \eqref{constraints} is known known as the Einstein constraint 
equations; the first equation is the Hamiltonian constraint and the second
is the momentum constraint.

The Cauchy problem has a formulation as a second-order quasilinear system 
of hyperbolic PDEs called the Einstein equations.  In \cite{cb-cauchy}
Choquet-Bruhat proved that this problem is locally well-posed for 
sufficiently smooth initial data satisfying the constraint equations. 
Subsequent investigations have reduced the amount of 
regularity required from the initial data, but development of the 
low regularity theory of the constraint equations has lagged behind
that of the evolution problem.
For example, the local well-posedness theorem of \cite{well-posed-hyp} 
for the Einstein equations
permits initial data $(g,K)$ in $H^s\loc\times H^{s-1}\loc$ with $s>5/2$.
But solutions of the constraint equations with this level of regularity 
were not known to exist until recently \cite{const-asympt-flat}.
The need for a theory of rough solutions of the 
constraint equations became especially prominent after the appearance of 
\cite{kr-rough-annals} which proved an a priori estimate for the
the time of existence of solutions of the vacuum Einstein equations
depending on the size of $(\partial g,K)$ in $H^{s-1}\times H^{s-1}$
with $s>2$.

Constructions of rough solutions of the constraint equations have
been obtained in \cite{cb-low-reg} on compact manifolds and in
\cite{const-asympt-flat} and more generally in 
\cite{maxwell-ahc} on asymptotically Euclidean manifolds. These works 
provide solutions with metrics in $H^2$.  More specifically, these
solutions were constructed in $\W kp\loc$ spaces where $k\ge 2$ is an
integer and $k>3/p$.  The restriction $k>3/p$ is interesting because
it is analogous to condition $s>3/2$ for $H^s$ spaces.  Since $s=3/2$
is the scaling limit for the Einstein equations, this level of
regularity is a natural lower bound for for seeking solutions of the
constraint equations.  These considerations motivated work
in \cite{maxwell-rough} which gave a construction of all constant mean
curvature (CMC) asymptotically Euclidean solutions of the constraint
equations with metrics in $H^s\loc$ for real values of $s>3/2$.  
In this paper we
extend the low regularity construction from
\cite{maxwell-rough} to compact manifolds. In particular,
we show that the conformal method can be used to construct all CMC vacuum
solutions $(g,K)$ of the constraint equations belonging to $H^s\times
H^{s-1}$ with $s>3/2$.

The constant mean curvature  conformal method of Lichnerowicz 
\cite{lich-conformal} and Choquet-Bruhat and York \cite{cb-york-held} 
is the basis of all known constructions of rough solutions of the
constraint equations. The method starts with conformal data $(M^n,g,S,\tau)$ 
where $M^n$ is a smooth, compact, connected manifold with $n\ge 3$,
$g$ is a metric on $M$, $S$ is a trace-free $(0,2)$-tensor, and 
$\tau$ is a scalar function.\footnote{We work with general
$n$-dimensional manifolds since some of our auxiliary results are 
meaningful outside the context of the constraint equations.}
We then seek a solution $(\tilde g, \tilde K)$ of the constraint equations 
on $M$ of the form
\bea
\tilde g &= \phi^{2\dimc} g\\
\tilde K &= \phi^{-2}\left(\sigma+\ck W\right)+{\tau\over n}\tilde g
\eea
where the unknowns $(\phi,W)$ are a positive function 
and vector field respectively, $\dimc={2\over n-2}$ is a dimensional constant, 
and $\ck$ is the conformal Killing operator defined by
\be
\ck W_{ab} = \nabla_a W_b + \nabla_bW_a - {2\over n} \div W g_{ab}.
\ee
It follows that $(\tilde g, \tilde K)$ solves the constraint equations
so long as
\beal{conf-sys}
-a\Lap_g \phi + R_g \phi &= \abs{S+\ck W}_g^2\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1} \\
\div \ck W &=b \phi^{-n\dimc} \;d\tau-\div S
\eeal
where $R_g$ is the scalar curvature of $g$ and  $a=2\dimc+4$ and 
$b={n-1\over n}$ are constants.  The function $\tau$ specifies the
mean curvature $\tr_{\tilde g}\tilde K$, and the conformal method
is most successful when $\tau$ is assumed to be constant (see, however,
\cite{const-non-cmc-compact}\cite{cb-low-reg}\cite{isenberg-omurchadha-noncmc} for
results in the non-CMC case).  When $\tau$ is constant,
system \eqref{conf-sys} decouples and the second equation becomes
\bel{ck}
\div \ck W =-\div S
\eel
In the smooth setting, equation \eqref{ck} is always solvable and
leads to a decomposition of the set of smooth symmetric, 
trace-free $(0,2)$-tensors $\calS$
\bel{iyork-split}
\calS=\im\ck\oplus \ker\div
\eel
known as York splitting.  Tensors in $\calS\cap\ker\div$ are
known as transverse-traceless, and specifying conformal data $(M,g,S,\tau)$ 
for the CMC conformal method is equivalent to specifying $(M,g,\sigma,\tau)$, 
where $\sigma$ is the transverse-traceless part of $S$. The study of
system \eqref{conf-sys} reduces to determining conditions when the
Lichnerowicz equation
\bel{lich}
-a\Lap_g \phi + R_g \phi = \abs{\sigma}^2_g\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1}
\eel
has a positive solution.  
For smooth metrics it is known\cite{cb-york-held}\cite{const-compact} that 
\eqref{lich} is solvable if and only if one of the following conditions hold:
\bl
\li{1.} $\sigma\not\equiv 0$ and $\tau\neq 0$,
\li{2.} $\lambda_g>0$, $\sigma\not\equiv 0$ and $\tau=0$,
\li{3.} $\lambda_g=0$, $\sigma\equiv 0$ and $\tau= 0$,
\li{4.} $\lambda_g<0$, $\sigma\equiv 0$ and $\tau\neq 0$.

\el
Here $\lambda_g$ is  the Yamabe invariant defined by
\be
\lambda_g = \inf_{f\in C^{\infty}(M)\atop f\not\equiv0} 
{\int_M a\abs{\nabla f}^2_g + R_g f^2\dV_g\over ||f||_{L^{2^*}}^2}.
\ee

Our main result, Theorem \ref{thm:lich}, shows that these conditions are
also necessary and sufficient for the solvability of \eqref{lich} when
$g\in H^s$ and $\sigma\in H^{s-1}$ with $s>n/2$.  We also show in
Theorem \ref{thm:splitting} that the York decomposition \eqref{iyork-split} 
holds for symmetric, trace-free $(0,2)$-tensors in $H^{s-1}$ and hence the
CMC conformal method fully extends to the rough setting.

Aside from lowering the regularity of solutions, we also obtain some
technical improvements to the previous construction of rough solutions
on compact manifolds obtained in \cite{cb-low-reg}.  
Solutions were found in that reference in the Yamabe negative
case ($\lambda_g<0)$ only under an additional hypothesis on the
singularities of $\sigma$, namely $\sigma\in H^{s-1}\cap L^\infty$.
Also, the case $\lambda_g>0$, $\tau=0$ and $\sigma\not\equiv0$ when
$\sigma$ has zeros was not treated.  The zeros and singularities of
$\sigma$ pose difficulties for the method of sub- and supersolutions
used to find solutions of \eqref{lich}.  Our approach to finding sub-
and supersolutions of \eqref{lich}, discussed in Section
\ref{sec:ham}, circumvents these problems by means of 
an initial conformal transformation to a metric with the singularities and
zeros of $\sigma$ already encoded in the scalar curvature.  This
strategy has been used previously for smooth conformal data,
but the choice of conformal change used here is new
and has interest even in the context of smooth solutions.

\section{Considerations for Rough Metrics}
Before turning to the specifics of solving the constraint equations,
we start with an outline of the rules for working with rough metrics.
Throughout this paper, let $(M^n,g)$ be a compact, connected
Riemannian manifold with $n\ge 3$ and $g\in H^s$.    Our definitions and
notation for Sobolev spaces follows \cite{taylor-I}.
The value $s=n/2$ is a dividing
point between metrics that are natural for geometric analysis and
those that are not.  For example, the Laplacian of a metric in $H^s$
with $s<n/2$ doesn't necessarily take $H^2$ to $L^2$ or even $H^1$ to
$H^{-1}$.  This is a consequence of the fact that a metric in $H^s$
with $s<n/2$ is not necessarily in $L^\infty$.  On the other hand, a
metric in $H^s$ with $s>n/2$ can be nearly as good as a smooth
metric. The following well-known lemma is of central importance.
\lem\label{lem:mult}
Suppose $M^n$ is a compact manifold, 
$\sigma \le \min (s_1,s_2)$, $s_1+s_2\ge 0$ 
and $\sigma < s_1+s_2-{n\over 2}$.
Then pointwise multiplication extends to a continuous bilinear map
\be
H^{s_1}(M)\times 
H^{s_2}(M)\rightarrow 
H^{\sigma}(M).
\ee
\NoProof
Lemma \ref{lem:mult} implies that
if $u\in H^s$ with $s>n/2$, then multiplication by $u$
takes $H^\sigma$ to $H^\sigma$ for every $\sigma\in [-s,s]$.
In particular, $H^s$ is an algebra.  This fact allows us to have well
defined conformal changes of metric, for example. If $\phi$ is a positive 
function in $H^s$, and if $g$ is a metric in $H^s$, then Lemma
\ref{lem:mult} implies $\phi^\alpha g$ is again rough metric in $H^s$ for
any positive integer $\alpha$.  To allow arbitrary real exponents $\alpha$, 
and to work with other nonlinearities, we require the following consequence of
Lemma \ref{lem:mult}.
\lem\label{lem:nonlin}
Suppose $M^n$ is a compact manifold and $F:I\ra\Reals$ is smooth
on some interval $I$. If $s>n/2$ and $u\in H^s(M)$ satisfies $u(x)\in I$
for all $x\in M$, then $F(u)\in H^s(M)$. 
\Proof
Working in a coordinate chart we can assume that $u$ is defined on $\Rn$
and need only show that $F(u)\in H^s\loc$.  Let $k$ be the largest 
integer such that $k\le s$.  If $\alpha$
is a multi-index such that $\abs{\alpha}=k$, then $\partial_\alpha F(u)$
is a sum of terms of the form 
$F^{(r)}(u) \partial_{\alpha_1}u\cdots\partial_{\alpha_r}u$ where
$\sum_{i=1}^r \abs{\alpha_i}= k$.  From repeated applications of
Lemma \ref{lem:mult} we have 
$\partial_{\alpha_1}u\cdots\partial_{\alpha_r}u\in H^\sigma\loc$ for any
$\sigma < s - k + (r-1) (s-{n\over 2})$.  Since $F^{r}(u)\in L^\infty\loc$, this 
shows that $F(u)\in H^k\loc$. To obtain the full regularity of
$F(u)$, consider any derivative $\partial_i F(u) = F'(u)\partial_i u$.
Since $F'$ is smooth, we have $F'(u)\in H^k\loc$ and from Lemma
\ref{lem:mult} we have $F'(u)\partial_iu\in H^{\sigma_1-1}$ for every
$\sigma_1\le s$ such that $\sigma_1<k+s-{n\over 2}$.  
Hence $F(u)\in H^{\sigma_1}\loc$. Repeating this argument shows that
$F(u)\in H^{\sigma_2}\loc$ for every $\sigma_2\le s$ such that 
$\sigma_2<k+2\left(s-{n\over 2}\right)$. A finite number of further repetitions
shows that $F(u)\in H^s\loc$.
\EOProof

