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+\documentclass[ a4paper, oneside]{amsart}
+
+%\listfiles
+
+
+\RequirePackage{amsmath}
+\RequirePackage{bm}
+\RequirePackage{amssymb}
+\RequirePackage{upref}
+\RequirePackage{amsthm}
+\RequirePackage{enumerate}
+%\RequirePackage{pb-diagram}
+\RequirePackage{amsfonts}
+\RequirePackage[mathscr]{eucal}
+\RequirePackage{verbatim}
+\RequirePackage{xr}
+
+
+\def\@thm#1#2#3{%
+ \ifhmode\unskip\unskip\par\fi
+ \normalfont
+ \trivlist
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+ \let\thm@swap\@gobble
+ \let\thm@indent\indent % no indent
+ \thm@headfont{\scshape}% heading font bold
+ %\thm@notefont{\fontseries\mddefault\upshape}%
+ \thm@notefont{}%
+ \thm@headpunct{.}% add period after heading
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+ \thm@preskip\topsep
+ \thm@postskip\thm@preskip
+ #1% style overrides
+ \@topsep \thm@preskip % used by thm head
+ \@topsepadd \thm@postskip % used by \@endparenv
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+ \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
+ \else
+ \refstepcounter{#2}%
+ \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
+ \fi
+ \@tempa
+}
+
+
+
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+%Redefined commands
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+
+
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+
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+
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+\newskip\Cbigskipamount
+\Cbigskipamount=\bigskipamount
+
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+\spt=0.5pt
+
+
+%%special roster macro
+
+\newcommand\citem{\cvs\advance\itemno by
+1{(\romannumeral\the\itemno})\hskip3pt}
+\newcommand{\bitem}{\cvm\nd\advance\itemno by
+1{\bf\the\itemno}\hspace{\cma}}
+\newcommand\cendroster{\cvm\itemno=0}
+
+
+%New counts
+
+\newcount\itemno
+\itemno=0
+
+%Labels
+
+\newcommand{\las}[1]{\label{S:#1}}
+\newcommand{\lass}[1]{\label{SS:#1}}
+\newcommand{\lae}[1]{\label{E:#1}}
+\newcommand{\lat}[1]{\label{T:#1}}
+\newcommand{\lal}[1]{\label{L:#1}}
+\newcommand{\lad}[1]{\label{D:#1}}
+\newcommand{\lac}[1]{\label{C:#1}}
+\newcommand{\lan}[1]{\label{N:#1}}
+\newcommand{\lap}[1]{\label{P:#1}}
+\newcommand{\lar}[1]{\label{R:#1}}
+\newcommand{\laa}[1]{\label{A:#1}}
+
+%Referencing
+
+\newcommand{\rs}[1]{Section~\ref{S:#1}}
+\newcommand{\rss}[1]{Section~\ref{SS:#1}}
+\newcommand{\rt}[1]{Theorem~\ref{T:#1}}
+\newcommand{\rl}[1]{Lemma~\ref{L:#1}}
+\newcommand{\rd}[1]{Definition~\ref{D:#1}}
+\newcommand{\rc}[1]{Corollary~\ref{C:#1}}
+\newcommand{\rn}[1]{Number~\ref{N:#1}}
+\newcommand{\rp}[1]{Proposition~\ref{P:#1}}
+\newcommand{\rr}[1]{Remark~\ref{R:#1}}
+\newcommand{\raa}[1]{Axiom~\ref{A:#1}}
+\newcommand{\re}[1]{\eqref{E:#1}}
+
+
+%Index
+\newcommand{\ind}[1]{#1\index{#1}}
+
+
+
+
+
+
+\RequirePackage{upref}
+\RequirePackage{amsthm}
+%\usepackage{amsfonts}
+%\usepackage{amsintx}
+\RequirePackage{enumerate}%\begin{enumerate}[(i)]
+
+%%\usepackage{showkeys}
+\setlength{\textwidth}{4.7in}%JDG
+\setlength{\textheight}{7.5in}
+
+\usepackage{germanquotes}
+
+\theoremstyle{plain}
+\newtheorem{thm}{Theorem}[section]
