Comptes Rendus Mécanique (CRM)

Michaël Darche

Raphaël Assier

Sébastien Guenneau

Bruno Lombard

Marie Touboul

Scattering of transient waves by an interface with time-modulated jump conditions

Jean Varnier

Doppler effect considered as a distortion of time, application to an engineering case

Serge Soula

La face cachée des orages

Larry Murcia Terranova

Emilio Turco

Anil Misra

Francesco dell’Isola

Computational identification of double-bending stiffness: from Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua

Yakov Shlapentokh-Rothman

Weak cosmic censorship, trapped surfaces, and naked singularities for the Einstein vacuum equations

Arthur Touati

Burnett’s conjecture in general relativity

Sergiu Klainerman

The black hole stability problem

Piotr T. Chruściel

Comptes RendusMécaniquePiotr T. ChruścielOn YvonneVolume 353 (2025), p. 411-414Online since: 13 February 2025https://doi.org/10.5802/crmeca.272This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mécanique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1873-7234Comptes Rendus. Mécanique2025, Vol. 353, p. 411-414https://doi.org/10.5802/crmeca.272Historical Commentary / Commentaire historiqueOn YvonneÀ propos d’YvonnePiotr T. Chruściel ,aa University of Vienna, Faculty of Physics, Boltzmanngasse 5, A1090 Vienna, AustriaE-mail: piotr.chrusciel@univie.ac.atAbstract. It is a pleasure and honor to present here a personal perspective on Yvonne Choquet-Bruhat’sscientific work.Résumé. C’est un plaisir et un honneur de présenter ici une perspective personnelle sur le travail scientifiqued’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 relativitégénérale.Manuscript received 9 October 2024, accepted 11 October 2024.To make this note less subjective, let me start by mentioning the citation list of YvonneChoquet-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 OUPoffices when she was preparing her book for publication ⌣̈.) In this list the book is followedby the milestone papers“Théorème d’existence pour certains systèmes d’équations aux dérivées partielles nonlinéaires”, 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 opérationslinéaires” 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 forgranted that our universe is described by evolution equations, with initial data prescribed onan initial data surface. The fact that this is the case for general relativity was not known untilYvonne’s “Théorême d’existence . . . ” [1]. There are several difficulties here. The first is thediffeomorphism invariance of the Einstein equations, which implies that solutions of Einsteinequations can only be unique up to diffeomorphisms. The first discovery in [1] was, that this canbe 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 ofthis for a large class of equations with the relevant structure. Last but not least, one needs toISSN (electronic): 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/412 Piotr T. Chruścielshow 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 equationscan be used to solve this problem. A self-contained presentation of these constructions usingmodern techniques can be found in [2].The local constructions of [1] immediately raise the question of existence and properties ofglobal solutions of Einstein equations. The second milestone on which mathematical generalrelativity rests is the already-mentioned paper with Robert Geroch [3]. The paper unravels thefundamental role of the notion of global hyperbolicity in this context, and presents the proofthat every set of, say vacuum, general relativistic initial data evolves to a unique spacetime whichis maximal in the set of globally hyperbolic spacetimes inducing the given initial data. A niceconceptual 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, suchas initial data for the Kerr black holes or for the Bianchi IX cosmological models, for which themaximal 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 veryspecial 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 isnot a minor task. Among the most significant achievements on this topic in the class of spatiallycompact 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 Fuchsianpartial differential equations, reviewed in her already mentioned monograph [5], remains one ofthe 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 foranalysing 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 keygoals of mathematical general relativity is to construct exhaustive families of their solutions, andto 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 thegeneral relativistic constraint equations.One of the fundamental properties of asymptotically flat solutions of the constraint equationsis positivity of mass [12, 13]. In a pioneering work with Marsden, Choquet-Bruhat provespositivity for near-Minkowskian initial data [14]. The arguments used there are used nowadaysin state-of-the-art gluing theorems for the constraint equations [15, 16].The general relativistic constraint equations on compact manifolds become quite a headachewhen matter fields transform in an inconvenient way under conformal transformations. Anexample is provided by the scalar field, which has been addressed by Yvonne and collaboratorsin [17]. More on the general relativistic constraint equations can be found in [18, 19], seealso [20–24] and references therein.Choquet-Bruhat’s pioneering work with Daniel Bancel, “Existence, uniqueness, and localstability 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. Herprophetic “Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivéespartielles non linéaires” J. Math. Pures Appl. (9) 48 (1969), 117–158, has been given new lifein recent papers [25,26], which come close to settling Burnett’s conjecture [27] concerning a classof high-frequency limits of solutions of Einstein’s equations.Piotr T. Chruściel 413I had the pleasure and honor to coauthor a series of papers with Yvonne on the characteristicCauchy 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] weobserve that all the constraints can be reduced to linear equations, the only obstruction to globalexistence being positivity of a conformal factor. This should be compared to the much moreintricate structure of the already-mentioned spacelike constraints. Natural extensions of thiswork can be found in [29, 30]. It should be mentioned that the role of the characteristic Cauchyproblem in general relativity has been steadily growing, see [31–34].Amongst the various honors received, Yvonne was the first female member of the FrenchAcademy of Sciences. She has been included in the list of the 85 original and important femalecontributors 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 CornellUniversity, she was awarded the 2003 Dannie Heineman Prize for Mathematical Physics,“for theirseparate as well as joint work in proving the existence and uniqueness of solutions to Einstein’sgravitational field equations so as to improve numerical solution procedures with relevance torealistic physical solutions”. The prize is presented in recognition of outstanding publications inthe field of mathematical physics. In 2006 Yvonne Choquet-Bruhat delivered the Emmy NoetherLecture at the International Congress of Mathematicians in Madrid on the topic of “Mathematicalproblems in General Relativity”. Her biography has been included on the web site ‘Biographies ofwomen mathematicians”, URL https://mathwomen.agnesscott.org/women/bruhat.htm, next toHypathia, Sofia Kovalevskaya, Emmy Noether, and Miryam Mirzakhani. A volume in her honorcan 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, EuropeanMathematical 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”), AustralianNational University Press, Canberra, 1991.[7] Y. Choquet-Bruhat, V. Moncrief, “Future global in time Einsteinian spacetimes with U(1) isometry group”, Ann. HenriPoincaré 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 prescribedsupport 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 compactmanifolds”, 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 functionspaces, 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, EuropeanMathematical 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).

On Yvonne

Jacques Smulevici

The stability of Minkowski space and its influence on the mathematical analysis of General Relativity

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