Consider the Laplacian $\Lap_g$ of a rough metric $g$.  In local coordinates
we have
$$\Lap_g u = {1\over \sqrt{\det g}} \partial_i ( {\sqrt{\det g}} g^{ij}\partial_j u ).
$$
It follows from Lemmas \ref{lem:mult} and \ref{lem:nonlin} that 
the Laplacian has the form in local coordinates
$$
a^{ij}\partial_i\partial_j +b^i\partial_i
$$
where $a\in H^{s}\loc$ and $b^i\in H^{s-1}\loc$.  Using Lemma \ref{lem:mult}
it follows that $\Lap_g:H^{\sigma}\ra H^{\sigma-2}$ for any 
$\sigma\in [2-s,s]$.  The following class of differential operators 
encodes the regularity of the coefficients of the Laplacian.
\Definition
Let $A$ be a linear differential operator that in any coordinate
chart has the form
\be
A = \sum_{\abs{\alpha}\le m} a^\alpha \partial_\alpha,
\ee
where $a^\alpha$ is a $\Reals^{k\times k}$ valued function.
We say that 
\be
A\in \calL^{m,s}
\ee
if  $a^\alpha\in H^{s-m+\abs\alpha}\loc$ for all $\abs{\alpha}\le m$.
\EODefinition
An analogous definition holds for differential operators on sections of vector
bundles.  From Lemma \ref{lem:mult} we have the following mapping properties.
\lem\label{lem:opmap}
Suppose $m\le 2s$ and $\sigma\in[m-s,s]$.  
If $A\in\calL^{m,s}$, then $A$ defines a continuous map
\be
A:H^{\sigma}\rightarrow H^{\sigma-m}.
\ee
\NoProof
We say $A$ is elliptic if for each $x$, the constant coefficient operator
$\sum_{\abs\alpha=m}a^\alpha(x)\partial_\alpha$ is elliptic in the usual sense.
Using Lemmas \ref{lem:mult} and \ref{lem:nonlin} it is easy to check 
that the Laplacian $\Lap_g$, conformal 
Laplacian $-a\Lap_g+R_g$, and vector Laplacian $\vl=\div\ck$ all belong to 
$\calL^{2,s}$ when $s>n/2$ and are all elliptic.   

The following interior elliptic estimate was proved in \cite{maxwell-rough}.
We use the notation $A\lesssim B$ to mean $A< c B$ for a
positive constant $c$ where $c$ does not depend on the variable functions and 
parameters of $A$ and $B$.  For example, in estimate \eqref{elliptic-local-est}
below the implicit constant is independent of the function $u$ but can
can depend on the open sets $U$, $V$, the operator $A$, and so forth.
\prop\label{prop:elliptic-local} 
Let $U$ and $V$ be open sets in $\Rn$ with $U\cpctsubset V$, and
suppose $s>n/2$ and  $\sigma\in(m-s,s]$.
If $A\in \calL^{m,s}$ is elliptic, then for every $u\in H^\sigma$ we have
\bel{elliptic-local-est}
||u||_{\H \sigma(U)} \lesssim || A u ||_{H^{\sigma-m}(V)} + 
||u||_{H^{m-s}(V)}.
\eel
\NoProof
Using coordinate charts and cutoff functions, we obtain as an immediate
consequence a-priori estimates on compact manifolds.
\cor\label{cor:elliptic-ests}
Suppose $(M^n,g)$ is a compact Riemannian manifold with $g\in H^s$ and $s>n/2$.
If $A$ is a second order elliptic differential operator in $\calL^{2,s}$,
then 
\be
||X||_{H^s(M)} \lesssim ||A X||_{H^{s-2}(M)} + ||X||_{L^2(M)}
\ee
for all $X\in H^s(M)$.
\NoProof

We now turn to duality properties and integration by parts for rough
metrics.  We would like to claim, for example, that the equation
$$
\int_M (-a\Lap_g u + R_g u)v \dV_g = 
\int_M a\abs{\nabla u}^2_g + R_g uv\dV_g
$$
that holds for the conformal Laplacian of a smooth metric
also holds for rough metrics.  We need to show that integration by parts
holds and that the term $\int_M R_g uv \dV_g$ can be suitably interpreted
for rough metrics (note that $R_g$ need not be a locally integrable
function when $n=3$).  

Suppose $g$ is a rough metric in $H^s$ with $s>n/2$ and 
$\hat g$ is a smooth background  metric on $M$.  We have a bilinear 
form defined for  $u,v\in C^\infty(M)$ by
\bel{bilin}
\ip<u,v>_{(M,\hatg)} =\int_M uv \dV_{\hat g}
\eel
which extends for any $\sigma\in \Reals$ to a continuous bilinear map
\be
\ip<\cdot,\cdot>_{(M,\hatg)}: H^{\sigma} \times 
H^{-\sigma}(M)\ra \Reals
\ee
and induces an isomorphism
\be
H^\sigma(M)=\left(H^{-\sigma}(M)\right)^*.
\ee
In applications we would prefer use the rough metric $g$ instead
of the smooth metric $\hat g$ to induce this isomorphism.  This
can be done for a restricted range of Sobolev spaces.  From
Lemma \ref{lem:nonlin}, we have $\dV_g = h\dV_{\hat g}$ where $h>0$ and 
$h\in H^s(M)$. Multiplication by $h$ is continuous on on $H^\sigma$ for every 
$\sigma\in[-s,s]$, so we can define for 
$u\in H^{\sigma}$ and $v\in H^{-\sigma}$
\be
\ip<u,v>_{(M,g)}= \ip<hu,v>_{(M,\hatg)}.
\ee
With this definition, we have the following easy consequence.
\lem\label{lem:duality} Suppose $(M^n,g)$ is a compact Riemannian manifold 
with $g\in H^s$ and $s>n/2$ and suppose  $\sigma\in[-s,s]$. Then
for every $u\in H^\sigma$ and $v\in H^{-\sigma}$,
$$
\abs{\ip<u,v>_{(M,g)}}\lesssim ||u||_{H^\sigma}||v||_{H^{-\sigma}}.
$$
Moreover, $H^\sigma(M)$ is naturally isomorphic to 
$\left(H^{-\sigma}(M)\right)^*$ through the pairing
$\ip<\cdot,\cdot>_{(M,g)}$. Finally, for smooth functions $u$ and $v$,
\be
\ip<u,v>_{(M,g)}=\int_M uv\dV_g.
\ee
\NoProof
These ideas readily extend to sections of vector bundles as well.
We show here explicitly how this is done for sections of the tangent
bundle. The smooth background metric $\hat g$ gives rise to a 
bilinear form defined for smooth vector fields $X$ and $Y$ by
\be
\ip<X,Y>_{(M,\hatg)} =\int_M \ip<X,Y>_g \dV_{\hat g}.
\ee
This map extends to a continuous bilinear form of less regular vector fields
\be
\ip<\cdot,\cdot >_{(M,\hatg)} : H^\sigma\times H^{-\sigma} \ra \Reals
\ee
for every $\sigma\in\Reals$.
Now if $X$ is a $H^\sigma$ vector field with $\sigma\in[-s,s]$, we
can define $X_*\in H^\sigma$ by
$$
X_*^a= h \hat g^{ab} g_{bc} X^c
$$
where as before $\dV_g=h\dV_{\hat g}$.
We then define for vector fields  $X$ and $Y$ in $H^\sigma$ and 
$H^{-\sigma}$ respectively
$$
\ip<X,Y>_{(M,g)} = \ip<X_*,Y>_{(M,\hat g)},
$$
and we note that if $X$ and $Y$ are smooth then
$$
\ip<X,Y>_{(M,g)} = \int_M \ip<X,Y>_g \dV_g.
$$

To establish an integration by parts formula, suppose that $u\in H^{\sigma}$
with $\sigma\in[2-s,s]$, $v\in H^{2-\sigma}$ , and suppose that $v$ is
supported in the domain of some coordinate chart.  We then have in local
coordinates
\def\gbar{\overline{g}}
$$
\ip<\nabla u,\nabla v>_{(M,g)} = \sum_{i,j} \ip< \sqrt{\det g} g^{ij} \partial_i u, \partial_j v>_{(\Rn,\gbar)}
$$
where the bilinear form on the right hand side is the usual pairing 
of $H^{\sigma-1}$ against $H^{1-\sigma}$ for
the flat Euclidean metric.  The standard integration by parts formula
$$
\ip< \partial_i f, g>_{(\Rn,\gbar)} = \ip< f, -\partial_i g>_{(\Rn,\gbar)}
$$
is valid for $f\in H^\tau(\Rn)$ and $g$ compactly supported in 
$H^{1-\tau}(\Rn)$, for any $\tau\in\Reals$.  Hence 
$$
\sum_{i,j} \ip< \sqrt{\det g} g^{ij} \partial_i u, \partial_j v>_{\Rn,\gbar}
=
\sum_{i,j} \ip< -\partial_j\sqrt{\det g} g^{ij} \partial_i u, v>_{\Rn,\gbar} = \ip< -\Lap_g u, v >_{(M,g)}.
$$
A partition of unity argument then establishes
\be
\ip<\nabla u,\nabla v>_{(M,g)} = \ip<-\Lap_g u, v>_{(M,g)}
\ee
for every $u\in H^\sigma(M)$ and $v\in H^{2-\sigma}(M)$ with $\sigma\in [2-s,s]$.
A similar approach also shows that integration by parts also holds for the
vector Laplacian. 
\lem\label{lem:ibp} 
Suppose $(M^n,g)$ is a compact Riemannian manifold with $g\in H^s$ with
$s>n/2$.  Then for functions $u\in H^\sigma$ and $v\in H^{2-\sigma}$ with
$\sigma\in[2-s,s]$ we have
\be
\ip<\nabla u,\nabla v>_{(M,g)} = \ip<-\Lap_g u, v>_{(M,g)}.
\ee
For vector fields $X\in H^\sigma$ and $Y\in H^{2-\sigma}$ with $\sigma\in[2-s,s]$
we have 
\be
\ip<\ck X,\ck Y>_{(M,g)} = \ip<-\vl X, Y>_{(M,g)}.
\ee
\NoProof

Finally, we show that maximum principles hold for rough metrics.
These are used most importantly to ensure that the conformal
factors we construct are positive. We say that a distribution
$V\in H^{\sigma}$ with $\sigma\in [-s,s]$ satisfies $V\ge 0$ if 
$$
\ip<V,\phi>_{(M,g)}\ge 0
$$
for every smooth function $\phi\ge 0$.