+\newtheorem{lem}[thm]{Lemma}
+\newtheorem{prop}[thm]{Proposition}
+\newtheorem{cor}[thm]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{rem}[thm]{Remark}
+\newtheorem{definition}[thm]{Definition}
+\newtheorem{example}[thm]{Example}
+\newtheorem{ex}[thm]{Exercise}
+
+\swapnumbers
+\theoremstyle{remark}
+\newtheorem{case}{Case}
+
+\numberwithin{equation}{section}
+
+%\renewcommand{\qed}{q.e.d.}
+
+\usepackage{xr-hyper}
+\usepackage{url}
+\usepackage[hyperindex=true, pdfauthor= Claus\ Gerhardt, pdftitle= LM-Volume, bookmarks=true, extension= pdf, colorlinks=true, plainpages=false,hyperfootnotes=true, debug=false, pagebackref]{hyperref}
+
+\newcommand{\anl}{\htmladdnormallink}
+
+%\listfiles
+\begin{document}
+%\larger[1]
+\title{Estimates for the volume of a Lorentzian manifold}
+
+% author one information
+\author{Claus Gerhardt}
+\address{Ruprecht-Karls-Universit\"at, Institut f\"ur Angewandte Mathematik,
+Im Neuenheimer Feld 294, 69120 Heidelberg, Germany}
+%\curraddr{}
+\email{gerhardt@math.uni-heidelberg.de}
+\urladdr{\url{http://www.math.uni-heidelberg.de/studinfo/gerhardt/}}
+%\thanks{}
+
+% author two information
+%\author{}
+%\address{}
+%\curraddr{}
+%\email{}
+%\thanks{}
+%
+\subjclass[2000]{35J60, 53C21, 53C44, 53C50, 58J05}
+\keywords{Lorentzian manifold, volume estimates, cosmological spacetime, general relativity, constant mean curvature, CMC hypersurface}
+\date{April 18, 2002}
+%
+% at present the "communicated by" line appears only in ERA and PROC
+%\commby{}
+
+%\dedicatory{}
+
+\begin{abstract} We prove new estimates for the volume of a Lorentzian
+mani\-fold and show especially that cosmological spacetimes with crushing
+singularities have finite volume.
+\end{abstract}
+\maketitle
+\thispagestyle{empty}
+
+\setcounter{section}{-1}
+\section{Introduction}
+
+\cvb
+Let $N$ be a $(n+1)$-dimensional Lorentzian manifold and suppose that $N$ can be
+decomposed in the form
+
+\begin{equation}\lae{0.1}
+N=N_0\uu N_-\uu N_+,
+\end{equation}
+
+\cvm
+\nd where $N_0$ has finite volume and $N_-$ resp. $N_+$ represent the critical
+past resp. future Cauchy developments with not necessarily a priori bounded
+volume. We assume that $N_+$ is the future Cauchy development of a Cauchy
+hypersurface $M_1$, and $N_-$ the past Cauchy development of a hypersurface
+$M_2$, or, more precisely, we assume the existence of a time function $x^0$,
+such that
+
+\begin{equation}
+\begin{aligned}
+N_+&={x^0}^{-1}([t_1,T_+)),&\qq M_1=\{x^0=t_1\}&,\\
+N_-&={x^0}^{-1}((T_-,t_2]),&\qq M_2=\{x^0=t_2\}&,
+\end{aligned}
+\end{equation}
+
+\cvm
+\nd and that the Lorentz metric can be expressed as
+
+\begin{equation}\lae{0.3}
+d\bar s^2=e^{2\psi}\{-{dx^0}^2+\s_{ij}(x^0,x)dx^idx^j\},
+\end{equation}
+
+\cvm
+\nd where $x=(x^i)$ are local coordinates for the space-like hypersurface $M_1$
+if $N_+$ is considered resp. $M_2$ in case of $N_-$.
+
+The coordinate system $(x^\al)_{0\le\al\le n}$ is supposed to be future
+directed, i.e. the \tit{past} directed unit normal $(\nu^\al)$ of the level sets
+
+\begin{equation}
+M(t)=\{x^0=t\}
+\end{equation}
+
+\cvm
+\nd is of the form
+
+\begin{equation}\lae{0.5}
+(\nu^\al)=-e^{-\psi}(1,0,\ldots,0).
