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authorPeter Breitenlohner <peb@mppmu.mpg.de>2009-07-09 11:52:21 +0000
committerPeter Breitenlohner <peb@mppmu.mpg.de>2009-07-09 11:52:21 +0000
commitc262054e6041bec309cd4ee58dc9c22cc1f40fae (patch)
tree5d55d17bc9d1027ce8afbbb9211d2cda2f9820e7 /Build
parentab471595e495323c12b4fcf31856f62269405a83 (diff)
synctex: more native line ends
git-svn-id: svn://tug.org/texlive/trunk@14199 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Build')
-rw-r--r--Build/source/texk/web2c/Makefile.in9
-rw-r--r--Build/source/texk/web2c/synctexdir/ChangeLog7
-rw-r--r--Build/source/texk/web2c/synctexdir/am/synctex.am10
-rw-r--r--Build/source/texk/web2c/synctexdir/synctex_main.c6
-rw-r--r--Build/source/texk/web2c/synctexdir/tests/LM-Volume-manuscript.tex1015
5 files changed, 1032 insertions, 15 deletions
diff --git a/Build/source/texk/web2c/Makefile.in b/Build/source/texk/web2c/Makefile.in
index 52f3b03c4c3..8631e2a5aa8 100644
--- a/Build/source/texk/web2c/Makefile.in
+++ b/Build/source/texk/web2c/Makefile.in
@@ -1227,8 +1227,8 @@ EXTRA_DIST = PROJECTS cftests config.h cpascal.h help.h tangle.web \
alephdir/aleph.version alephdir/eomega.ech alephdir/etex.ch \
alephdir/omega.ch0 alephdir/omega.ch1 alephdir/omega.ch2 \
alephdir/omxml.ch alephdir/aleph.mk synctexdir/ChangeLog \
- synctexdir/INSTALL synctexdir/Makefile synctexdir/Makefile.in \
- synctexdir/README.txt synctexdir/synctex_parser_readme.txt \
+ synctexdir/INSTALL synctexdir/README.txt \
+ synctexdir/synctex_parser_readme.txt \
synctexdir/synctex_parser_version.txt synctexdir/tests \
synctexdir/synctex-common.h synctexdir/synctex-convert.sh \
synctexdir/synctex.c synctexdir/synctex.defines \
@@ -1239,7 +1239,8 @@ EXTRA_DIST = PROJECTS cftests config.h cpascal.h help.h tangle.web \
synctexdir/synctex-e-mem.ch1 synctexdir/synctex-e-rec.ch0 \
synctexdir/synctex-e-rec.ch1 synctexdir/synctex-pdf-rec.ch2 \
synctexdir/synctex-xe-mem.ch2 synctexdir/synctex-xe-rec.ch2 \
- synctexdir/synctex-xe-rec.ch3 libmd5/md5.test fmtutil.in \
+ synctexdir/synctex-xe-rec.ch3 synctexdir/Makefile \
+ synctexdir/Makefile.in libmd5/md5.test fmtutil.in \
Makefile.in.orig configure.in.orig metapost.mk
DISTCLEANFILES = $(EXTRA_LIBRARIES) tangle.c tangle.h tangle.p \
tangle-web2c stamp-tangle tangleboot.c tangleboot.h ctangle.c \
@@ -2040,7 +2041,7 @@ synctex_DEPENDENCIES = $(ZLIB_DEPEND)
@XETEX_SYNCTEX_TRUE@ synctexdir/synctex-common.h \
@XETEX_SYNCTEX_TRUE@ synctexdir/synctex-xetex.h
-@XETEX_SYNCTEX_FALSE@synctex_convert_xetex = && echo "warning: SyncTeX for e-TeX is NOT enabled"
+@XETEX_SYNCTEX_FALSE@synctex_convert_xetex = && echo "warning: SyncTeX for XeTeX is NOT enabled"
@XETEX_SYNCTEX_TRUE@synctex_convert_xetex = && $(SHELL) $(srcdir)/synctexdir/synctex-convert.sh xetex
libmd5_a_CPPFLAGS = -I$(srcdir)/libmd5
libmd5_a_SOURCES = libmd5/md5.c libmd5/md5.h
diff --git a/Build/source/texk/web2c/synctexdir/ChangeLog b/Build/source/texk/web2c/synctexdir/ChangeLog
index 9741299e4cc..c31415b14ce 100644
--- a/Build/source/texk/web2c/synctexdir/ChangeLog
+++ b/Build/source/texk/web2c/synctexdir/ChangeLog
@@ -1,8 +1,13 @@
+2009-07-09 Peter Breitenlohner <peb@mppmu.mpg.de>
+
+ * synctex_main.c: make 'synctex help update' work as advertised.
