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\DOI{10.5802/crmeca.272}
\datereceived{2024-10-09}
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\dateposted{2025-02-13}
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\CDRsetmeta{articletype}{historical-commentary}

\title{On Yvonne}
\alttitle{\`A propos d'Yvonne}
 
\author{\firstname{Piotr T.} \lastname{Chru\'{s}ciel}\CDRorcid{0000-0001-8362-7340}}
\address{University of Vienna, Faculty of Physics, Boltzmanngasse 5,
A1090 Vienna, Austria}
\email{piotr.chrusciel@univie.ac.at}

\keywords{\kwd{Mathematical general relativity} 
\kwd{Einstein equations} \kwd{General relativistic Cauchy problems}}

\altkeywords{\kwd{Relativit\'e g\'en\'erale math\'ematique} \kwd{\'Equations
d'Einstein} \kwd{Probl\`emes de Cauchy en relativit\'e g\'en\'erale}}

\begin{abstract} 
It is a pleasure and honor to present here a personal perspective on
Yvonne Choquet-Bruhat's scientific work.
\end{abstract}

\begin{altabstract} 
C'est un plaisir et un honneur de pr\'esenter ici une perspective
personnelle sur le travail scientifique d'Yvonne Choquet-Bruhat.  
\end{altabstract}

\maketitle

\end{noXML}

{\vspace*{1pc}}

\xsection{}

To make this note less subjective, let me start by mentioning the
citation list  of Yvonne Choquet-Bruhat on MathSciNet, which  is opened
by her 2009 book
\smallskip

 ``General relativity and the Einstein equations'' 
\smallskip
 
\noindent published by
Oxford University Press. (I had the pleasure to accompany Yvonne to the
OUP offices when she was preparing her book for publication
$\ddot\smile$.)  In this list the book  is followed by  the milestone papers
\smallskip

``Th\'eor\`eme d'existence pour certains syst\`emes d'\'equations aux
d\'eriv\'ees partielles non lin\'eaires'',  Acta Math. 88 (1952), 141--225, 
\smallskip

\noindent
and
\smallskip

 ``Global aspects of the Cauchy problem in general relativity''
 with Robert Geroch,  Commun.\ Math.\ Phys.\ 14 (1969), 329--335. 
\smallskip

\noindent These last two papers are to mathematical relativity as what
Banach's ``Th\'eorie des op\'erations lin\'eaires'' is  to functional
analysis. 

Indeed, her brilliant proof, that Einstein equations possess a well
posed Cauchy problem, is the foundation on which mathematical general
relativity is built. Nowadays we take it for granted that our universe
is described by evolution equations, with initial data prescribed on an
initial data surface. The fact that this is the case for general
relativity was not known until Yvonne's ``Th\'eor\^eme 
d'existence\,\ldots''~\cite{FouresBruhat52}. There are several difficulties here. The
first is the diffeomorphism invariance of the Einstein equations, which
implies that solutions of Einstein equations can only be unique up to
diffeomorphisms. The first discovery  in~\cite{FouresBruhat52} was,
that this can be cured by introducing harmonic coordinates. The next
difficulty is to prove that the resulting ``harmonically-reduced''
equations  admit local solutions.  Her paper provides the first proof
of this for a large class of equations with the relevant structure.
Last but not least, one needs to show that a solution of the reduced
equations provide a solution of the  equations of real interest, namely
the Einstein equations. It is shown in~\cite{FouresBruhat52} how the
general relativistic constraint equations can be used to solve this
problem. A self-contained presentation of these constructions using
modern techniques can be found in~\cite{RingstroemBook}.

The local constructions of~\cite{FouresBruhat52} immediately raise the
question of existence and properties of global solutions of Einstein
equations. The second milestone on which mathematical general
relativity rests is the already-mentioned paper with Robert
Geroch~\cite{ChoquetBruhatGeroch69}.  The paper unravels the
fundamental role of the notion of global hyperbolicity in this context,
and presents the proof that every set of, say vacuum, general
relativistic initial data evolves to a unique   spacetime which is
maximal in the set of globally hyperbolic spacetimes inducing the given
initial data. A nice conceptual simplification of the proof can be
found in~\cite{SbierskiMGHD}.

It turns out that this is not the end of the story: there exist well
behaved initial data sets, such as initial data for the Kerr black
holes or for the Bianchi IX cosmological models, for which the maximal
globally hyperbolic developments can be extended beyond a Cauchy
horizon in a non-unique way. The contents of the \emph{cosmic
censorship conjectures} is that this happens only in very special
circumstances; more on this can be found in~\cite{YvonneBook,SCC}. To
resolve this one needs to understand the global structure of generic
solutions of the general relativistic Cauchy problem, which is not a
minor task. Among the most significant achievements on this topic in
the class of  spatially compact spacetimes with one Killing vector are
the results presented by Yvonne and Vince Moncrief
in~\cite{YvonneVinceUone}.  In this context it should be added that
Yvonne Choquet-Bruhat's analysis of Fuchsian partial differential
equations, reviewed in her already mentioned
monograph~\cite{YvonneBook},  remains one of the best tools to explore
the dynamics of cosmological spacetimes near singularities.\looseness=-1