\lem\label{lem:maxpr} Suppose $(M^n,g)$ is a compact Riemannian manifold 
with $g\in H^s$ and $s>n/2$.  Suppose also that $V\in H^{s-2}$ and
that $V\ge 0$ with $V\not\equiv 0$. If  $u\in H^s$ satisfies
\bel{maxprsys}
-\Delta u +V u  \le 0,
\eel
then $u\le 0$.  
\Proof
Let $v=u^{(+)}$. One readily finds from Lemma 
\ref{lem:mult}, since $u^{(+)}\in H^1$, that $uv$ belongs to $H^1$.
Since $s-2 \ge -1$ and since $V\in H^{s-2}$, we can apply $V$ to $uu^{(+)}$.  
Since $V\ge0 $, and since $uv\ge 0$, we have $\ip <V,uv>_{(M,g)} \ge 0$. 
Finally, since $u$ satisfies \eqref{maxprsys} we obtain
\be
\ip<-\Delta u,v>_{(M,g)} \le - \ip< V, uv >_{(M,g)} \le 0.
\ee
Integrating by parts and using the fact that 
$$
\nabla v = 
%\cases \nabla u &\text{if } u>0\\
%                   0   &\text{if } u\le 0\\
%\endcases
\cases x & y = 1 \\
\endcases
$$
we conclude $v$ is constant.  To show
that $v\equiv0$ we see that if not, then $u^{(+)}=v$ is a positive
constant and hence $u$ is a positive constant.  Since
$-\Delta u +V u = Vu \ge 0$, and  since $-\Delta u +V u \le 0$
we conclude $V\le0$ and $V\ge 0$.  But then $V\equiv0$, a contradiction.
\EOProof

A version of the following strong maximum principle was proved in
\cite{maxwell-rough} for asymptotically Euclidean metrics.  Since
the argument there was purely local, using only the regularity of the
metric and the connectivity of the manifold, the same proof yields 
a strong maximum principle for compact manifolds.

\prop\label{prop:strmaxpr} Suppose $(M^n,g)$ is a compact Riemannian manifold
with $g\in H^s$ and $s>n/2$.  Suppose that $u\in H^s(M)$ is nonnegative 
and that $V\in H^{s-2}(M)$ satisfies
\be
-\Lap_g u +V u \ge 0.
\ee
If $u(x)=0$ at some point $x\in M$, then $u$ vanishes identically.
\NoProof

\section{The Yamabe Invariant of a Rough Metric}
\seclabel{sec:conformal}
The Yamabe invariant for smooth compact manifolds, defined by
\be
\lambda_g = \inf_{f\in C^{\infty}(M)\atop f\not\equiv0} 
{\int_M a\abs{\nabla f}^2_g + R_g f^2\dV_g\over ||f||_{L^{2^*}}^2},
\ee
where $2^*=2n/(n-2)$, plays an essential role in
classifying CMC solutions of the constraint equations on compact manifolds.
To understand properties of the Yamabe invariant for rough metrics, we
consider the quadratic bilinear form
$$
A(f,h) = \int_M a\ip<\nabla f,\nabla h> + R_g f h\dV_g.
$$  
When $g$ is smooth then $A$ is obviously continuous on $H^1\times H^1$.  This
is also true when $g$ is rough, so long as we interpret 
$\int_M R_g f h\dV_g$ as $\ip<R_g,fh>_{(M,g)}$. To see this, we note that
the term $\int_M a\ip<\nabla f,\nabla h> \dV_g$ poses no difficulties and
we need only consider the scalar curvature term. From Lemmas 
\ref{lem:mult} and \ref{lem:ibp} we have
\bel{scalar-term-strong}
\abs{\ip<R_g,fh>_{(M,g)}} \lesssim ||R_g||_{H^{s-2}}||f||_{H^{1}}||h||_{H^{1}}
\eel
since $1 + 1 -{n\over 2} > 2-s$. This establishes the continuity of $A$ on
$H^1$.  In fact, we can say more than this.  Pick $\eta$ such that
$0<\eta<1$ and such that $2\eta < s-{n\over 2}$.  Then
$(1-\eta)+(1-\eta) -{n\over 2}> 2 - s$ as well, and we 
conclude from Lemma \ref{lem:mult}
that
\bel{scalar-term-weak}
\abs{\ip<R_g,fh>_{(M,g)}} \lesssim ||R_g||_{H^{s-2}}||f||_{H^{1-\eta}}
||h||_{H^{1-\eta}}.
\eel
Hence the scalar curvature term is continuous on $H^{1-\eta}\times H^{1-\eta}$.

We would like to show that $\lambda_g\neq-\infty$, which we do by
considering the first eigenvalue of the conformal Laplacian.
Let 
$$
J_p(f)= {A(f,f)\over ||f||_{L^p}^2},
$$ 
where $||f||_{L^p}$ is the metric dependent norm defined by
$$
||f||_{L^p}^p= \int_M \abs{f}^p\dV_g.
$$
We have 
$$
\lambda_g = \inf_{f\in C^{\infty}(M)\atop f\not\equiv0} J_{2^*}(f)
$$
and the first eigenvalue of the conformal Laplacian is defined by
$$
\mu_h = \inf_{f\in C^{\infty}(M)\atop f\not\equiv0} J_2(f).
$$
If $\lambda_g<0$ then it is easy to see that $\mu_g\le \lambda_g$
and therefore $\lambda_g\neq-\infty$ if $\mu_g\neq-\infty$.
Our estimate for a lower bound for $\mu_g$ starts with the following 
estimate for $A$.

\lem\label{lem:Aest} Let $(M^n,g)$ be a compact Riemannian manifold
with $g\in H^s$ and $s>n/2$.  Then there are positive constants $C_1$ 
and $C_2$ such that for every $f\in H^1$
$$
A(f,f) \ge C_1 ||f||_{H^1}^2 - C_2 ||f||_{L^2}^2
$$
\Proof
Pick $\eta$ as was done in deriving inequality \eqref{scalar-term-weak}.
From interpolation we have for any $\epsilon>0$ that 
$$
||f||_{H^{1-\eta}}\le C(\eta,\epsilon)||f||_{L^2} + \epsilon||f||_{H^1}.
$$
From inequality \eqref{scalar-term-weak} we then have
\bel{mubound1}
\abs{\ip<R_g,f^2>_{(M,g)}} \le C(\eta, \epsilon,||R_g||_{H^{s-2}})||f||_{L^2}^2
+\epsilon^2||f||_{H^1}^2.
\eel
We also note that 
\bel{mubound2}
||f||_{H^1}^2\lesssim \int_M a\abs{\nabla f}^2_g \dV_g + ||f||_{L^2}^2.
\eel
Since
$$
A(f,f) \ge \int_M a\abs{\nabla f}^2_g \dV_g - \abs{\ip<R_g,f^2>_{(M,g)}}
$$
we obtain from \eqref{mubound1} and \eqref{mubound2}, taking $\epsilon$ 
sufficiently small, that
$$
A(f,f) \gtrsim ||f||_{H^1}^2-
\left(C(\eta,\epsilon,||R_g||_{H^{s-2}})+1\right)||f||_{L^2}^2. 
$$
\EOProof

It follows immediately that $\lambda_g$ and $\mu_g$ are both finite.

\lem\label{lem:finite-inf} Let $(M^n,g)$ be a compact Riemannian manifold
with $g\in H^s$ and $s>n/2$.  Then $\lambda_g$ and $\mu_g$ are both finite.
\Proof
Since $||f||_{L^2}^2\lesssim ||f||_{L^{2^*}}^2$, it follows that if 
$\lambda_g=-\infty$ then $\mu_g=-\infty$. So it is enough to show 
that $\mu_g$ is finite.  From Lemma \ref{lem:Aest}  we have
$$
\int_M a\abs{\nabla f}^2_g +R_gf^2\dV_g \ge C_1||f||_{H^1}^2-C_2 ||f||_{L^2}^2 
\ge -C_2 ||f||_{L^2}^2.
$$
Dividing by $||f||_{L^2}^2$ we conclude that
$$
\mu_g \ge -C_2.
$$
\EOProof

\prop\label{prop:conformal-class} 
Let $(M^n,g)$ be a compact Riemannian manifold
with $g\in H^s$ and $s>n/2$.  Then the following are equivalent:
\bl
\li{1.} There exists a positive function $\phi\in H^s$ such that 
$\tilde g = \phi^{2\dimc} g$ satisfies $R_{\tilde g}$ is continuous and
positive (resp. zero, resp. negative).
\li{2.} $\lambda_g$ is positive (resp. zero, resp. negative).
\li{3.} $\mu_g$ is positive (resp. zero, resp. negative).
\el

\Proof
We start by proving Condition 3 implies Condition 1.  We claim that there 
is function
$f$ in $H^1$, not identically zero, such that $J_2(f)=\mu_g$.
To see this, let $\{f_k\}$ be a sequence of smooth functions such that 
$||f_k||_{H^1}=1$ and $J_2(f_k)$ converges to $\mu_g$. 
Since the sequence is bounded in $H^1$, we can assume without loss of 
generality that it converges weakly to some $f\in H^1$. 
From Lemma \ref{lem:Aest} we have
$$
J_2(f_k) \ge
{ C_1 ||f_k||_{H^1}^2-C_2 ||f_k||_{L^2}^2\over ||f_k||^2_{L^2}} =
{ C_1 \over ||f_k||_{L^2}^2}-C_2,
$$
Since $f_k\ra f$ strongly in $L^2$, and since the sequence $J_2(f_k)$ is 
bounded, we conclude that $f$ is not identically zero.  It is easy
to see that $J_2$ is weakly lower semicontinuous on $H^1\setminus\{0\}$ and
hence we conclude that $J_2(f)=\mu_g$.  

Since $f$ is a minimizer in $H^1$ of $J_2(f)$ it satisfies 
$$
-a\Lap_g f + R_g f=\mu_g f
$$ 
in $H^{-1}$.  
Proposition \ref{prop:lapmap} then implies 
$f\in H^s$. We can assume without loss of generality that
$f\ge0$ since $\abs{f}\in H^1$ and $J_2(f)=J_2(\abs{f})$.
So from the strong maximum principle
Proposition \ref{prop:strmaxpr} we conclude that $f>0$ everywhere.  
Letting $\phi=f$ and $\tilde g = \phi^{2\dimc} g$, it follows that 
$R_{\tilde g} = \phi^{-2\dimc-1}(-a\Lap_g \phi + R_g\phi) = \phi^{-2\dimc} \mu_g
$
Since $\phi$ is continuous and positive, we conclude that 
$R_{\tilde g}$ is continuous and everywhere has the same sign as $\mu_g$.