+ * tests/LM-Volume-manuscript.tex: Mac -> native line ends.
+
2009-07-08 Peter Breitenlohner <peb@mppmu.mpg.de>
* am/synctex.am: distribute new files.
* synctex_parser_local.h: add newline at end of file.
- * synctex_parser_readme.txt: Mac -> native line end.
+ * synctex_parser_readme.txt: Mac -> native line ends.
2009-07-08 Jerome Laurens <jerome.laurens@u-bourgogne.fr>
diff --git a/Build/source/texk/web2c/synctexdir/am/synctex.am b/Build/source/texk/web2c/synctexdir/am/synctex.am
index d8fc6eb9e4c..3eab195d4fd 100644
--- a/Build/source/texk/web2c/synctexdir/am/synctex.am
+++ b/Build/source/texk/web2c/synctexdir/am/synctex.am
@@ -26,8 +26,6 @@ dist_man_MANS += synctexdir/man1/synctex.1 synctexdir/man5/synctex.5
EXTRA_DIST += \
synctexdir/ChangeLog \
synctexdir/INSTALL \
- synctexdir/Makefile \
- synctexdir/Makefile.in \
synctexdir/README.txt \
synctexdir/synctex_parser_readme.txt \
synctexdir/synctex_parser_version.txt \
@@ -197,7 +195,7 @@ synctex_convert_xetex = && $(SHELL) $(srcdir)/synctexdir/synctex-convert.sh xete
else !XETEX_SYNCTEX
-synctex_convert_xetex = && echo "warning: SyncTeX for e-TeX is NOT enabled"
+synctex_convert_xetex = && echo "warning: SyncTeX for XeTeX is NOT enabled"
endif !XETEX_SYNCTEX
@@ -209,3 +207,9 @@ EXTRA_DIST += \
synctexdir/synctex-xe-rec.ch2 \
synctexdir/synctex-xe-rec.ch3
+## Not used
+##
+EXTRA_DIST += \
+ synctexdir/Makefile \
+ synctexdir/Makefile.in
+
diff --git a/Build/source/texk/web2c/synctexdir/synctex_main.c b/Build/source/texk/web2c/synctexdir/synctex_main.c
index dbdb4272fb6..fdcece591a8 100644
--- a/Build/source/texk/web2c/synctexdir/synctex_main.c
+++ b/Build/source/texk/web2c/synctexdir/synctex_main.c
@@ -91,6 +91,7 @@ int main(int argc, char *argv[]);
void synctex_help(char * error,...);
void synctex_help_view(char * error,...);
void synctex_help_edit(char * error,...);
+void synctex_help_update(char * error,...);
int synctex_view(int argc, char *argv[]);
int synctex_edit(int argc, char *argv[]);
@@ -110,6 +111,9 @@ int main(int argc, char *argv[])
} else if(0==strcmp("edit",argv[arg_index])) {
synctex_help_edit(NULL);
return 0;
+ } else if(0==strcmp("update",argv[arg_index])) {
+ synctex_help_update(NULL);
+ return 0;
}
}
synctex_help(NULL);
@@ -752,7 +756,7 @@ void synctex_help_update(char * error,...) {
"\n"
"usage: synctex update -o output [-d directory] [-m number] [-x dimension] [-y dimension]\n"
"\n"
- "-o output is the full or relative path of an existing file,.\n"
+ "-o output is the full or relative path of an existing file,\n"
" either the real synctex file you wish to update\n"
" or a related file: foo.tex, foo.pdf, foo.dvi...\n"
"-d directory is the directory containing the synctex file, in case it is different from the directory of the output.\n"
diff --git a/Build/source/texk/web2c/synctexdir/tests/LM-Volume-manuscript.tex b/Build/source/texk/web2c/synctexdir/tests/LM-Volume-manuscript.tex
index ba4ddbd48fd..2853bd550e2 100644
--- a/Build/source/texk/web2c/synctexdir/tests/LM-Volume-manuscript.tex
+++ b/Build/source/texk/web2c/synctexdir/tests/LM-Volume-manuscript.tex
@@ -1,6 +1,1009 @@
