Solving the constraint equation for general free data
Abstract. We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo low modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres. Our method can be applied to construct various types of initial data, including data with arbitrarily fast decay, borderline decay, and short-pulse type data. This is joint work with Sergiu Klainerman.
Armand COUDRAY (Tours)
Existence and uniqueness of solutions to the conformal constraint equations
Abstract. I will present the results obtained in collaboration with Romain Gicquaud concerning the classical construction of initial data using the conformal method, which was originally proposed by Holst, Nagy and Tsogtgerel and later refined by Maxwell. This method transforms the usual constraint equation for initial data in general relativity into a set of two coupled nonlinear elliptic PDEs. Our work revisits the standard proof by removing certain assumptions. In particular, I will explain how our proof guarantees the uniqueness of solutions to the equations of the conformal method as soon as a bound is imposed on the physical volume and how it provides an explicit construction of solutions.
Romain GICQUAUD (Tours)
On the topology of vacuum initial data in general relativity
Abstract. Parameterizing (vacuum) initial data is a major challenge in general relativity. The most studied method, the conforma method, has been successful in parameterizing the CMC vacuum initial data but beyond it, very limited is known despite decades of efforts. One potential reason is that the parametrizations we are employing are too naive in the sense that the set of seed data has a topology that is too simple compared to the vacuum initial data. Building on earlier works by Jonathan Glöckle and Berndt Amman, we show that there exist compact manifolds of dimension greater than 5 with infinitely many non-trivial homotopy groups. This is a joint work with Jonathan Glöckle.
Bruno LE FLOCH (LPTHE, Sorbonne)
Optimal localization for the Einstein constraints
Abstract. We establish the existence of asymptotically Euclidean initial data sets to Einstein’s vacuum constraints exhibiting both gravitational shielding and arbitrarily low decay. We resolve a conjecture of Carlotto and Schoen on gluing two solutions across an asymptotically conical domain: in the interpolation region we establish optimal estimates at and beyond harmonic decay, together with ADM invariant estimates. Starting from a seed data set that asymptotically solves the constraints, we project it to an exact solution, whose difference from the seed inherits the natural optimal radial decay rate.
Zhongkai TAO (IHES & Mittag-Leffler, Stockholm)
An h-principle for general relativistic initial data sets
Abstract. I will talk about a flexibility result on the Einstein constraint equations. It uses the Bogovskii-type operators introduced in Mao-Oh-Tao and Isett-Mao-Oh-Tao and generalizes the obstruction-free gluing theorem. This is joint work with Jonathan Luk, Yuchen Mao and Sung-Jin Oh.
Arthur TOUATI (Bordeaux)
Spacelike initial data for black hole stability
Abstract. I will present a joint work with Allen Juntao Fang and Jérémie Szeftel on the resolution of the constraint equations in the context of black hole stability. After having presented the motivations coming from the evolution problem and reviewing the literature on the constraint equations, I will show how a bit of Fredholm theory and the careful construction of a boundary condition for the elliptic system can help avoiding linear obstructions (the famous KIDS) and thus produce initial black hole perturbations with arbitrary decay.
Organized as an event of the ANR project:
Einstein’s constraints: past, present, and future
Romain GICQUAUD (Tours), Erwann DELAY (Avignon), and Philippe G. LeFloch (Sorbonne, Paris)
lecture room15-25–104 (Jussieu) (changement de salle !)
14h Fréderic HELEIN (Université Paris Cité)
Théories de Kaluza-Klein sans hypothèses de fibration a priori
Abstract. Je présenterai une action lagrangienne sur un espace de champs dont les points critiques produisent des solutions du système d’équations d’Einstein-Yang-Mills, dans l’esprit des théories de Kaluza-Klein. La nouveauté est le fait que l’on fait l’économie de l’hypothèse d’existence d’une structure de fibré principal : les champs sont définis sur une variété Y de dimension 4+r, où r est la dimension du groupe de structure (par exemple 3 pour SU(2)). Si ce dernier groupe est compact et simplement connexe, à chaque solution des équations d’Euler-Lagrange correspond une variété pseudo-riemannienne X de dimension 4 (qui peut être interprétée comme étant notre espace-temps) de telle façon que Y acquière une structure de fibré principal au-dessus de X équipé d’une connexion. De plus la métrique sur X et la connexion sur Y sont solutions du système d’Einstein-Yang-Mills. Si le groupe de structure est U(1) (qui correspond au système d’Einstein-Maxwell) la situation est légèrement dégénérée et des hypothèses supplémentaires sont nécessaires.