We next show Condition 1 implies Condition 2.  Since $\lambda_g$ is 
a conformal invariant,
we can assume without loss of generality that $R_g$ is continuous and has
constant sign.  If $R_g$ is negative, then $J_{2^*}(1)<0$ and hence 
$\lambda_g<0$. If $R_g\ge0$ then clearly $\lambda_g\ge 0$.  
But if $R_g=0$,then $J_{2^*}(1)=0$ so
$\lambda_g=0$.  If $R_g>0$, then
\be
J_{2^*}(f)\gtrsim ||f||_{H^1}^2 \gtrsim ||f||_{L^{2^*}}^2.
\ee
and hence $\lambda_g>0$.  On the other hand, if $R_g=0$,
then  $J_{2^*}(1)=0$ and hence $\lambda_g=0$.  


Finally, we show that Condition 2 implies Condition 3.  If $\lambda_g<0$, 
then $A(f,f)<0$
for some smooth function $f$, so $\mu_g<0$ also.  On the other hand,
if $\lambda_g \ge 0$, then $A(f,f)\ge 0$ for every smooth function $f$
and $\mu_g\ge0$ also. Since $||f||_{L^2}^2\lesssim ||f||_{L^6}^2$ we
conclude that $J_{2^*}(f)\lesssim J_{2}(f)$ for every smooth
non-vanishing function $f$. Hence if $\lambda_g>0$, then $\mu_g>0$.  On
the other hand, from previous implications we have shown that if
$\mu_g>0$, then $\lambda_g>0$. So if $\lambda_g=0$ then $\mu_g=0$,
which competes the proof.
\EOProof

We conclude by showing that the sign of $\lambda_g$ can be detected
if $(M,g)$ has scalar curvature with $R_g\ge 0$ or $R_g\le0$ even
when $R_g$ is not continuous.
\cor\label{cor:zero} Suppose $(M,g)$ is a compact Riemannian manifold 
with $g\in H^s$ and $s>n/2$.  If $R_g\le 0$ then $\lambda_g\le 0$ and 
$\lambda_g=0$ if and only if $R_g=0$.  If $R_g\ge 0$ then $\lambda_g\ge 0$ and
$\lambda_g=0$ if and only if $R_g=0$.
\Proof
Suppose $R_g\le 0$. Computing $J_{2^*}(1)$ we find $\lambda_g\le 0$.
If $\lambda_g=0$ then $\ip<R_g,c>_{(M,g)}=0$ for any constant $c$.  
Since $R_g\le0$, it follows that for any smooth function $f$
$$
0=\ip<R_g,\min_M f>_{(M,g)} \ge \ip<R_g,f>_{(M,g)} \ge \ip<R_g,\max_M f>_{(M,g)} 
= 0
$$ 
and from Lemma \ref{lem:duality} we conclude $R_g=0$.  

If $R_g\ge 0$ then it is clear that $\lambda_g\ge 0$. If
$R_g\ge 0$ and $\lambda_g=0$ then $\mu_g=0$ also.  Let
$\{f_n\}$ be a sequence of non-negative functions in $H^1$ such that
$J_2(f_n)\ra \mu_g=0$ and such that $||f_n||_{L^2}=1$.  Then we have
$||f_n||_{H^1}$ is bounded and some subsequence converges weakly in
$H^1$ to a function $f$ such that $||f||_{L^2}=1$.  From the weak 
lower semicontinuity of $J_2$ we have $J_2(f)=0$ and hence 
$\nabla f=0$ almost everywhere.  So $f$ is a constant and moreover
$\ip<R_g,f^2>=0$.  This implies $\ip<R_g,c>=0$ for any constant $c$,
and using the fact that $R_g\ge 0$ a similar argument as before 
implies $R_g=0$.

Finally, we note that if $R_g=0$, then Proposition \ref{prop:conformal-class} 
implies $\lambda_g=0$.  
\EOProof

\section{The Hamiltonian Constraint}
\seclabel{sec:ham}
In this section we determine conditions under which the Lichnerowicz equation 
\bel{lichh}
-a\Lap_g \phi + R_g\phi = \abs{\sigma}^2_g\phi^{-2\dimc-3}-b\tau^2\phi^{\dimc+1}
\eel
is solvable.
Although we only require solutions when $\tau$ is constant,
it might be useful for future considerations of non-CMC rough solutions
to treat here the case of a general function $\tau$.  Hence we seek 
necessary and sufficient properties of the conformal data $(M,g,\sigma,\tau)$ 
leading to the existence of a positive solution of \eqref{lichh},
including the case where $\tau$ is not constant.

A standard approach to finding
a solution to \eqref{lichh} is to first make a conformal change 
to a metric with amenable scalar curvature and then to seek constant sub- and 
super solutions $\phi_-$ and $\phi_+$  such that $0<\phi_-\le\phi_+$.
Having done this, a barrier argument proves that there exists a solution 
$\phi$ with $0<\phi_-\le \phi\le \phi_+$. Propositions \ref{prop:subsup} and 
\ref{prop:subsup-high}  of Section \ref{sec:subsup} show that 
this same barrier argument works for rough metrics.  So we concentrate
here of finding sub- and supersolutions. 

To illustrate the potential for difficulty in finding constant
sub and supersolutions for rough conformal data, 
consider the case 
$\lambda_g>0$, and let us suppose for simplicity that the scalar 
curvature $R_g\equiv 1$. We then seek constants $\phi_-\le \phi_+$ such that
\bea
\phi_+ &\ge \abs{\sigma}^2_g\phi_+^{-2\dimc-3}-b\tau\phi_+^{\dimc+1}\\
\phi_- &\le \abs{\sigma}^2_g\phi_-^{-2\dimc-3}-b\tau\phi_-^{\dimc+1}.
\eea
If the conformal data are smooth, $\tau$ is constant, and $\sigma$ is 
non-vanishing, then the 
existence of such  constants follows easily from the asymptotic behaviour 
of the functions
$y\mapsto y^{-2\dimc-3}$ and $y\mapsto -y^{\dimc+1}$.  But if $\sigma$
vanishes somewhere we cannot find a positive constant such that
$$
\phi_- < \abs{\sigma}^2_g\phi_-^{-2\dimc-3}-b\tau\phi_-^{\dimc+1}
$$
everywhere on $M$. Similarly, if $\sigma$ has singularities,
then we typically cannot find a constant supersolution either.  Since the
difficulties posed by zeros are similar to those posed by singularities, we
consider how the problem of zeros is treated for smooth conformal data.

Suppose that $(M,g)$ is smooth, $R_g\equiv 1$, $\tau\equiv 0$ and 
suppose $\sigma\not\equiv 0$ has zeros. 
A solution to the Lichnerowicz equation in 
this case was found in \cite{const-compact} by finding a non-constant 
subsolution. Specifically, $\phi_-$ can be taken to be the  solution of
$$
-a\Lap_g\phi_- = \abs{\sigma}^2_g A
$$
where $A=\max\{1,\max_M{\abs{\sigma}^2_g}\}$.  An argument using the 
monotonicity
of the function $y\mapsto y^{-2\dimc-3}$ and the maximum principle then 
shows that  $\phi_-$ is a subsolution.  Although this argument cannot be 
adapted directly to the situation where $\sigma$ might have singularities,
there is another way to understand the role that this subsolution plays.  
Suppose we ignore the constant $A$ and simply solve
$$
-a\Lap_g\phi_1+\phi_1 = \abs{\sigma}^2_g.
$$
By the maximum principle we have $\phi_1>0$, so we can use
$\phi_1$ as a conformal factor.  Setting $g_1=\phi_1^{2\dimc}g$ 
we find that the scalar curvature of of $g_1$ is 
\be
R_{g_1}=\phi_1^{-2\dimc-1}(-a\Lap_g\phi_1+\phi_1) = 
\phi_1^{-2\dimc-1} \abs{\sigma}^2_g =\phi_1^{2\dimc+3}\abs{\sigma_1}_{g_1}
\ee
where $\sigma_1=\phi_1^{-2}\sigma$. Solving the Lichnerowicz equation 
is then equivalent to solving
\be
-a\Lap_{g_1} \phi + R_{g_1}\phi = \abs{\sigma_1}_{g_1}\phi^{-2\dimc-3}.
\ee
Since the zeros of $\sigma_1$  are incorporated
into $R_{g_1}$, finding constant sub- and supersolutions is easy.  Moreover,
there is no obstacle to finding constant sub- and supersolutions 
even if $\sigma_1$ has singularities.

This motivates our approach for solving \eqref{lichh} when $\lambda_g\ge0$.
We start by making a conformal change
to a metric with scalar curvature encoding both the zeros and 
the singularities of the conformal data, and then we find constant
sub- and supersolutions.  The specific choice of conformal change
replaces specialized arguments used previously for subcases 
of $\lambda_g\ge0$ with a single unifying technique.

\prop\label{prop:yamabe-pos} Suppose $(M^n,g)$ is a compact Riemannian
manifold with $g\in H^s$, $s>n/2$ and $\lambda_g\ge0$.
Suppose also  that $\tau$ is a scalar function in $H^{s-1}$ and $\sigma$ 
is a transverse-traceless tensor in $H^{s-1}$.  Then there 
exists a positive solution $\phi\in H^s$ of the Lichnerowicz equation 
if and only if one of the following conditions holds:
\bl
\li{1.} $\sigma\not\equiv 0$ and $\tau\not\equiv 0$.
\li{2.} $\lambda_g>0$, $\sigma\not\equiv 0$ and $\tau\equiv 0$.
\li{3.} $\lambda_g=0$, $\sigma\equiv 0$ and $\tau\equiv 0$.