+\end{equation}
+
+\cvm
+If we assume the mean curvature of the slices $M(t)$ with respect to the past
+directed normal---cf. \ci[Section 2]{cg8} for a more detailed explanation of our
+conventions---is strictly bounded away from zero, then, the following volume
+estimates can be proved
+
+\bt\lat{0.1}
+Suppose there exists a positive constant $\e_0$ such that
+
+
+\begin{align}
+H(t)&\ge \e_0&\A\,t_1\le t< T_+&,\lae{0.6}\\
+\intertext{and}
+H(t)&\le-\e_0&\A\,T_-<t\le t_2&,\lae{0.7}
+\end{align}
+
+\cvm
+\nd then
+
+\begin{align}
+\abs{N_+}&\le \frac1{\e_0}\abs{M(t_1)},\\
+\intertext{and}
+\abs{N_-}&\le \frac1{\e_0}\abs{M(t_2}.
+\end{align}
+
+These estimates also hold locally, i.e. if $E_i\su M(t_i)$, $i=1,2$, are measurable
+subsets and $E_1^+,E_2^-$ the corresponding future resp. past directed
+cylinders, then,
+
+\begin{align}
+\abs{E_1^+}&\le\frac1{\e_0}\abs{E_1},\lae{0.10}\\
+\intertext{and}
+\abs{E_2^-}&\le\frac1{\e_0}\abs{E_2}.
+\end{align}
+\et
+
+\cvb
+\section{Proof of \rt{0.1}}\las{1}
+
+\cvb
+In the following we shall only prove the estimate for $N_+$, since the other case
+$N_-$ can easily be considered as a future development by reversing the time
+direction.
+
+\cvm
+Let $x=x(\xi)$ be an embedding of a space-like hypersurface and $(\nu^\al)$ be
+the past directed normal. Then, we have the Gau{\ss} formula
+
+\begin{equation}
+x^\al_{ij}=h_{ij}\nu^\al.
+\end{equation}
+
+\cvm
+\nd where $(h_{ij})$ is the second fundamental form, and the Weingarten equation
+
+\begin{equation}
+\nu^\al_i=h^k_ix^\al_k.
+\end{equation}
+
+
+\cvm
+We emphasize that covariant derivatives, indicated simply by indices, are
+always \tit{full} tensors.
+
+\cvm
+The slices $M(t)$ can be viewed as special embeddings of the form
+
+\begin{equation}
+x(t)=(t,x^i),
+\end{equation}
+
+\cvm
+\nd where $(x^i)$ are coordinates of the \tit{initial} slice $M(t_1)$. Hence, the
+slices $M(t)$ can be considered as the solution of the evolution problem
+
+\begin{equation}\lae{1.4}
+\dot x=-e^\psi \nu, \qq t_1\le t<T_+,
+\end{equation}
+
+\cvm
+\nd with initial hypersurface $M(t_1)$, in view of \re{0.5}.
+
+\cvm From the equation \re{1.4} we can immediately derive evolution equations
+for the geometric quantities $g_{ij}, h_{ij}, \nu$, and $H=g^{ij}h_{ij}$ of $M(t)$, cf.
+e.g.
+\ci[Section 4]{cg4}, where the corresponding evolution equations are derived in
+Riemannian space.
+
+\cvm
+For our purpose, we are only interested in the evolution equation for the metric,
+and we deduce
+
+\begin{equation}
+\dot g_{ij}=\spd{\dot x_i}{x_j}+\spd{x_i}{\dot x_j}=- 2e^\psi h_{ij},
+\end{equation}
+
+\cvm
+\nd in view of the Weingarten equation.
+
+\cvm
+Let $g=\det(g_{ij})$, then,
+
+\begin{equation}\lae{1.6}
+\dot g= g g^{ij}\dot g_{ij}=-2e^\psi H g,
+\end{equation}
+
+\cvm
+\nd and thus, the volume of $M(t), \abs{M(t)}$, evolves according to
+
+\begin{equation}\lae{1.7}
+\frac d{dt} \abs{M(t)}=\int_{M(t_1)}\frac d{dt}\sqrt g=-\int_{M(t)}e^\psi H,
+\end{equation}
+
+\cvm
+\nd where we shall assume without loss of generality that $\abs{M(t_1}$ is finite,
+otherwise, we replace $M(t_1)$ by an arbitrary measurable subset of $M(t_1)$
+with finite volume.