15h30 Ruben ZEITOUN (ENS Lyon)
Spectral action and the resolvent of the wave operator on asymptotically de Sitter spaces
Abstract. It is well know that for compact Riemannian manifolds the trace of the complex powers of the Laplace-Beltrami operator is linked to the scalar curvature of the manifold. Nguyen Viet Dang and Michal Wrochna showed that this is also possible for asymptotically Minkowski spaces, and therefore in Lorentzian signature, based on recent progress in the field. The goal of this lecture is to study such a link in asymptotically de Sitter spaces. More precisely we investigated the elaboration of such a link with the Feynman propagator and not the resolvent. Then we studied the self-adjointness of the Laplace-Beltrami operator which is the first step to the establishment of such a link.
Thursday May 28, 2026
lecture room15-16–309 (Jussieu)
14h Zhongkai TAO (IHES)
Bogovskii-type operators and the flexibility of initial data in general relativity
Abstract. The initial data of the vacuum Einstein equations on a spacelike hypersurface satisfy the Einstein constraint equations. It is an underdetermined PDE system. We exploit the under-determinedness to find nontrivial solutions of the Einstein constraint equations with interesting properties. A key ingredient is the construction of Bogovskii-type operators for underdetermined PDEs satisfying a robust symbol condition. The Bogovskii-type operators have favorable support and regularizing properties, allowing us to manipulate the solutions flexibly. This is based on joint work (partly in progress) with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Sung-Jin Oh (UC Berkeley)
15h30 Weidong ZHANG (LJLL, Sorbonne and Xi’an Jiaotong)
Global nonlinear stability of self-gravitating massive Dirac fields
Abstract. I will present a work in collaboration with P.G. LeFloch and Yue Ma on the nonlinear stability of Minkowski spacetime. We consider the massive Einstein-Dirac system and investigate the global evolution problem when the initial data set is sufficiently close to data describing a spacelike and asymptotically Euclidean slice in Minkowski spacetime. We establish the existence of a globally hyperbolic development, which remains asymptotic to Minkowski spacetime in future timelike, null, and spacelike directions. Previous results on this problem have been limited to the massless Einstein-Dirac system. Our analysis follows closely the asymptotically hyperboloidal-Euclidean framework introduced by LeFloch and Yue Ma for the massive Klein-Gordon-Einstein system. The structure specific to spinor fields and the Dirac equation necessitates significantly new elements in the proof. In contrast with prior approaches, our treatment of spinor fields and the Dirac equation is fully gauge-invariant, relying on the formalism of Lorentz Clifford algebras, principal fiber bundles, and Dirac forms.
Carrollian geometry and application to gravitational dynamics
Abstract. The Poincaré group is at the heart of Einstein’s description of spacetime. Its non-relativistic limit, the Galilean group, is adapted to physics at velocities that are small compared with the speed of light. There is another limit, called the Carrollian limit, obtained formally as a zero-speed-of-light limit. Despite its exotic nature, this limit has found a vast field of applications in the study of the asymptotic symmetries of certain types of spacetimes, in the description of the horizons of black holes, and in the formulation of the ultra-relativistic limit of fluids. In this presentation, I will provide an overview of the fundamental principles of Carrollian geometry and explain how these can be used to derive the Bondi flux balance equations on the null boundary of asymptotically flat spacetimes.
15h30 Jingbo WAN (Laboration J-L Lions, Sorbonne)
Wave decay on slowly-rotating extremal Kerr-Newman black holes
Abstract. I will discuss the decay of solutions to the wave equation on slowly-rotating Kerr-Newman black holes. The behavior of waves on extremal spherically symmetric black holes, as well as that of axisymmetric waves on rotating extremal black holes, has seen remarkable recent developments. I will explain how these ideas can be extended to the slowly-rotating Kerr-Newman case. In particular, I will highlight a close analogy between the structure of the wave operator near null infinity and its structure near the extremal horizon. This is a joint work with Allen Juntao Fang and Elena Giorgi.
Thursday March 19, 2026
lecture room 15-16–309 (Jussieu)
14h Xuantao CHEN (Laboration J-L Lions, Sorbonne)
Solving the constraint equation for general free data
Abstract. We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres, which we solve by an iteration procedure. Our method provides a large class of exterior solutions of the constraint equations that can be matched to given interior solutions, according to the existing gluing techniques. This is joint work with S. Klainerman.