\el
\Proof
We wish to solve
\bel{lich-start}
-a\Lap_g\phi + R_g \phi = \abs{\sigma}^2_g\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1}.
\eel
From Proposition \ref{prop:conformal-class} there exists a conformal
factor $\phi_1\in H^s$ such that $g_1=\phi_1^{2\dimc} g$ has
continuous scalar curvature $R_{g_1}$ with the same sign as $\lambda_g$.  
If $\lambda_g=0$, $\sigma\equiv 0$ and $\tau\equiv 0$, then $\phi_1$
is a solution of \eqref{lich-start} and we are done.  So we can restrict
our attention to Cases 1 and 2 where $\sigma\not\equiv 0$.
Letting $\sigma_1=\phi_1^{-2}\sigma$, it follows that there exists
a solution of \eqref{lich-start} if and only if there is a solution
of
\bel{lich-mid}
-a\Lap_{g_1}\phi + R_{g_1} \phi = 
\abs{\sigma_1}_{g_1}^2\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1}.
\eel
From Proposition \ref{prop:lapmap}, since 
$\left(R_{g_1}+b\tau^2\right) \phi_2$ is non-negative and not
identically zero, there is a solution $\phi_2\in H^s$ of
\be
-a\Lap_{g_1}\phi_2 + \left(R_{g_1}+{2\over 3}\tau^2\right) \phi_2 = 
\abs{\sigma_1}_{g_1}^2
\ee
Since $\sigma_1\not\equiv 0$, we have moreover from the maximum principles
Lemma \ref{lem:maxpr} and Proposition \ref{prop:strmaxpr} that $\phi_2>0$. Let
$g_2=\phi_2^{2\dimc} g_1$ and $\sigma_2=\phi_2^{-2}\sigma_1$.  
As before, there is a solution of \eqref{lich-start} if and only if there
is a solution of
\bel{lich-end}
-a\Lap_{g_2}\phi + R_{g_2} \phi = 
\abs{\sigma_2}_{g_2}^2\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1}.
\eel
We are now in a setting where we can compute constant sub- and supersolutions.
Since
\bea
R_{g_2}&=  \left( \abs{\sigma_1}_{g_1}^2-b\tau^2\phi_2\right)\phi_2^{-2\dimc-1} \\
& = \abs{\sigma_2}_{g_2}^2\phi_2^{2\dimc+3}-b\tau^2\phi_2^{-2\dimc},
\eea
a constant $\phi_+$ is a supersolution of \eqref{lich-end} if
\be
\left[\abs{\sigma_2}_{g_2}^2\phi_2^{2\dimc+3} - b\tau^2\phi_2^{-2\dimc}\right]\phi_+
\ge \abs{\sigma_2}_{g_2}^2\phi_+^{-2\dimc-3}-b \tau^2\phi_+^{2\dimc+1}.
\ee
Pick $\phi_+$ such that $\phi_+^{2\dimc+4} \ge \max_{M} \phi_2^{-2\dimc-3}$ and 
such that $\phi_+^{2\dimc}\ge \max_{M} \phi_2^{-2\dimc}$.  This is possible since
$\phi_2\in H^s$ and is hence continuous. Then
\bea
\abs{\sigma_2}_{g_2}^2\phi_2^{2\dimc+3}\phi_+ &\ge \abs{\sigma_2}_{g_2}^2\phi_+^{-2\dimc-3}\\
-b\tau^2\phi_2^{-2\dimc}\phi_+ &\ge -b\tau^2\phi_+^{2\dimc+1}\\
\eea
and therefore $\phi_+$ is a supersolution.  Similarly, pick a positive constant
$\phi_-$ such
that $\phi_-^{2\dimc+4}\le\min_{M}\phi_2^{-2\dimc-3}$ and 
$0<\phi_-^{2\dimc}\le \min_M \phi_2^{-2\dimc}$.
Then $\phi_-$ is a subsolution and it is easy to see that $\phi_-\le\phi+$.
So from Propositions \ref{prop:subsup} and \ref{prop:subsup-high}
proved in Section \ref{sec:subsup}
there exists a positive solution $\phi\in H^s$ of \eqref{lich-end}.

To prove the converse direction, we need to show that if there is a solution
with $\lambda_g\ge 0$, and if at least one of $\tau\equiv 0$ or 
$\sigma\equiv 0$, then either $\lambda_g>0$ and $\sigma\not\equiv 0$ or 
$\lambda_g=0$ and $\sigma$ and $\tau$ both vanish identically.  This
is a consequence of Proposition \ref{prop:conformal-class}.  If a positive
solution $\phi$ of \eqref{lichh} exists, we can set $g' = \phi^{2\dimc} g$
and $\sigma'=\phi^{-2}\sigma$ to obtain
$$
R_{g'} = \abs{\sigma'}^2_{g'}-b\tau^2.
$$
If $\sigma$ and therefore $\sigma'\equiv 0$, then Corollary \ref{cor:zero} 
implies $\lambda_{g'}\le 0$. Since $\lambda_{g} \ge 0$ we have 
$\lambda_g=\lambda_{g'}=0$ and $\tau^2\equiv 0$. On the other hand, 
if $\tau^2\equiv0$ and $\sigma\not\equiv 0$, then 
Corollary \ref{cor:zero} implies $\lambda_g>0$.
\EOProof

Existence in the case $\lambda_g<0$ is more delicate.  Assuming
the scalar curvature of $g$ is negative and continuous, we
want to follow the approach of Proposition \ref{prop:yamabe-pos} 
and solve an equation such as
\bel{lich-start-neg}
-a\Lap_g\phi + R_g \phi +b\tau^2 \phi_1 = \abs{\sigma}^2_g
\eel
to make a conformal change to a metric with scalar curvature encoding
the zeros and singularities of $\tau$ and $\sigma$.  But $R_g< 0$ 
and we are not assured that we can solve this equation.  
On the other hand, if $\tau\not\equiv0$ we can solve
\be
-a\Lap_g\phi +b\tau^2 \phi_1 = \abs{\sigma}^2_g
\ee
as a substitute. Assuming that $\sigma\not\equiv 0$, then $\phi_1$ will be
positive and we can set $g_1=\phi_1^{2\dimc} g$. 
The resulting scalar curvature 
$$
R_{g_1} = \phi_1^{-2\dimc-1}\left(\abs{\sigma}_g^2+R_g\phi_1-b\tau^2\phi_1\right)
$$
is problematic. Since $R_{g}$ is strictly negative the term $R_g-b\tau^2$ does 
not vanish when $\tau^2$ does. This prevents us from finding
constant supersolutions.  On the other hand, if $R_g$ were a multiple of 
$-\tau^2$ then there would be no such difficulty.  We now show that
the existence of a conformal change to a metric with scalar curvature
proportional to $-\tau^2$ is the precise condition required to solve
\eqref{lichh} in the negative Yamabe case.

\prop\label{prop:yamabe-neg} Suppose $(M^n,g)$ is a compact Riemannian
manifold with $g\in H^s$, $s>n/2$ and $\lambda_g<0$.  Suppose also that
$\tau$ is a function in  $H^{s-1}$ and $\sigma$ is a transverse-traceless 
tensor in $H^{s-1}$.   Then there exists a positive solution $\phi\in H^s$ 
of the Lichnerowicz equation if and only if there is a positive function  
$\phi_1\in H^{s}$ such that $g_1 = \psi^{2\dimc}_1 g$ 
has scalar curvature $R_{\tilde g_1} = -b\tau^2$.
\Proof
In the case $\sigma\equiv0$ there is nothing to prove we see can
assume that $\sigma\not\equiv 0$.  First suppose that
there exists a conformal factor $\phi_1$ such that $g_1=\phi_1^{2\dimc} g$ 
satisfies $R_{g_1}=-b\tau^2$. Since $\lambda_g<0$ this implies in
particular that $\tau\not\equiv 0$.  Solving the Lichnerowicz equation
then reduces to solving
\be
-a\Lap_{g_1}\psi -b\tau^2\psi = \abs{\sigma_1}_{g_1}^2\psi^{-2\dimc-3}
-b\tau^2\psi^{2\dimc+1}
\ee
Let $\phi_2$ be the unique solution in $H^s$ of
\be
-a\Lap_{g_1}\phi_2 +b\tau^2\phi_2 = \abs{\sigma_1}_{g_1}^2.
\ee
Since $\tau^2\not\equiv 0$ and since $\sigma\not\equiv 0$, the
solution exists and is positive. Letting $g_2=\phi_2^{2\dimc}g_1$ we have
\bea
R_{g_2} &=\phi_2^{-2\dimc-1}(-2b\tau^2\phi_2+\abs{\sigma_1}_{g_1}^2)\\
&=-2b\tau^2 \phi_2^{-2\dimc}+\abs{\sigma_2}_{g_2}^2 \phi_2^{2\dimc+3}.
\eea
We can now solve 
\bel{yneg-last}
-a\Lap_{g_2}\psi + \abs{\sigma_2}_{g_2}^2 \phi_2^{7}\psi
-2b\tau^2\phi_2^{-2\dimc}\psi = \abs{\sigma_2}_{g_2}^2\psi^{-2\dimc-3}
-b\tau^2\psi^{2\dimc+1} .
\eel
by finding constant sub- and supersolutions.   

Let $\phi_+$ satisfy $\phi_+^{2\dimc+4} \ge 
\max_M \phi_2^{-2\dimc-3}$
and $\phi_+^{2\dimc} \ge \max_M 2\phi_2^{-2\dimc}$. Then, arguing as in Proposition
\ref{prop:yamabe-pos} we have $\phi_+$ is a supersolution.
Let $\phi_-$ satisfy $\phi_-^{2\dimc+4} \le \min_M \phi_2^{-2\dimc-3}$
and $\phi_-^{2\dimc} \le \min_M 2\phi_2^{-2\dimc}$.  Then $\phi_-$ is a subsolution
with $\phi_-\le\phi+$.  Hence by Proposition \ref{prop:subsup} or
Proposition \ref{prop:subsup-high} there 
exists a solution to \eqref{yneg-last}.  