+
+\cvm
+Now, let $T\in [t_1, T_+)$ be arbitrary and denote by $Q(t_1,T)$ the
+cylinder
+
+\begin{equation}\lae{1.8}
+Q(t_1,T)=\set{(x^0,x)}{t_1\le x^0\le T},
+\end{equation}
+
+\cvm
+\nd then,
+
+\begin{equation}\lae{1.9}
+\abs{Q(t_1,T)}=\int_{t_1}^T\int_Me^\psi,
+\end{equation}
+
+\cvm
+\nd where we omit the volume elements, and where, $M=M(x^0)$.
+
+\cvm
+By assumption, the mean curvature $H$ of the slices is bounded from below by
+$\e_0$, and we conclude further, with the help of \re{1.7},
+
+\begin{equation}
+\begin{aligned}
+\abs{Q(t_1,T)}&\le\frac 1{\e_0} \int_{t_1}^T\int_Me^\psi H\\
+&=\frac1{\e_0}\{\abs{M(t_1)}-\abs{M(T)}\}\\
+&\le \frac1{\e_0}\abs{M(t_1)}.
+\end{aligned}
+\end{equation}
+
+
+\cvm
+Letting $T$ tend to $T_+$ gives the estimate for $\abs {N_+}$.
+
+\cvm
+To prove the estimate \re{0.10}, we simply replace $M(t_1)$ by $E_1$.
+
+\cvb
+If we relax the conditions \re{0.6} and \re{0.7} to include the case $\e_0=0$, a
+volume estimate is still possible.
+
+\cvm
+\bt
+If the assumptions of \rt{0.1} are valid with $\e_0=0$, and if in addition the
+length of any future directed curve starting from $M(t_1)$ is bounded by a
+constant $\ga_1$ and the length of any past directed curve starting from $M(t_2)$
+is bounded by a constant $\ga_2$, then,
+\begin{align}
+\abs{N_+}&\le \ga_1\abs{M(t_1)}\\
+\intertext{and}
+\abs{N_-}&\le \ga_2\abs{M(t_2)}.
+\end{align}
+\et
+
+\cvm
+\bp
+As before, we only consider the estimate for $N_+$.
+
+\cvm
+From \re{1.6} we infer that the volume element of the slices $M(t)$ is decreasing
+in $t$, and hence,
+\begin{equation}\lae{1.13}
+\sqrt{g(t)}\le \sqrt{g(t_1)}\qq\A\,t_1\le t.
+\end{equation}
+
+\cvm
+Furthermore, for fixed $x\in M(t_1)$ and $t>t_1$
+\begin{equation}\lae{1.14}
+\int_{t_1}^te^\psi\le \ga_1
+\end{equation}
+because the left-hand side is the length of the future directed curve
+\begin{equation}
+\ga(\tau)=(\tau,x)\qq t_1\le\tau\le t.
+\end{equation}
+
+\cvm
+Let us now look at the cylinder $Q(t_1,T)$ as in \re{1.8} and \re{1.9}. We have
+\begin{equation}
+\begin{aligned}
+\abs{Q(t_1,T)}&=\int_{t_1}^T\int_{M(t_1)}e^\psi\sqrt{g(t,x)}\le
+\int_{t_1}^T\int_{M(t_1)}e^\psi\sqrt{g(t_1,x)}\\[\cma]
+&\le \ga_1\int_{M(t_1)}\sqrt{g(t_1,x)}=\ga_1\abs{M(t_1)}
+\end{aligned}
+\end{equation}
+by applying Fubini's theorem and the estimates \re{1.13} and \re{1.14}.
+\ep
+
+\cvb
+\section{Cosmological spacetimes}\las{2}
+
+\cvb
+A cosmological spacetime is a globally hyperbolic Lorentzian manifold $N$ with
+compact Cauchy hypersurface $\so$, that satisfies the timelike convergence
+condition, i.e.
+
+\begin{equation}
+\bar R_{\al\bet}\nu^\al\nu^\bet\ge 0 \qq \A\,\spd\nu\nu=-1.