-\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 \let\thmheadnl\relax \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 \thm@headsep 5\p@ plus\p@ minus\p@\relax \thm@preskip\topsep \thm@postskip\thm@preskip #1% style overrides \@topsep \thm@preskip % used by thm head \@topsepadd \thm@postskip % used by \@endparenv \def\@tempa{#2}\ifx\@empty\@tempa \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}% \else \refstepcounter{#2}% \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}% \fi \@tempa } %Redefined commands %Greek Letters \newcommand{\al}{\alpha} \newcommand{\bet}{\beta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta } \newcommand{\e}{\epsilon} \newcommand{\ve}{\varepsilon} \newcommand{\f}{\varphi} \newcommand{\h}{\eta} \newcommand{\io}{\iota} \newcommand{\tht}{\theta} \newcommand{\ka}{\kappa} \newcommand{\lam}{\lambda} \newcommand{\m}{\mu} \newcommand{\n}{\nu} \newcommand{\om}{\omega} \newcommand{\p}{\pi} \newcommand{\vt}{\vartheta} \newcommand{\vr}{\varrho} \newcommand{\s}{\sigma} \newcommand{\x}{\xi} \newcommand{\z}{\zeta} \newcommand{\C}{\varGamma} \newcommand{\D}{\varDelta} \newcommand{\F}{\varPhi} \newcommand{\Lam}{\varLambda} \newcommand{\Om}{\varOmega} \newcommand{\vPsi}{\varPsi} \newcommand{\Si}{\varSigma} %New Commands \newcommand{\di}[1]{#1\nobreakdash-\hspace{0pt}dimensional}%\di n \newcommand{\nbdd}{\nobreakdash--} \newcommand{\nbd}{\nobreakdash-\hspace{0pt}} \newcommand{\ce}[1]{$C^#1$\nbd{estimate}} \newcommand{\ces}[1]{$C^#1$\nbd{estimates}} \newcommand{\fm}[1]{F_{|_{M_#1}}} \newcommand{\fmo}[1]{F_{|_{#1}}}%\fmo M \newcommand{\fu}[3]{#1\hspace{0pt}_{|_{#2_#3}}} \newcommand{\fv}[2]{#1\hspace{0pt}_{|_{#2}}} \newcommand{\cchi}[1]{\chi\hspace{0pt}_{_{#1}}} \newcommand{\so}{{\mc S_0}} %\newcommand\sql[1][u]{\sqrt{1-|D#1|^2}} \newcommand{\const}{\tup{const}} \newcommand{\slim}[2]{\lim_{\substack{#1\ra #2\\#1\ne #2}}} \newcommand{\pih}{\frac{\pi}{2}} \newcommand{\msp[1]}[1]{\mspace{#1mu}} \newcommand{\low}[1]{{\hbox{}_{#1}}} %Special Symbols \newcommand{\R}[1][n+1]{{\protect\mathbb R}^{#1}} \newcommand{\Cc}{{\protect\mathbb C}} \newcommand{\K}{{\protect\mathbb K}} \newcommand{\N}{{\protect\mathbb N}} \newcommand{\Q}{{\protect\mathbb Q}} \newcommand{\Z}{{\protect\mathbb Z}} \newcommand{\eR}{\stackrel{\lower1ex \hbox{\rule{6.5pt}{0.5pt}}}{\msp[3]\R[]}} \newcommand{\eN}{\stackrel{\lower1ex \hbox{\rule{6.5pt}{0.5pt}}}{\msp[1]\N}} \newcommand{\eO}{\stackrel{\lower1ex \hbox{\rule{6pt}{0.5pt}}}{\msc O}} %Special math symbols \DeclareMathOperator{\arccot}{arccot} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\Grad}{Grad} \DeclareMathOperator*{\es}{ess\,sup} \DeclareMathOperator{\graph}{graph} \DeclareMathOperator{\sub}{sub} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\lc}{lc} \DeclareMathOperator{\osc}{osc} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\rec}{Re} \DeclareMathOperator{\imc}{Im} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Diff}{Diff} \DeclareMathOperator{\rg}{rg} \newcommand\im{\implies} \newcommand\ra{\rightarrow} \newcommand\xra{\xrightarrow} \newcommand\rra{\rightrightarrows} \newcommand\hra{\hookrightarrow} \newcommand{\nea}{\nearrow} \newcommand{\sea}{\searrow} \newcommand{\ua}{\uparrow} \newcommand{\da}{\downarrow} \newcommand{\rha}{\rightharpoondown} \newcommand{\wha}{\underset{w^*}\rightharpoondown} %PDE