15h30 Georgios MOSCHIDIS (EPFL, Lausanne)
Naked singularities with finite blue-shift for the Einstein–massless Vlasov system
Abstract. In his celebrated proof of the weak cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system, Christodoulou exploited the following property of that specific matter model: Naked singularities, when they arise, exhibit infinite blue-shift along the null geodesics terminating at the singularity. This behaviour is consistent with self-similarity: Even for more general spherically symmetric matter models, it can be shown that self-similar naked singularities must exhibit infinite blue-shift. Whether, for these more general models, all naked singularities have the infinite blue shift property (and hence are potentially subject to an instability mechanism analogous to that introduced by Christodoulou) still remains an open question. In this talk, I will present the construction of a spherically symmetric solution to the Einstein-massless Vlasov system which contains a locally naked singularity with finite total blue-shift along its past null cone. The initial data giving rise to this solution have limited differentiability, but belong to a regularity class above the scale invariant threshold.
Thursday February 19, 2026
lecture room 15-25-322(nouvelle salle)
14h Jack BORTHWICK (Institut de Mathématiques de Jussieu, Sorbonne)
Geometry at time-like infinity and massive fields
Abstract. This talk explores recent geometric constructions in the setting of projectively compact Ricci-flat Einstein manifolds, with an emphasis on the role of projective geometry. I will discuss how these constructions encode aspects of the geometry at infinity, and present some initial ideas and results on their potential relevance for the analysis of massive classical particle fields a « timelike » infinity.
15h30 Itsvan KADAR (ETH, Zürich)
Matching conditions for scattering solutions of scalar wave equations on extremal black holes
Abstract. The existence of spacetimes describing multiple black holes and their asymptotic properties—such as their late-timebehavior—is an exciting open area of mathematical research. In this talk, I will report on some preliminary steps towards addressing this problem. As a toy model, we study a nonlinear scalar wave equation on a multi–extremal- black-hole spacetime with a prescribed, polynomially decaying, smooth radiation field at null infinity. We construct smooth solutions in the spacetime by identifying appropriate initial data on a spacelike hypersurface inside the black hole regions. This is joint ongoing work with Yannis Angelopoulos.
Thursday December 18, 2025
lecture room 15-25-322(nouvelle salle)
14h Flavio ROSSETTI (L’Aquila)
Strong cosmic censorship for de Sitter black holes
Abstract. We will discuss modern formulations of the strong cosmic censorship conjecture (SCCC) and possible resolutions supported by rigorous non-linear results for the spherically symmetric Einstein-Maxwell-scalar field system. We will show that the presence of a positive cosmological constant suggests a violation of the SCCC at a fundamental level of regularity. Indeed, the blueshift mechanism occurring at the Cauchy horizon can be counter-balanced by the dispersive effects encoded in the exponential Price law along (cosmological) black hole event horizons. On the other hand, we show that, if non-smooth black hole solutions are allowed, then the aforementioned violations are non-generic in a positive co-dimension sense.
15h30 Gemma HOOD (Leipzig)
A scattering construction for nonlinear wave equations on Kerr-Anti de Sitter spacetimes
Abstract. Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici, 2013), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain ‘close’ and dissipate sufficiently fast.
Thursday November 27, 2025
lecture room 15-25-101 (Jussieu)
14h Mahdi HAGHSHENAS (Imperial College, London)
Boundedness and decay of waves on decelerated FLRW spacetimes
Abstract. After outlining the stability problem for Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, we study the wave equation —as a proxy for the Einstein equations— on decelerated FLRW spacetimes with non-compact, flat spatial sections. We demonstrate how dispersion and expansion affect the long-time behavior of waves. In particular, we present uniform energy bounds and integrated local energy decay estimates across the full decelerated expansion range. Furthermore, we describe a hierarchy of r-weighted energy estimates, in the spirit of the Dafermos–Rodnianski method, which lead to energy decay estimates.
15h30 Pau FIGUERAS (Queen Mary, London)
The initial value problem for higher derivative theories of gravity
Abstract. General relativity can be thought of as a low energy (classical) effective field theory (EFT) of gravity. As such, on general grounds, it is expected that it should receive higher derivative corrections. However, the equations of motion of such higher derivative theories are higher than second order; in particular, they have more than two time derivatives and hence they are plagued with runaway solutions that are unphysical. Furthermore, being higher than second order, it is not clear how to formulate the initial value problem and thus extract their predictions consistently with the EFT expansion. In this talk, I will review the various approaches to this old problem and I will present our recent proposal called “regularization”. As I will show, regularisation allows to formulate the initial value problem for a very general class of higher derivative theories in a manifestly well-posed way, it is covariant and it does not require any fine tuning.