+\end{equation}
+
+\cvm
+If there exist crushing singularities, see \ci{es} or \ci{cg1} for a definition, then,
+we proved in
+\ci{cg1} that
+$N$ can be foliated by spacelike hypersurfaces $M(\tau)$ of constant mean
+curvature $\tau$, $-\un<\tau<\un$,
+
+\begin{equation}
+N=\uuu_{0\ne\tau\in \R[]}M(\tau)\uu{\msc C}_0,
+\end{equation}
+
+
+\cvm
+\nd where $\msc C_0$ consists either of a single maximal slice or of a whole
+continuum of maximal slices in which case the metric is stationary in $\msc
+C_0$. But in any case $\msc C_0$ is a compact subset of $N$.
+
+\cvm
+In the complement of $\msc C_0$ the mean curvature function $\tau$ is a regular
+function with non-vanishing gradient that can be used as a new time function, cf.
+\ci{cg6} for a simple proof.
+
+\cvm
+Thus, the Lorentz metric can be expressed in Gaussian coordinates $(x^\al)$ with
+$x^0=\tau$ as in \re{0.3}. We choose arbitrary $\tau_2<0<\tau_1$ and de\-fine
+
+\begin{equation}
+\begin{aligned}
+N_0&=\set{(\tau,x)}{\tau_2\le\tau \le \tau_1},\\
+N_-&=\set{(\tau,x)}{-\un<\tau \le \tau_2},\\
+N_+&=\set{(\tau,x)}{\tau_1\le \tau<\un}.
+\end{aligned}
+\end{equation}
+
+\cvm
+Then, $N_0$ is compact, and the volumes of $N_-, N_+$ can be estimated by
+
+\begin{align}
+\abs{N_+}&\le \frac1{\tau_1}\abs{M(\tau_1)},\\
+\intertext{and}
+\abs{N_-}&\le \frac1{\abs{\tau_2}}\abs{M(\tau_2)}.
+\end{align}
+
+\cvm
+Hence, we have proved
+
+\bt
+A cosmological spacetime $N$ with crushing singularities has finite volume.
+\et
+
+\cvb
+\br
+Let $N$ be a spacetime with compact Cauchy hypersurface and suppose that a
+subset
+$N_-\su N$ is foliated by constant mean curvature slices $M(\tau)$ such that
+
+\begin{equation}
+N_-=\uuu_{0<\tau\le \tau_2}M(\tau)
+\end{equation}
+
+\cvm
+\nd and suppose furthermore, that $x^0=\tau$ is a time function---which will be
+the case if the timelike convergence condition is satisfied---so that the metric
+can be represented in Gaussian coordinates $(x^\al)$ with $x^0=\tau$.
+
+\cvm
+Consider the cylinder $Q(\tau,\tau_2)=\{\tau\le x^0\le \tau_2\}$ for some
+fixed $\tau$. Then,
+
+\begin{equation}
+\abs{Q(\tau,\tau_2)}=\int_\tau^{\tau_2}\int_Me^\psi=\int_\tau
+^{\tau_2}H^{-1}\int_MH e^\psi,
+\end{equation}
+
+\cvm
+\nd and we obtain in view of \re{1.7}
+
+\begin{equation}
+\tau^{-1}_2\{\abs {M(\tau)}-\abs{M(\tau_2)}\}\le\abs{Q(\tau,\tau_2)},
+\end{equation}
+
+\cvm
+\nd and conclude further
+
+\begin{equation}
+\lim_{\tau\ra 0}\msp[2]\abs{M(\tau)}\le \tau_2\abs{N_-}+\abs{M(\tau_2)},
+\end{equation}
+
+\nd i.e.
+
+\begin{equation}
+\lim_{\tau\ra 0}\msp[2]\abs{M(\tau)}=\un\im \abs{N_-}=\un.
+\end{equation}
+\er
+
+\cvb
+\section{The Riemannian case}
+
+\cvb
+Suppose that $N$ is a Riemannian manifold that is decomposed as in \re{0.1} with
+metric
+
+
+\begin{equation}
+d\bar s^2=e^{2\psi}\{{dx^0}^2+\s_{ij}(x^0,x)dx^idx^j\}.