For the converse direction, let us first make a conformal change to
a metric with continuous negative scalar curvature via a conformal
factor $\phi_1$.  Let $g_1=\phi_1^{2\dimc}g$ and for future reference set
$\sigma_1=\phi_1^{-2}\sigma$.
We wish to find a solution of the equation
\bel{const-neg}
-a\Lap_{g_1} \phi + R_{g_1} \phi = - b\tau^2\phi^{2\dimc+1}.
\eel
To do this we seek non-constant sub- and supersolutions.
Since there exists a positive solution of the Lichnerowicz equation,
there also exists a positive solution $\phi_+$ of
$$
-a\Lap_{g_1} \phi_+ + R_{g_1} \phi_+ = \abs{\sigma_1}_{g_1}\phi_+^{2\dimc+3}
-b\tau^2\phi_+^{2\dimc+1} \ge -b\tau^2\phi_+^{2\dimc+1} .
$$
So $\phi_+$ is a supersolution of \eqref{const-neg}.
To find a subsolution we consider the family of equations
$$
-a\Lap_{g_1}\phi_\epsilon - R_{g_1}\phi_\epsilon = -R_{g_1} - \epsilon \tau^2.
$$
Since $R_{g_1}<0$ there is a unique solution to this equation for each
$\epsilon$. When $\epsilon=0$ the solution is identically 1.
So taking $\epsilon$ sufficiently small, noting that convergence in
$H^s$ implies uniform convergence, we can ensure that 
$\phi_\epsilon>{1\over 2}$ everywhere.  Fix such an $\epsilon$.
We claim that $\phi_-=\eta\phi_\epsilon$ if $\eta$ is chosen sufficiently
small.
Indeed, pick $\eta$ such that $\eta\phi_\epsilon < \phi_+$ and
such that $\eta^{2\dimc}\le {\epsilon\over b}\min_M \phi_\epsilon^{-2\dimc-1}$.
Then
\bea
-a\Lap_{g_1}\phi_- + R_{g_1}\phi_- &= \eta( R_{g_1}(2\phi_\epsilon-1)-\epsilon\tau^2)\\
&\le -\eta \epsilon\tau^2\\
&\le -b\tau^2(\eta\phi_\epsilon)^{2\dimc+1}\\
&\le -b\tau^2\phi_-^{2\dimc+1}.
\eea
Hence $\phi_-$ is a subsolution with $0<\phi_- \le \phi_+$ and we conclude 
there exists a positive solution of \eqref{const-neg}.
\EOProof

The apparently new observation of Proposition \ref{prop:yamabe-neg} is that
the study of the Lichnerowicz equation in the Yamabe negative case 
completely reduces to the study of the prescribed scalar curvature problem.
In the smooth category there exists an integral condition
\cite{yamabe-neg-psc} we can use to detect if we can
make a conformal change to a metric with scalar curvature $-b\tau^2$.   
Suppose $g$ and $\tau$ are smooth, $\lambda_g<0$, $R_g$ is a negative 
constant, and $\Vol_g(M) =1$ (since we can always make a conformal
change to a metric with negative constant scalar curvature and $\Vol_g(M)=1$,
there is no restriction on $g$ here other than $\lambda_g<0$ and smoothness).
Then there exists a conformal change to a metric with scalar 
curvature $-b\tau^2$ if and only if
\bel{psc-neg}
\inf _{f\in {\cal A}} { \int_M a \abs{\nabla f}^2_g\;dV_g \over \int f^2\dV_g } 
> -R_g
\eel
where 
$$
{\cal A} = \{f\in H^1: \hbox{$f\ge 0$, $f\not\equiv 0$, and $\int_M \tau^2 f\dV_g =0$}\}.
$$
Since condition \eqref{psc-neg} is well defined even for rough metrics,
it is a good candidate for a necessary and sufficient condition even
in the rough category.  It remains to be seen, however, if this is true.
We prove here a weaker result that suffices for finding CMC solutions
of the constraint equations.

\prop\label{prop:bounded-below} Suppose $(M^n,g)$ is a compact Riemannian
manifold with $g\in H^s$, $s>n/2$ and $\lambda_g<0$. 
Suppose also that $\tau$ a function in $H^{s-1}$ such that $\tau\ge c$ 
for some positive constant $c$.  Then there exists a positive function
$\phi\in H^s$ such that $\tilde g =\phi^{2\dimc} g$ has scalar curvature 
$R_{\tilde g} = -b\tau^2$.
\Proof
We undertake a series of conformal changes designed to build a metric
with scalar curvature that is negative,  bounded away from zero, and 
singular wherever $\tau$ is.  We first make a conformal change through 
a function $\phi_1$ given
by Proposition \ref{prop:conformal-class} such that $g_1=\phi_1^{2\dimc}g$
has continuous negative scalar curvature $R_{g_1}$. Next, let
$v\in H^s$ be the solution of
$$
-a\Lap_{g_1} v -R_{g_1}v+\tau^2 v = -R_{g_1}
$$
which exists and is positive since $R_{g_1}<0$.
Setting $\phi_2=1+v$ we have
$$
-a\Lap_{g_1} \phi_2 +R_{g_1}\phi_2= (2R_{g_1}-\tau^2)v.
$$
Hence the scalar curvature of $g_2=\phi_2^{2\dimc}g_1$ is
$$
R_{g_2}=\phi_2^{-2\dimc-1}v(2 R_{g_1}-\tau^2).
$$
Since $R_{g_1}$, $v$, and $\phi_2$ are all continuous, and since $R_{g_1}$ is 
negative, we conclude there exist constants $c_1\ge c_2>0$ such that
\bel{scalar-bounds}
-c_1(1+\tau^2)\le R_{g_2} \le -c_2(1+\tau^2).
\eel
To prove the proposition it is enough to find a positive solution of
\bel{negpsc-last}
-a\Lap_{g_2}\phi+R_{g_2}\phi = -b\tau^2\phi^5
\eel
which we do by finding constant sub- and supersolutions.  Pick
constants $\phi_-$ and $\phi_+$ such that
\be
0<\phi_-^{2\dimc}<{c_2\over b}
\ee
and
\be
\phi_+^{2\dimc}>{c_1\over b}\left(1+{1\over \min(\tau^2)}\right).
\ee
From \eqref{scalar-bounds} it follows that $\phi_-$ is a subsolution and 
$\phi_+$ is a supersolution such that $\phi_-\le \phi_+$. Hence there exists a 
positive solution $\phi\in H^s$ of \eqref{negpsc-last}.
\EOProof

The uniqueness of any solution of the Lichnerowicz equation follows, as
usual, from the fact that the functions $y\mapsto y^{-2\dimc-3}$ and
$y\mapsto -y^{2\dimc}$ are monotone decreasing.
\prop\label{prop:lich-unique}  Suppose $(M^n,g)$ is a Riemannian manifold
with $g\in H^s$ and $s>n/2$.  
Suppose also that $\tau$ is a function and $\sigma$ is a transverse-traceless
tensor in $H^{s-1}$.  If $\phi_1$ and $\phi_2$ are solutions of \eqref{lichh},
then either
\bl
\li{1.} $\phi_1=\phi_2$, or
\li{2.} $\sigma\equiv0$, $\tau\equiv 0$, $\lambda_g=0$, and 
$\phi_1=c\phi_2$ for some positive constant $c$.

\el
\Proof
Let $g_1=\phi_1^{2\dimc}$, $\sigma_1=\phi_1^{-2}\sigma$, and $\phi=\phi_2/\phi_1$.
Then $\phi$ solves
$$
-a\Lap_{g_1} \phi + \abs{\sigma_1}^2_{g_1}\phi-b\tau^2\phi = 
\abs{\sigma_1}^2_{g_1}\phi^{-2\dimc-3}-b\tau^2\phi^{2\dimc+1}.
$$
Hence $\phi-1$ satisfies
\bel{unique1}
-a\Lap_{g_1} (\phi-1) + \left[\abs{\sigma_1}^2_{g_1}-b\tau^2\right](\phi-1) = 
\abs{\sigma_1}^2_{g_1}(\phi^{-2\dimc-3}-1)-b\tau^2(\phi^{2\dimc+1}-1).
\eel
Since $(\phi-1)^{(+)}\in H^1$, and since 
$-a\Lap_{g_1}(\phi-1)\in H^{s-2}\subset H^{-1}$, we can compute using
Lemma \ref{lem:ibp}
\bea
\ip<-a\Lap_{g_1}(\phi-1),(\phi-1)^{(+)}>_{(M,g_1)} 
&= a\ip<\nabla(\phi-1),\nabla(\phi-1)^{(+)}>_{(M,g_1)}\\
& = 
\int_{\phi> 1} a\ip<\nabla(\phi-1),\nabla(\phi-1)>_{g_1}\dV_g.
\eea
Hence
\beal{unique2}
&\int_{\phi>1} a\ip<\nabla(\phi-1),\nabla(\phi-1)>_{g_1} \dV_g = \\
&\qquad\qquad\qquad
\int_{\phi> 1}  
\abs{\sigma_1}^2_{g_1}(\phi^{-2\dimc-3}-\phi)(\phi-1)+b\tau^2(\phi-\phi^{2\dimc+1})(\phi-1)
\dV_g
\eeal
If $\phi>1$ then $(\phi^{-2\dimc-3}-\phi)(\phi-1)<0$ and 
$(\phi-\phi^{2\dimc+1})(\phi-1)<0$. So the integral on the right hand side
of \eqref{unique2} must be non-positive.  Since the integral on the 
left-hand side is non-negative, we conclude
\bean
\int_{\phi>1} a\ip<\nabla(\phi-1),\nabla(\phi-1)>_{g_1} \dV_{g_1} &= 0
\label{unique-plus1}\\
\int_{\phi> 1}  
\abs{\sigma_1}^2_{g_1}(\phi^{-2\dimc-3}-\phi)(\phi-1)+b\tau^2(\phi-\phi^{2\dimc+1})(\phi-1)\dV_{g_1}
&=0.
\label{unique-plus2}\\
\eean
A similar argument using $(\phi-1)^{(-)}$ as a test function shows
\bean
\int_{\phi<1} a\ip<\nabla(\phi-1),\nabla(\phi-1)>_{g_1} \dV_{g_1} &= 0
\label{unique-neg1}\\
\int_{\phi< 1}  
\abs{\sigma_1}^2_{g_1}(\phi^{-2\dimc-3}-\phi)(\phi-1)+b\tau^2(\phi-\phi^{2\dimc+1})(\phi-1)\dV_{g_1}
&=0.
\label{unique-neg2}
\eean
From the continuity of $\phi-1$ and equations \eqref{unique-plus1} and
\eqref{unique-neg1} we conclude $\phi$ is constant and therefore $\phi_1$ is a
constant multiple of $\phi_2$.  Suppose that $\phi_1\neq\phi_2$, so
$\phi\neq 1$. Then from equation \eqref{unique-plus2} or \eqref{unique-neg2} 
we obtain
\be
\int_M  
\abs{\sigma_1}^2_{g_1}(\phi^{-2\dimc-3}-\phi)(\phi-1)+b\tau^2(\phi-\phi^{2\dimc+1})(\phi-1)
\dV_{g_1} = 0
\ee
and therefore $\abs{\sigma_1}^2_{g_1}\equiv 0$, $\tau^2\equiv 0$ and 
$\lambda_g=0$.
\EOProof

We can now establish in the rough setting the necessary and 
sufficient conditions under which the Lichnerowicz equation is
solvable when $\tau$ is constant.
\thm\label{thm:lich}
Suppose $(M^n,g)$ is a compact Riemannian manifold with $g\in H^s$ and $s>n/2$.
Suppose also that $\tau$ is a constant and $\sigma$ is a transverse-traceless
tensor in $H^{s-1}$.  Then there exists a positive solution $\phi\in H^s$
of the Lichnerowicz equation \eqref{lichh} if and only if one of the 
following conditions holds:
\bl
\li{1.} $\sigma\not\equiv 0$ and $\tau\neq 0$.
\li{2.} $\lambda_g>0$, $\sigma\not\equiv 0$ and $\tau=0$.
\li{3.} $\lambda_g=0$, $\sigma\equiv 0$ and $\tau= 0$.
\li{4.} $\lambda_g<0$, $\sigma\equiv 0$ and $\tau\neq 0$.

