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Comptes Rendus
Mécanique
Piotr T. Chruściel
On Yvonne
Volume 353 (2025), p. 411-414
Online since: 13 February 2025
https://doi.org/10.5802/crmeca.272
This article is licensed under the
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C EN T R E
MER S ENN E
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Comptes Rendus. Mécanique
2025, Vol. 353, p. 411-414
https://doi.org/10.5802/crmeca.272
Historical Commentary / Commentaire historique
On Yvonne
À propos d’Yvonne
Piotr T. Chruściel ,a
a University of Vienna, Faculty of Physics, Boltzmanngasse 5, A1090 Vienna, Austria
E-mail: piotr.chrusciel@univie.ac.at
Abstract. It is a pleasure and honor to present here a personal perspective on Yvonne Choquet-Bruhat’s
scientific work.
Résumé. C’est un plaisir et un honneur de présenter ici une perspective personnelle sur le travail scientifique
d’Yvonne Choquet-Bruhat.
Keywords. Mathematical general relativity, Einstein equations, General relativistic Cauchy problems.
Mots-clés. Relativité générale mathématique, Équations d’Einstein, Problèmes de Cauchy en relativité
générale.
Manuscript received 9 October 2024, accepted 11 October 2024.
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
“General relativity and the Einstein equations”
published by Oxford University Press. (I had the pleasure to accompany Yvonne to the OUP
offices when she was preparing her book for publication ⌣̈.) In this list the book is followed
by the milestone papers
“Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non
linéaires”, Acta Math. 88 (1952), 141–225,
and
“Global aspects of the Cauchy problem in general relativity” with Robert Geroch, Commun.
Math. Phys. 14 (1969), 329–335.
These last two papers are to mathematical relativity as what Banach’s “Théorie des opérations
linéaires” 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 “Théorême d’existence . . . ” [1]. 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 [1] 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
ISSN (electronic): 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
412 Piotr T. Chruściel
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 [1] 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 [2].
The local constructions of [1] 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 [3]. 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 [4].
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 cosmic censorship conjectures is that this happens only in very
special circumstances; more on this can be found in [5, 6]. To resolve this one needs to under-
stand 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 Mon-
crief in [7]. In this context it should be added that Yvonne Choquet-Bruhat’s analysis of Fuchsian
partial differential equations, reviewed in her already mentioned monograph [5], remains one of
the best tools to explore the dynamics of cosmological spacetimes near singularities.
Needless to say, these are not the only contributions of Yvonne: MathSciNet lists 239 publi-
cations, including 11 books. One finds there seminal papers on the constraint equations in gen-
eral relativity, including the development, with Demetrios Christodoulou, of the framework for
analysing these equations on asymptotically flat manifolds [8]. These constraint equations con-
stitute 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 “con-
formal method”, some aspects of which have been intensively studied in the mathematics liter-
ature in the context of the Yamabe problem [9–11] mainly on compact manifolds. The paper [8]
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 [12, 13]. In a pioneering work with Marsden, Choquet-Bruhat proves
positivity for near-Minkowskian initial data [14]. The arguments used there are used nowadays
in state-of-the-art gluing theorems for the constraint equations [15, 16].
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 [17]. More on the general relativistic constraint equations can be found in [18, 19], see
also [20–24] 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 approchées pour des systèmes d’équations aux dérivées
partielles non linéaires” J. Math. Pures Appl. (9) 48 (1969), 117–158, has been given new life
in recent papers [25,26], which come close to settling Burnett’s conjecture [27] concerning a class
of high-frequency limits of solutions of Einstein’s equations.
Piotr T. Chruściel 413
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 [28] 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 [29, 30]. It should be mentioned that the role of the characteristic Cauchy
problem in general relativity has been steadily growing, see [31–34].
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 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 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 https://celebratio.org/ChoquetBruhat_Y/article/1108/.
References
[1] Y. Fourès-Bruhat, “Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires”,
Acta Math. 88 (1952), p. 141-225.
[2] H. Ringström, The Cauchy problem in General Relativity, ESI Lectures in Mathematics and Physics, European
Mathematical Society (EMS), Zürich, 2009, MR 2527641 (2010j:83001).
[3] Y. Choquet-Bruhat, R. Geroch, “Global aspects of the Cauchy problem in general relativity”, Commun. Math. Phys.
14 (1969), p. 329-335, MR MR0250640 (40 #3872).
[4] J. Sbierski, “On the existence of a maximal Cauchy development for the Einstein equations: a dezornification”, Ann.
Henri Poincaré 17 (2016), p. 301-329, MR 3447847.
[5] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Mathematical Monographs, Oxford Uni-
versity Press, Oxford, 2009, MR 2473363.
[6] P. T. Chruściel, On Uniqueness in the Large of Solutions of Einstein Equations (“Strong Cosmic Censorship”), Australian
National University Press, Canberra, 1991.
[7] Y. Choquet-Bruhat, V. Moncrief, “Future global in time Einsteinian spacetimes with U(1) isometry group”, Ann. Henri
Poincaré 2 (2001), p. 1007-1064, MR 1877233.