commands \newcommand\pa{\partial} \newcommand\pde[2]{\frac {\partial#1}{\partial#2}} \newcommand\pd[3]{\frac {\partial#1}{\partial#2^#3}} %e.g. \pd fxi \newcommand\pdc[3]{\frac {\partial#1}{\partial#2_#3}} %contravariant \newcommand\pdm[4]{\frac {\partial#1}{\partial#2_#3^#4}} %mixed \newcommand\pdd[4]{\frac {{\partial\hskip0.15em}^2#1}{\partial {#2^ #3}\,\partial{#2^#4}}} %e.g. \pdd fxij, Abl. zweiter Ordnung \newcommand\pddc[4]{\frac {{\partial\hskip0.15em}^2#1}{\partial {#2_ #3}\,\partial{#2_#4}}} \newcommand\PD[3]{\frac {{\partial\hskip0.15em}^2#1}{\partial #2\,\partial#3}} %e.g \PD fxy \newcommand\df[2]{\frac {d#1}{d#2}} \newcommand\sd{\vartriangle} \newcommand\sq[1][u]{\sqrt{1+|D#1|^2}} \newcommand\sql[1][u]{\sqrt{1-|D#1|^2}} \newcommand{\un}{\infty} \newcommand{\A}{\forall} \newcommand{\E}{\exists} %Set commands \newcommand{\set}[2]{\{\,#1\colon #2\,\}} \newcommand{\uu}{\cup} \newcommand{\ii}{\cap} \newcommand{\uuu}{\bigcup} \newcommand{\iii}{\bigcap} \newcommand{\uud}{ \stackrel{\lower 1ex \hbox {.}}{\uu}} \newcommand{\uuud}[1]{ \stackrel{\lower 1ex \hbox {.}}{\uuu_{#1}}} \newcommand\su{\subset} \newcommand\Su{\Subset} \newcommand\nsu{\nsubset} \newcommand\eS{\emptyset} \newcommand{\sminus}[1][28]{\raise 0.#1ex\hbox{$\scriptstyle\setminus$}} \newcommand{\cpl}{\complement} \newcommand\inn[1]{{\stackrel{\msp[9]\circ}{#1}}} %Embellishments \newcommand{\ol}{\overline} \newcommand{\pri}[1]{#1^\prime} \newcommand{\whn}[1]{\widehat{(#1_n)}} \newcommand{\wh}{\widehat} %Logical commands \newcommand{\wed}{\wedge} \newcommand{\eqv}{\Longleftrightarrow} \newcommand{\lla}{\Longleftarrow} \newcommand{\lra}{\Longrightarrow} \newcommand{\bv}{\bigvee} \newcommand{\bw}{\bigwedge} \newcommand{\nim}{{\hskip2.2ex\not\hskip-1.5ex\im}} \DeclareMathOperator*{\Au}{\A} \DeclareMathOperator*{\Eu}{\E} \newcommand\ti{\times } %Norms \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\absb}[1]{\Bigl|#1\Bigr|} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\normb}[1]{\Big\lVert#1\Big\rVert} \newcommand{\nnorm}[1]{| \mspace{-2mu} |\mspace{-2mu}|#1| \mspace{-2mu} |\mspace{-2mu}|} \newcommand{\spd}[2]{\protect\langle #1,#2\protect\rangle} %Geometry \newcommand\ch[3]{\varGamma_{#1#2}^#3} \newcommand\cha[3]{{\bar\varGamma}_{#1#2}^#3} \newcommand{\riem}[4]{R_{#1#2#3#4}} \newcommand{\riema}[4]{{\bar R}_{#1#2#3#4}} \newcommand{\cod}{h_{ij;k}-h_{ik;j}=\riema\al\bet\ga\de\n^\al x_i^\bet x_j^\ga x_k^\de} \newcommand{\gau}[1][\s]{\riem ijkl=#1 \{h_{ik}h_{jl}-h_{il}h_{jk}\} + \riema \al\bet\ga\de x_i^\al x_j^\bet x_k^\ga x_l^\de} \newcommand{\ric}{\h_{i;jk}=\h_{i;kj}+\riem lijk\msp \h^l} %Font commands \newcommand{\tbf}{\textbf} \newcommand{\tit}{\textit} \newcommand{\tsl}{\textsl} \newcommand{\tsc}{\textsc} \newcommand{\trm}{\textrm} \newcommand{\tup}{\textup}% text upright \newcommand{\mbf}{\protect\mathbf} \newcommand{\mitc}{\protect\mathit} \newcommand{\mrm}{\protect\mathrm} \newcommand{\bs}{\protect\boldsymbol} \newcommand{\mc}{\protect\mathcal} \newcommand{\msc}{\protect\mathscr} %Miscellaneous \providecommand{\bysame}{\makeboc[3em]{\hrulefill}\thinspace} \newcommand{\la}{\label} \newcommand{\ci}{\cite} \newcommand{\bib}{\bibitem} \newcommand{\cq}[1]{\glqq{#1}\grqq\,} \newcommand{\cqr}{\glqq{$\lra$}\grqq\,} \newcommand{\cql}{\glqq{$\lla$}\grqq\,} \newcommand{\bt}{\begin{thm}} \newcommand{\bl}{\begin{lem}} \newcommand{\bc}{\begin{cor}} \newcommand{\bd}{\begin{definition}} \newcommand{\bpp}{\begin{prop}} \newcommand{\br}{\begin{rem}} \newcommand{\bn}{\begin{note}} \newcommand{\be}{\begin{ex}} \newcommand{\bes}{\begin{exs}} \newcommand{\bb}{\begin{example}} \newcommand{\bbs}{\begin{examples}} \newcommand{\ba}{\begin{axiom}} \newcommand{\et}{\end{thm}} \newcommand{\el}{\end{lem}} \newcommand{\ec}{\end{cor}} \newcommand{\ed}{\end{definition}} \newcommand{\epp}{\end{prop}} \newcommand{\er}{\end{rem}} \newcommand{\en}{\end{note}} \newcommand{\ee}{\end{ex}} \newcommand{\ees}{\end{exs}} \newcommand{\eb}{\end{example}} \newcommand{\ebs}{\end{examples}} \newcommand{\ea}{\end{axiom}} \newcommand{\bp}{\begin{proof}} \newcommand{\ep}{\end{proof}} \newcommand{\eps}{\renewcommand{\qed}{}\end{proof}} \newcommand{\bal}{\begin{align}} %\newcommand{\eal}{\end{align}} \newcommand{\bi}[1][1.]{\begin{enumerate}[\upshape #1]} \newcommand{\bia}[1][(1)]{\begin{enumerate}[\upshape #1]} \newcommand{\bin}[1][1]{\begin{enumerate}[\upshape\bfseries #1]} \newcommand{\bir}[1][(i)]{\begin{enumerate}[\upshape #1]} \newcommand{\bic}[1][(i)]{\begin{enumerate}[\upshape\hspace{2\cma}#1]} \newcommand{\bis}[2][1.]{\begin{enumerate}[\upshape\hspace{#2\parindent}#1]} \newcommand{\ei}{\end{enumerate}} % comma is raised when components are quotients \newcommand\ndots{\raise 0.47ex \hbox {,}\hskip0.06em\cdots % \raise 0.47ex \hbox {,}\hskip0.06em} %Layout commands \newcommand{\clearemptydoublepage}{\newpage{\pagestyle{empty}\cleardoublepage}} \newcommand{\q}{\quad} \newcommand{\qq}{\qquad} \newcommand{\vs}[1][3]{\vskip#1pt} \newcommand{\hs}[1][12]{\hskip#1pt} \newcommand{\hp}{\hphantom} \newcommand{\vp}{\vphantom} \newcommand\cl{\centerline} \newcommand\nl{\newline} \newcommand\nd{\noindent} \newcommand{\nt}{\notag} % %my private skips; set to 0 to restore default \newskip\Csmallskipamount \Csmallskipamount=\smallskipamount \newskip\Cmedskipamount \Cmedskipamount=\medskipamount \newskip\Cbigskipamount \Cbigskipamount=\bigskipamount \newcommand\cvs{\vspace\Csmallskipamount} \newcommand\cvm{\vspace\Cmedskipamount} \newcommand\cvb{\vspace\Cbigskipamount} \newskip\csa \csa=\smallskipamount \newskip\cma \cma=\medskipamount \newskip\cba \cba=\bigskipamount \newdimen\spt \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 %------------------------------------------------------------------------------ \ No newline at end of file
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+\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}}
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+\newcommand{\raa}[1]{Axiom~\ref{A:#1}}
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+
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+\RequirePackage{upref}
+\RequirePackage{amsthm}
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+%\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
+%------------------------------------------------------------------------------