\el
In Cases 1, 2, and 4 the solution is unique.  In Case 3
the solution is unique up to multiplication by a positive constant. 
\Proof
The proof of the proposition in the case $\lambda_g\ge 0$ follows
directly from Propositions \ref{prop:yamabe-pos} and \ref{prop:lich-unique}.
The existence of a solution if $\lambda_g<0$ and $\tau\neq 0$
follows from Propositions \ref{prop:yamabe-neg} and \ref{prop:bounded-below};
its uniqueness is a consequence of Proposition \ref{prop:lich-unique}.
So it only remains to show that if $\lambda_g<0$ and if there is a solution,
then $\tau\ne 0$.  But this is a consequence of Corollary \ref{cor:zero},
arguing as at the end of Proposition \ref{prop:yamabe-pos}.
\EOProof
\Remark
It is easy to extend these results to so-called scaled matter sources.
In the presence of scaled sources, the Lichnerowicz equation becomes
\bel{lich-matter}
-a\Lap_g \phi + R_g\phi = \abs{\sigma}^2_{g}\phi^{-2\dimc-3}+\rho\phi^{-1-\dimc}-b\tau\phi^{\dimc+1}
\eel
where $\rho\in H^{s-2}$ and $\rho\ge 0$.  The previous theorems carry over
to this case by replacing the condition $\sigma\equiv0$ (resp. 
$\sigma\not\equiv0$) with 
$\sigma\equiv0$ and $\rho\equiv 0$ 
(resp. $\sigma\not\equiv0$ or $\rho\not\equiv 0$).  For example, the
initial step of Proposition \ref{prop:yamabe-neg} when $\lambda_g\ge0$
and $\sigma\not \equiv 0$ or $\rho\not\equiv0$ is to solve
$$
-a\Lap_g \phi + b\tau^2\phi =  \abs{\sigma}^2_g+\rho
$$
rather than 
$$
-a\Lap_g \phi + b\tau^2\phi =  \abs{\sigma}^2_g.
$$
We have only used features of the term $\abs{\sigma}^2_g\phi^{-2\dimc-3}$
that are shared by the term $\rho\phi^{-\dimc\-1}$.
\EORemark

\section{The Momentum Constraint}

The treatment of the momentum constraint for rough metrics 
follows easily from the a priori estimates of 
Corollary \ref{cor:elliptic-ests} and standard arguments.  We have the 
following properties of the vector Laplacian $\vl=\div\ck$.

\prop\label{prop:vl}
Suppose $(M^n,g)$ is a compact Riemannian manifold with $g\in H^s$ and $s>n/2$.
Then $\vl:H^s\ra H^{s-2}$ is Fredholm of index 0.  The kernel of $\vl$
consists of conformal Killing fields in $H^s$ and the equation
$$
\vl X = Y
$$
with $Y\in H^{s-2}$ is solvable if and only if
$$
\ip<Y,Z>_{(M,g)}=0
$$
for every conformal Killing field $Z\in H^s(M)$.  
\Proof
The estimates of Corollary \ref{cor:elliptic-ests} imply that $\vl$ is 
semi-Fredholm. 
On the other hand, the vector Laplacian of a smooth metric is well known to 
have index 0.  Approximating $g$ with a sequence of smooth metrics, using the
fact that the index of a semi-Fredholm map is locally constant, it follows
that the index of $\vl$ is also 0.  If $\vl X =0$, then integration
by parts yields $0=-\ip<\vl X, X>_{(M,g)}=\ip<\ck X,\ck X>_{(M,g)}$ so $X$ is a 
conformal Killing field.  Finally, the conditions under which the 
equation $\vl X=Y$ is solvable follow from the fact that $\vl$ is 
Fredholm and self-adjoint.
\EOProof

The York decomposition of symmetric, trace-free, (0,2)-tensors is
a standard consequence of Proposition \ref{prop:vl}.

\thm\label{thm:splitting}  Suppose $(M^n,g)$ is a Riemannian manifold of 
with $g\in H^s$ and $s>n/2$.  
Let $\calS$ be the space of symmetric trace-free (0,2) tensors
in $H^{s-1}$.  Then
$$
\calS = \im \ck \oplus \ker \div
$$
where $\ck$ acts on $H^s$ vector fields and $\div$ acts on $\calS$.
\Proof
Suppose $\sigma\in\calS$.  If $X$ is a conformal Killing field,
then $\ip<\div\sigma,X>_{(M,g)}= -\ip<\sigma,\ck X>_{(M,g)}= 0$. So
from Proposition \ref{prop:vl} there exists a vector field $W\in H^s$ such
that $\vl W=\div\sigma$.  Since $\sigma =\ck W + (\sigma-\ck W)$,
and since $\div(\sigma-\ck W)=0$, we have proved that 
$\calS=\im\ck + \ker\,\div$.  To show that the sum is a direct sum,
it suffices to show that $\im\ck \cap \ker\div$  is trivial.
If $\sigma\in \im\ck \cap \ker\div $, then $\sigma=\ck W$ for some $W\in H^s$.
Since $\div\sigma=0$, we have $\vl W=0$ and hence $W$ is a conformal
Killing field.  So $\sigma=\ck W = 0$.
\EOProof

\section{Sub- and Supersolutions}
\seclabel{sec:subsup}
We now turn to the existence theorem for the semilinear problem
\beal{fund-sys-two}
-\Lap_g u & = F(x,u)\\
\eeal
where $F$ has the form
\be
F(x,y)=\sum_{j=1}^l F_j(x)G_j(y).
\ee
We start with the following properties of the linearized operator.

\prop\label{prop:lapmap}
Suppose $(M^n,g)$ is a compact Riemannian manifold with $g\in H^s$
and $s>n/2$. Suppose also that $V\in H^{s-2}$ and let $\calL=-\Lap_g +V$.
Then $\calL:H^\sigma\ra H^{\sigma-2}$ for any $\sigma\in[2-s,s]$ is
Fredholm with index 0 and we have the estimate
\bel{lap-est-ker}
||u||_{H^\sigma} \lesssim ||\calL u||_{H^{\sigma-2}} + ||u||_{H^{2-s}}.
\eel
The kernel of $\calL$ is a finite dimensional subspace of $H^s$.  If
$V\ge 0$ and $V\not\equiv 0$, then the kernel of $\calL$ is trivial
and we have the estimate
\bel{lap-est}
||u||_{H^\sigma} \lesssim ||\calL u||_{H^{\sigma-2}}.
\eel
If $f\in H^{\sigma-2}$, the equation
$$
\calL u = f
$$
admits a solution if and only if $\ip<f,v>_{(M,g)}=v$ for every $v\in\ker \calL$.
Finally, if $u\in H^\sigma$ for some $\sigma\in [2-s,s]$, and if
$\calL u \in H^{\tau-2}$ for some $\tau\in[\sigma,s]$, then
$u\in H^\tau$.
\Proof
First consider the Laplacian $\Lap_g$.  The estimates of 
Corollary \ref{cor:elliptic-ests} imply that $\Lap_g$ is semi-Fredholm
acting on $H^\sigma$ with $\sigma\in(2-s,s]$.  
On the other hand, the Laplacian of a smooth metric is well known to have
index 0.  Approximating $g$ with a sequence of smooth metrics, using the
fact that the index of a semi-Fredholm map is locally constant, it follows
that the index $\Lap_g$ is also 0. With the help of Lemma \ref{lem:mult}
it is easy to see that $\calL$ is a compact perturbation of $-\Lap_g$
and we conclude $\calL$ acting
on $H^\sigma$ with $\sigma\in (2-s,s]$ is is also Fredholm with index 0.
Since $\calL$ is self-adjoint, it follows that $\calL$ is in fact 
Fredholm acting on $H^\sigma$ with $\sigma\in [2-s,s]$.  The
estimate \eqref{lap-est-ker} is trivial if $\sigma=2-s$ and follows
from Corollary \ref{cor:elliptic-ests} for $\sigma\in(2-s]$. Since the 
index of $\calL$ is 0, and since $\calL$ is self adjoint, the dimension 
of the kernel of $\calL$ acting 
on $H^s$ is the  same as the dimension of the kernel of $\calL$ acting 
on $H^{2-s}$.  So these kernels coincide, which shows that the kernel of
$\calL$ acting on $H^\sigma$ for any $\sigma\in [2-s,s]$ is contained in
$H^s$; the kernel is finite dimensional since $\calL$ is Fredholm.  
If $V\ge0$ and $V\not\equiv 0$, then the weak maximum principle
Lemma \ref{lem:maxpr} implies the kernel of $\calL$ is trivial and
estimate \eqref{lap-est} follows since $\calL$ is invertible.
The conditions under which the equation
$$
\calL u =f
$$
is solvable now follow from standard Fredholm theory.  Finally, suppose 
$u\in H^\sigma$ with $\sigma\in[2-s,s]$ and suppose 
$$
\calL u = f
$$
with $f\in H^{\tau-2}$ where $\sigma\le \tau \le s$.  We want to show
that $u\in H^\tau$.  To see this, we note that since 
$\ip<f,v>_{(M,g)}=\ip<\calL u,v>_{(M,g)}=0$ for every $v\in\ker\calL$ we can 
solve
$$
\calL \hat  u = f
$$
for some  $\hat u\in H^\tau$ and therefore
$u-\hat u\in \ker \calL \subset H^s$.  Hence $u\in H^\tau$.
\EOProof

The existence theorem for the equation 
\bel{fund-sys-three}
-\Lap_g u  =\sum_{j=1}^l F_j(x)G_j(u).
\eel
is a technical refinement of the constructive approach used in many 
previous papers treating the constraint equations
\cite{cb-low-reg}\cite{const-asympt-flat}\cite{const-compact}.   
In \cite{maxwell-rough} the technique was
extended to asymptotically Euclidean manifolds $(M^n,g)$ of 
class $H^s$ with $s>n/2$.  That proof divided into two cases: 3-manifolds and
higher dimensional manifolds.  We follow the same approach here and
start with a result for the hard case of 3-manifolds
that treats a more general class of coefficient functions 
$F_j$ than those considered in \cite{maxwell-rough}.

\prop\label{prop:subsup} Suppose
\bl
\li{1.} $(M^3,g)$ is a compact Riemannian manifold with $g\in H^s$
and $s>3/2$,
\li{2.} $u_-,\;u_+\in H^s$ are a subsolution and a supersolution 
respectively of \eqref{fund-sys-two} such that $u_-\le u_+$,
\li{3.} each $F_j\in H^{s-2}(M)$ and is non-negative,
\li{4.} each $G_j$ is smooth on $I=[\inf(u_-),\sup(u_+)]$.  