Abstract. Modeled on the Anti-de Sitter space, asymptotically Anti-de Sitter spaces are defined as Lorentzian manifolds that possess a timelike conformal boundary. As a result, they are not globally hyperbolic. In order to find such spaces that also solve the Einstein equations (with a negative cosmological constant), it is therefore necessary to consider the Cauchy problem as an initial boundary value problem. In this talk, I will discuss the geometric boundary conditions that can be prescribed on the conformal boundary to ensure local existence and uniqueness of solutions in dimension 4. The first one, introduced by Friedrich in his pioneering 1995 work, consists in imposing the boundary conformal class and is known as the Dirichlet boundary condition. The second is a new family of geometric reflective boundary conditions involving both the boundary conformal class and the boundary stress-energy tensor. It can be regarded as the homogeneous Robin boundary conditions.
Abstract. I will present our recent work on the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system in 1+3 dimensions. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. The inclusion of the Vlasov field introduces several new challenges. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this entire regime. Our results extend the work of Fournodavlos, Rodnianski, and Speck from the Einstein-scalar field system to the physically more complex system with the Vlasov field. This is joint work with Xinliang An and Dawei Shen.
Starting at 8:55am on Wednesday, and closing at 5:15pm on Friday
ORGANIZERS
Luc Blanchet (Institut d’Astrophysique de Paris) Eric Gourgoulhon (LUX, Observatoire de Paris & CNRS) José-Luis Jaramillo (IMB, Université Bourgogne Europe) Bruno Le Floch (LPTHE, Sorbonne Université & CNRS) Philippe G. LeFloch (LJLL, Sorbonne Université & CNRS)
Abstract. Generic unitary irreducible representations (UIRs) of the Bondi-Metzner-Sachs (BMS) group are considered. They are shown to describe quantum superpositions of (Poincaré) particles propagating on inequivalent gravity vacua. This follows from reconsidering McCarthy’s classification of BMS group UIRs through a unique, Lorentz-invariant but non-linear, decomposition of supermomenta into hard and soft pieces.
Abstract. Current agnostic tests of gravity with gravitational waves are plagued by a lack of realistic deviations, making it difficult to interpret such detections with respect to specific theories. In this talk, I present a dictionary that identifies the scaling of deviations with the objects’ masses and the leading order post-Newtonian corrections in generic theories constructed through an Effective Field Theory approach based on curvature. In particular, I will demonstrate that a vast set of theories only deviates from General Relativity beginning at a relatively high order. I will also clarify some subtleties of the application of the PN-EFT formalism to higher-curvature EFT theories.
Jean-Pierre BOURGUIGNON (Nicolaas Kuiper Honorary Professor at IHÉS)
Abstract. In the middle of the 20th century, some major actors have succeeded in putting Differential Geometry much more centre stage than it used to be: Élie Cartan, Chern Shiing-Shen, Isadore M. Singer, Sir Michael Atiyah, Eugenio Calabi and of course André Lichnerowicz. This process had a lot to do with the development of new concepts and the appropriation for the field of new tools coming in particular from Analysis and Topology. In almost all cases Lichnérowicz played a key role through research articles and well appreciated books. The purpose of this lecture is two-fold: first, to describe this transformation with an emphasis on some issues to which Lichnerowics gave a lot of attention; second, to highlight some of the interactions I had with him with very clear, friendly and rewarding messages.
Abstract. On a Riemannian manifold of dimension three or higher, we introduce two differential operators acting on (fields of) trace-free symmetric 2-tensors. The first, a second-order operator, is a conformally covariant operator, similar to the Yamabe Laplacian on functions. It can be used to test the stability of certain Einstein metrics. The second, a fourth-order operator, acts as a machine for TT-tensors (symmetric 2-tensors that are both trace-free and divergence-free) on Einstein manifolds, as it allows any trace-free symmetric 2-tensor to be transformed into a TT-tensor, with many such tensors being obtained in this way. This operator can also be used to approximate a less regular TT-tensor by a smooth TT-tensor. On a Ricci-flat manifold, the restriction of these two operators to TT-tensors corresponds to the Lichnerowicz Laplacian and its square.