+\end{equation}
+
+\cvm
+The Gau{\ss} formula and the Weingarten equation for a hypersurface now have
+the form
+
+\begin{align}
+x^\al_{ij}&=-h_{ij}\nu^\al,\\
+\intertext{and}
+\nu^\al_i&=h^k_ix^\al_k.
+\end{align}
+
+
+\cvm
+As default normal vector---if such a choice is possible---we choose the outward
+normal, which, in case of the coordinate slices $M(t)=\{x^0=t\}$ is given by
+
+\begin{equation}
+(\nu^\al)=e^{-\psi}(1,0,\ldots,0).
+\end{equation}
+
+
+\cvm
+Thus, the coordinate slices are solutions of the evolution problem
+
+\begin{equation}
+\dot x=e^\psi \nu,
+\end{equation}
+
+\cvm
+\nd and, therefore,
+
+\begin{equation}
+\dot g_{ij}=2e^\psi h_{ij},
+\end{equation}
+
+\cvm
+\nd i.e. we have the opposite sign compared to the Lorentzian case leading to
+
+\begin{equation}
+\frac d{dt}\abs{M(t)}=\int_Me^\psi H.
+\end{equation}
+
+\cvm
+The arguments in \rs{1} now yield
+
+\bt
+\tup{(i)} Suppose there exists a positive constant $\e_0$ such that the mean
+curvature $H(t)$ of the slices $M(t)$ is estimated by
+
+\begin{align}
+H(t)&\ge \e_0&\A\,t_1\le t< T_+&,\\
+\intertext{and}
+H(t)&\le-\e_0&\A\,T_-<t\le t_2&,
+\end{align}
+
+\cvm
+\nd then
+
+\begin{align}
+\abs{N_+}&\le \frac1{\e_0}\lim_{t\ra T_+}\abs{M(t)},\\
+\intertext{and}
+\abs{N_-}&\le \frac1{\e_0}\lim_{t\ra T_-}\abs{M(t}.
+\end{align}
+
+\cvm
+\tup{(ii)} On the other hand, if the mean curvature $H$ is negative in $N_+$ and
+positive in $N_-$, then, we obtain the same estimates as \rt{0.1}, namely,
+
+\begin{align}
+\abs{N_+}&\le \frac1{\e_0}\abs{M(t_1)},\\
+\intertext{and}
+\abs{N_-}&\le \frac1{\e_0}\abs{M(t_2)}.
+\end{align}
+\et
+
+\cvb
+
+\begin{thebibliography}{99}
+\bib{es}
+D. Eardley \& L. Smarr, \emph{Time functions in numerical relativity: marginally
+bound dust collapse}, Phys. Rev. D \tbf{19} (1979) 2239\nbdd2259.
+
+
+\bib{cg1}
+C. Gerhardt, \emph{H-surfaces in Lorentzian manifolds}, Commun. Math. Phys.
+\tbf{89} (1983) 523\nbdd{553}.
+
+
+
+\bib{cg4}
+\bysame, \emph{Hypersurfaces of prescribed Weingarten curvature}, Math. Z.
+\tbf{224} (1997) 167\nbdd{194}.
+\url{http://www.math.uni-heidelberg.de/studinfo/gerhardt/MZ224,97.pdf}
+
+
+
+\bib{cg6}
+\bysame, \emph{On the foliation of space-time by constant mean curvature
+hypersurfaces}, preprint,
+\url{http://www.math.uni-heidelberg.de/studinfo/gerhardt/Foliation.pdf}
+
+
+\bib{cg8}
+\bysame, \emph{Hypersurfaces of
+prescribed curvature in Lorentzian manifolds}, Indiana Univ. Math. J. \tbf{49}
+(2000) 1125\nbdd1153.
+\url{http://www.math.uni-heidelberg.de/studinfo/gerhardt/GaussLorentz.pdf}]
+
+
+
+
+
+\bib{HE}
+S. W. Hawking \& G. F. R. Ellis, \emph{The large scale structure of space-time},
+Cambridge University Press, Cambridge, 1973.
+
+
+
+\end{thebibliography}
+\end{document}
+
+
+%------------------------------------------------------------------------------
+% End of journal.top
+%------------------------------------------------------------------------------