Needless to say, these are not the only contributions of Yvonne:
MathSciNet lists 239 publications, including 11 books. One finds there 
seminal papers on the constraint equations in
general relativity, including the development, with Demetrios
Christodoulou, of the framework for analysing these equations on
asymptotically flat
manifolds~\cite{choquet-bruhat:christodoulou:elliptic}.  These
constraint equations constitute a set of nonlinear equations that has 
to be satisfied by the initial data, and one of the key goals of
mathematical general relativity is to construct exhaustive families of
their solutions, and to understand the properties of these. For many
years a standard tool for this has been the ``conformal method'', some
aspects of which have been intensively studied in the mathematics
literature in the context of the Yamabe
problem~\cite{BrendleYamabe,KhuriMarquesSchoen,LeeParker} mainly on
compact manifolds. The
paper~\cite{choquet-bruhat:christodoulou:elliptic} provides key tools
to understand elliptic systems on asymptotically flat manifolds,
including the general relativistic constraint equations.  

One of the fundamental properties of asymptotically flat solutions of the
constraint equations is positivity of
mass~\cite{Witten:mass,SchoenYau81b}. In a pioneering work with
Marsden, Choquet-Bruhat proves positivity for near-Minkowskian initial
data~\cite{YvonneMarsden}. The arguments used there are used nowadays
in state-of-the-art gluing theorems for the constraint
equations~\cite{MaoOhTao,CzimekRodnianski}.

The general relativistic constraint equations on compact manifolds
become quite a headache when matter fields transform in an inconvenient
way under conformal transformations. An example is provided by the
scalar field, which has been addressed by Yvonne and collaborators
in~\cite{YvonneJimDaniel}. More on the general relativistic constraint
equations can be found in~\cite{CBY,CarlottoLRR}, see
also~\cite{HintzBH,Corvino,CorvinoSchoen2,ChDelay,CorvinoPasqualotto} 
and references therein.

Choquet-Bruhat's pioneering work with Daniel Bancel, ``Existence,
uniqueness, and local stability for the Einstein--Maxwell--Boltzman
system'', Comm. Math. Phys. 33 (1973), 83--96, provides the first 
rigorous treatment of the Boltzmann equation within Einsteinian
gravity. Her prophetic ``Ondes asymptotiques et approch\'ees pour des
syst\`emes d'\'equations aux d\'eriv\'ees partielles non lin\'eaires''  J.
Math. Pures Appl. (9) 48 (1969), 117--158, has  been given new life in
recent papers \cite{Huneau:2024dyd,Touati:2024xpw}, which come close to
settling Burnett's conjecture~\cite{BurnettConjecture} concerning a
class of high-frequency limits of solutions of Einstein's equations.

I had the pleasure and honor to coauthor a series of papers with Yvonne
on the characteristic Cauchy problem in general relativity. There we
present a variation of an approach of Rendall, perhaps making clearer
the constraints character of some of Einstein equations. In~\cite{CCG}
we  observe that all the constraints can be reduced to linear
equations, the only obstruction to global existence being positivity of
a conformal factor. This should be compared to the much more  intricate
structure of the  already-mentioned spacelike constraints. Natural
extensions of this work can be found in~\cite{ChPaetz,ChCong0}. It
should be mentioned that the role of the characteristic Cauchy problem
in general relativity has been steadily growing,
see~\cite{ChristodoulouBHFormation,Kehle:2022uvc,KlainermanRodnianskiBH2,ChristodoulouAnnals99}.

Amongst the various honors received, Yvonne  was the first female
member of the French Academy of Sciences. She has been included in the
list of the 85 original and important female contributors to the
physics of the 20th Century on the web site URL
\url{https://web.archive.org/web/20161007090528/http://cwp.library.ucla.edu/exp.html}.
Together with James York, from Cornell University, she was  awarded the
2003 Dannie Heineman Prize for Mathematical Physics,``for their
separate as well as joint work in proving the existence and uniqueness
of solutions to Einstein's gravitational field equations so as to
improve numerical solution procedures with relevance to realistic
physical solutions''. The prize is presented in recognition of
outstanding publications in the field of mathematical physics.   In
2006 Yvonne Choquet-Bruhat delivered the Emmy Noether Lecture at the
International Congress of Mathematicians in Madrid on the topic of
``Mathematical problems in General Relativity''. Her biography has been
included on the web site `Biographies of women mathematicians'', URL
\url{https://mathwomen.agnesscott.org/women/bruhat.htm}, next to
Hypathia, Sofia Kovalevskaya, Emmy Noether, and Miryam Mirzakhani. A
volume in her honor can be found on URL
\url{https://celebratio.org/ChoquetBruhat_Y/article/1108/}.

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