\el
Then there exists a solution $u\in H^s(M)$ of \eqref{fund-sys-two} such 
that $u_-\le u \le u_+$.
\Proof 
\def\LV{\mathop{L_V}}
\def\FV{F_V}
We first assume $3/2 < s \le 2$. Let
\bea
V(x)&=1+\sum_{j=1}^l F_j(x) \abs{\min_I G_j'}\\
\eea
so that $V\in H^{s-2}(M)$ is nonnegative and not identically
zero.  Let
$\FV(x,y)=F(x,y)+V(x)y$ and so that $\FV$ is non-decreasing in $y$.  
By this we mean that if $u,v\in H^s$
satisfy $u\ge v$, then $\FV(x,u)-\FV(x,v)\ge 0$ as a distribution.
Finally, let $\LV=-\Lap_g+V$ so that $\LV:H^s\ra H^{s-2}$ is an isomorphism.

We construct a sequence of functions as follows.  Let $u_0=u_+$,
and for $i\ge 1$ let $u_i$ be the solution of
\be
\LV u_{i+1} = \FV(x,u_i).
\ee
Now 
\be
\LV (u_{1}-u_0) \le \FV(x,u_0)- F(x,u_0) - V(x)u_0 = 0 
\ee
since $u_0$ is a super-solution.  Moreover,
\be
\LV (u_{1}-u_-) \ge \FV(x,u_+)- FV(x,u_-) \ge 0
\ee
since $u_+\ge u_-$.  From Lemma \ref{lem:maxpr}
we conclude $u_0\ge u_1 \ge u_-$.  Now suppose 
$u_0 \ge u_1 \ge \cdots \ge u_i \ge u_-.$
Then 
\be
\LV (u_{i+1}-u_i) = \FV(x,u_{i})- F(x,u_{i-1}) \le 0.
\ee
Hence $u_{i+1}\le u_i$.  Also,
\be
\LV (u_{i+1}-u_-) = \FV(x,u_{i})- F(x,u_{-}) \ge 0.
\ee
So $u_{i}\ge u_{i+1} \ge u_{-}$.  We obtain by induction for the
entire sequence
$u_+=u_0\ge u_1 \ge u_2 \ge \cdots \ge u_-$.

We claim the sequence $\{u_i\}_{i=1}^\infty$ is bounded in $H^s(M)$.
From Proposition \ref{prop:lapmap} we can estimate
\bel{subsupe1}
||u_{i+1}||_{H^s(M)}
\lesssim ||\FV(x,u_{i})||_{H^{s-2}(M)}.
\eel
Now $\FV(x,u_{i})$ is a sum of terms of the form
\be
F(x) G(u_{i})
\ee
where $F\in H^{s-2}(M)$.
Fix $s'\in (3/2,s)$. From Lemma \ref{lemma:one-deriv-mult} proved below we have
\be
||F(x) G(u_{i})||_{H^{s-2}} \lesssim ||F||_{H^{s-2}}
\left[ ||G(u_{i})||_{L^\infty} + 
||G'(u_{i})||_{L^\infty}||u_i||_{H^{s'}}\right]. 
\ee
Since $u_-\le u_i \le u_+$, we have uniform estimates for each of the
terms $||G(u_{i})||_{L^\infty}$ and $||G'(u_{i})||_{L^\infty}$. Hence
\bel{subsupe2}
||\FV(x,u_{i})||_{H^{s-2}(M)} \lesssim 1 + ||u||_{H^{s'}(M)}.
\eel
Now from interpolation we know that for any $\epsilon > 0$ there is a 
constant $C(\epsilon)$ such that
\bel{subsupe4}
||u_i||_{H^{s'}}\le C(\epsilon) ||u_i||_{L^2} + \epsilon ||u_i||_{H^s}.
\eel
Again, since $u_-\le u_i \le u_+$, we have uniform estimates for
$||u_i||_{L^2}$.  Combining \eqref{subsupe2} and \eqref{subsupe4},
taking $\epsilon$ sufficiently small, we obtain
\be
||u_{i+1}||_{H^s(M)} \le {1\over 2} ||u_{i}||_{H^{s}(M)}+C
\ee
for some constant $C$ independent of $i$.
Iterating this inequality we obtain a bound for all $i$
\be
||u_{i}||_{H^{s}} \le ||u_{+}||_{H^{s}}+ 2C.
\ee
Hence some subsequence of $\{u_i\}_{i=1}^\infty$ (and by monotonicity, the 
whole sequence) converges weakly in $H^s(M)$ to a limit $u_\infty$. 

It remains to see $u_\infty$ is a solution of \eqref{fund-sys-two}.
Now $u_i$ converges strongly to $u_\infty$ in $H^{s'}$ for any 
$s'<s$ and also converges uniformly on compact sets.
Hence for any $\phi\in C^\infty(M)$,
\bea
\int_M\left(\FV(x,u_i)-V(x)u_{i+1}\right)\phi\dV_g &\rightarrow 
\int_M F(x,u_\infty)\phi\dV_g\\
\int_M \ip<\nabla u_{i+1}, \nabla \phi> \phi\dV_g
&\rightarrow
\int_M \ip<\nabla u_\infty, \nabla \phi>\dV_g.
\eea
So
\be
\int_M \ip<\nabla u_\infty, \nabla \phi>\dV_g =
\int_M F(x,u_\infty)\phi\dV_g,
\ee
which shows that $u_\infty$ is a weak solution.  So 
$-\Lap_g u_\infty = F(x,u_\infty)$ in the sense of distributions.

To handle the case $s > 2$ we use a bootstrap.  First
suppose $4\ge s \ge 2$.  From the above we have a solution $u$ in 
$H^2$. Since $2 > n/2=3/2$ and since $2>s-2 \in [0,2]$, we know
from Lemma \ref{lem:nonlin} that $c(x)f(u)\in H^{s-2}$.  
Since $-\Lap_g u \in H^{s-2}$, 
Proposition \ref{prop:lapmap} implies $u\in H^s$. We now
obtain the result for all $s>3/2$ by iteration.
\EOProof

The following technical 
lemma completes the proof of Proposition \ref{prop:subsup}.

\lem\label{lemma:one-deriv-mult}
Suppose $(M^n,g)$ is a Riemannian manifold with $g\in H^s$ and $s>n/2$.  
Suppose also that $F$ is a smooth function, 
$\sigma\in[-1,1]$, and $s'\in(n/2,s)$.  
Then for every $u\in H^s$ and $v\in H^\sigma$  we have 
\be
||v f(u)||_{H^\sigma(M)}\lesssim ||v||_{H^\sigma}\left[
||f(u)||_{L^\infty(M)}+ ||f'(u)||_{L^\infty(M)}||u||_{H^{s'}(M)}\right].
\ee
\Proof
If $\sigma=1$, then
\bea
||v f(u)||_{H^1(M)} 
&\lesssim  ||vf(u)||_{L^2(M)}+ ||\nabla (vf(u)) ||_{L^2(M)}\\
&\lesssim  ||vf(u)||_{L^2(M)}+ 
||\nabla v f(u) ||_{L^2(M)}+
||v f'(u) \nabla u ||_{L^2(M)}\\
&\lesssim  ||v||_{H^1(M)}||f(u)||_{L^\infty(M)}+ 
||v f'(u) \nabla u ||_{L^2_(M)}.
\eea
To estimate the last term we note that since $v\in H^1$, it is also
in $L^p$ where
\be
{1\over p} = {1\over 2} - {1\over n}.
\ee
On the other hand, $\nabla u\in L^q$ where
\be
{1\over q} = {1\over 2} - {s' -1 \over n}.
\ee
Since
\bea
{1\over p} + {1\over q} &= {1\over 2} - {1\over n} +
{1\over 2} - {s' -1 \over n}\\
&=  {1\over 2} + \left( {1\over 2} - {s'\over n}\right)\\
&<  {1\over 2}
\eea
we conclude 
\bel{est-s-pos}
||v f'(u) \nabla u ||_{L^2(M)} \lesssim 
||v||_{H^1}||f'(u)||_{L^\infty(M)} ||u||_{H^{s'}}.
\eel
This proves the result in the case $s=1$.  

We prove the case $s=-1$ by duality.  If $w\in H^{1}(M)$,
then
\bea
\abs{\ip< v f(u), w>_{(M,g)}} &\lesssim 
||v||_{H^{-1}}||f(u)w||_{H^1(M)}\nonumber\\
& \lesssim ||v||_{H^{-1}}
\left[
||f(u)||_{L^\infty}(M)+ ||f'(u)||_{L^\infty}(M)||u||_{H^{s'}(M)}\right]
||w||_{H^1(M)}.
\eea
Hence
\be
||v f(u)||_{H^{-s}}\lesssim ||v||_{H^{-1}}
\left[
||f(u)||_{L^\infty}(M)+ ||f'(u)||_{L^\infty}(M)||u||_{H^{s'}(M)}\right].
\ee
We have therefore obtained the result for $s=-1$, and the result
for all $s\in [-1,1]$ now follows from interpolation.
\EOProof

The proof of Proposition \ref{prop:subsup} does not apply to manifolds
$M^n$ with $n>3$ since it depends, for example, on the fact that $H^2$
is an algebra.  From the point of view of the constraint equations, this
is not a serious drawback since the most interesting dimension is 
$n=3$.  Nevertheless, it would be nice to have an existence theorem in
higher dimensions. This can be done following the approach of 
\cite{maxwell-rough}.  The idea is to first obtain
existence in the space  $W^{k,p}$ where $k$ is the largest integer 
with $k<n$ and where ${1\over p} = {1\over 2} - {s-k\over n}$. 
The full regularity in $H^s$ then follows from an interpolation argument
and elliptic regularity. The proof from \cite{maxwell-rough} readily 
adapts to the compact setting using results and techniques 
from \cite{cb-low-reg} in place of \cite{maxwell-ahc} to obtain 
existence in $\W kp$.  The bootstrap argument then follows exactly as 
in \cite{maxwell-rough}; we omit the proof.

\prop\label{prop:subsup-high} Suppose
\bl
\li{1.} $(M^n,g)$ is a Riemannian manifold with $n>3$, $g\in H^s$ and 
$s>n/2$,
\li{2.} $u_-,\;u_+\in H^s$ are a 
subsolution and a supersolution 
respectively of \eqref{fund-sys-two} such that $u_-\le u_+$,
\li{3.} each $F_j\in H^{s-2}(M)$ and is non-negative,
\li{4.} each $G_j$ is smooth on $I=[\inf(u_-),\sup(u_+)]$.  

\el
Then there exists a solution $u\in H^s(M)$ of \eqref{fund-sys-two} such 
that $u_-\le u \le u_+$.
\NoProof

\Acknowledge
This work was sponsored in part by NSF grant DMS-0305048 and
the University of Washington Royalty Research Fund.
\EOAcknowledge
\bigskip

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\endmatter

rough-compact.bbl
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