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\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_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_-