[8] D. Christodoulou, Y. Choquet-Bruhat, “Elliptic systems in Hs,δ spaces on manifolds which are Euclidean at infinity”,
Acta. Math. 146 (1981), p. 129-150.
[9] S. Brendle, “Blow-up phenomena for the Yamabe equation”, J. Am. Math. Soc. 21 (2008), p. 951-979, MR 2425176
(2009m:53084).
[10] M. A. Khuri, F. C. Marques, R. M. Schoen, “A compactness theorem for the Yamabe problem”, J. Differ. Geom. 81
(2009), p. 143-196, MR 2477893 (2010e:53065).
[11] J. M. Lee, T. H. Parker, “The Yamabe problem”, Bull. Amer. Math. Soc. (N.S.) 17 (1987), p. 37-91, MR MR888880
(88f:53001).
[12] E. Witten, “A simple proof of the positive energy theorem”, Commun. Math. Phys. 80 (1981), p. 381-402.
[13] R. Schoen, S.-T. Yau, “The energy and the linear momentum of spacetimes in general relativity”, Commun. Math.
Phys. 81 (1981), p. 47-51.
[14] Y. Choquet-Bruhat, J. E. Marsden, “Solution of the local mass problem in general relativity”, Commun. Math. Phys.
51 (1976), p. 283-296, MR 478215.
[15] Y. Mao, S.-J. Oh, Z. Tao, “Initial data gluing in the asymptotically flat regime via solution operators with prescribed
support properties”, preprint, 2023, [math.AP], https://arxiv.org/abs/2308.13031.
414 Piotr T. Chruściel
[16] S. Czimek, I. Rodnianski, “Obstruction-free gluing for the Einstein equations”, preprint, 2022, [gr-qc], https://arxiv.
org/abs/2210.09663.
[17] Y. Choquet-Bruhat, J. Isenberg, D. Pollack, “The constraint equations for the Einstein-scalar field system on compact
manifolds”, Class. Quantum Grav. 24 (2007), p. 809-828, MR 2297268.
[18] Y. Choquet-Bruhat, J. York, “The cauchy problem”, in General Relativity (A. Held, ed.), Plenum Press, New York, 1980,
MR 583716 (82k:58028), p. 99-172.
[19] A. Carlotto, “The general relativistic constraint equations”, Living Rev. Relativ. 24 (2021), article no. 2.
[20] P. Hintz, “Gluing small black holes along timelike geodesics I: Formal solution”, preprint, 2023, [gr-qc], https:
//arxiv.org/abs/2306.07409.
[21] J. Corvino, “Scalar curvature deformation and a gluing construction for the Einstein constraint equations”, Commun.
Math. Phys. 214 (2000), p. 137-189, MR MR1794269 (2002b:53050).
[22] J. Corvino, R. Schoen, “On the asymptotics for the vacuum Einstein constraint equations”, J. Differ. Geom. 73 (2006),
p. 185-217, MR MR2225517 (2007e:58044).
[23] P. T. Chruściel, E. Delay, “On mapping properties of the general relativistic constraints operator in weighted function
spaces, with applications”, Mém. Soc. Math. France. 94 (2003), p. 1-103, MR MR2031583 (2005f:83008).
[24] J. Anderson, J. Corvino, F. Pasqualotto, “Multi-localized time-symmetric initial data for the Einstein vacuum equa-
tions”, J. Reine Angew. Math. 2024 (2024), no. 808, p. 67-110.
[25] C. Huneau, J. Luk, “High-frequency solutions to the Einstein equations”, Class. Quantum Grav. 41 (2024), no. 14,
article no. 143002, MR 4777534.
[26] A. Touati, “The reverse Burnett conjecture for null dusts”, preprint, 2024, [math.AP], https://arxiv.org/abs/2402.
17530.
[27] G. A. Burnett, “The high frequency limit in general relativity”, J. Math. Phys. 30 (1989), p. 90-96.
[28] Y. Choquet-Bruhat, P. T. Chruściel, J. M. Martín-García, “The light cone theorem”, Class. Quantum Grav. 26 (2009),
p. 135011, MR 2515694 (2010g:53131).
[29] P. T. Chruściel, T.-T. Paetz, “The many ways of the characteristic Cauchy problem”, Class. Quantum Grav. 29 (2012),
article no. 145006, MR 2949552.
[30] P. T. Chruściel, W. Cong, “Gluing variations”, Class. Quantum Grav. 40 (2023), article no. 165009.
[31] D. Christodoulou, The Formation of Black Holes in General Relativity, EMS Monographs in Mathematics, European
Mathematical Society (EMS), Zürich, 2008, MR MR2488976 (2009k:83010).
[32] C. Kehle, R. Unger, “Gravitational collapse to extremal black holes and the third law of black hole thermodynamics”,
preprint, 2022, [gr-qc], https://arxiv.org/abs/2211.15742.
[33] S. Klainerman, I. Rodnianski, “On the formation of trapped surfaces”, Acta Math. 208 (2012), p. 211-333, MR 2931382.
[34] D. Christodoulou, “The instability of naked singularities in the gravitational collapse of a scalar field”, Ann. Math. (2)
149 (1999), p. 183-217, MR MR1680551 (2000a:83086).


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