Abstract. We introduce the concept of k−future convex spacelike/null hypersurface Σ in an n + 1 dimensional spacetime and prove that no k−dimensional trapped submanifold can be tangent to Σ from its future side. As a consequence, k-dimensional closed trapped submanifolds cannot be found in open spacetime regions foliated by such hypersurfaces. In gravitational collapse scenarios, specific hypersurfaces of this kind act as past barriers for trapped submanifolds. Examples will be given of (3+1) spacetime regions containing trapped loops (k = 1) but no closed trapped surfaces (k = 2) and of how trapped loops could be used as an early indicator of black hole formation in numerical relativity.
Abstract. We consider solutions to critical and sub-critical semilinear elliptic PDEs on complete, noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are established for solutions to the Liouville equation on Riemannian surfaces. Our results are obtained via an appropriate P-function whose constancy implies the classification of both the solutions and the underlying manifold.
Abstract. A Kähler metric is called extremal if its scalar curvature is a Killing potential, i.e. is the moment relative to the Kähler form of a Hamiltonian Killing vector field; it is called toric extremal if the latter belongs to a maximal, effective Hamiltonian toric action preserving the whole Kähler structure. The presence of such a Kähler structure in the conformal class of a class of four-dimensional gravitational instantons of ALF type, including the Euclidean version of well-known Lorentzian spaces, as well as the one-parameter family of instantons discovered in 2011 by Yu Che and Edward Teo, plays a prominent role in its eventual complete classification, including a new description of the Chen–Teo instantons. This is a joint work with Olivier Biquard.
Abstract. Given a closed Riemannian manifold of dimension three, when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 such that its conformal infinity is the above Riemannian metric? This problem is motivated by the correspondance AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds. In this talk, I will discuss the compactness issue of asymptotically hyperbolic Einstein manifolds in dimension four, that is, how the compactness on conformal infinity leads to the compactness of the compactification of such manifolds under the suitable conditions on the topology and on some conformal invariants. As an application, I will discuss some recent progress on the existence result.
Abstract. The conformal method and its variants have long been among the most effective tools for constructing solutions to the Einstein constraint equations. In this talk, I will briefly review the method and its key achievements in generating large classes of initial data. I will then present recent results showing that the conformal method is not conformally covariant. This is an undesirable feature, which I will illustrate through explicit analytic constructions and numerical evidence.
Abstract. In recent years, it has been demonstrated that asymptotic symmetries of gravity (the so called BMS group) constrain the gravitational S-matrix. In particular, infrared divergences of the gravitational S-matrix are now understood to arise from to the impossibility of the usual Fock space of massless particles to ensure the conservation of the BMS charges. I will review these results taking the original perspective of representation theory. It is indeed natural to conjecture that asymptotic states suited for an infrared finite S-matrix should be unitary representations of the BMS group and thus BMS particles, rather than the usual Poincaré particles of Wigner. In a recent work with X. Bekaert and L. Donnay we constructed explicitly such BMS particles; this talk aims to serve as an introduction to X. Bekaert’s talk.
Emmanuel HUMBERT
Conformal eigenvalues of GJMS operators
Abstract. I will present recent results obtained in collaboration with R. Petrides and B. Premoselli. Our study focuses on the minimum (maximum) of the positive (negative) eigenvalues of the GJMS operator, considering metrics of volume one within a conformal class on a compact manifold. Specifically, we investigate the existence (or non-existence) of extremizers and explore their properties. This work generalizes previous results, extending the analysis of the second eigenvalue and the Yamabe operator to arbitrary orders and to GJMS operators.
Abstract. I shall outline a portrait of André Lichnerowicz (1915-1998). Professor at the Collège de France, he was a great mathematician who published in mathematical physics, as well as in differential geometry, from his thesis in 1939 until his death. He was a reformer of the teaching of mathematics in France as well as a philosopher. I shall underline his collaboration with Moshé Flato when they introduced the theory of deformation quantization. He supervised numerous students, many of whom went on to become well-known mathematicians. Ten years after his death, the Lichnerowicz prize was instituted to honor young researchers in Poisson geometry, a field he pioneered in his groundbreaking article of 1977.
Abstract. This talk will present a very broad family of scalar-tensor theories of gravity that contains a single scalar degree of freedom, in addition to the usual tensor modes. These theories, known as Degenerate Higher-Order Scalar-Tensor (DHOST) theories, include and extend traditional scalar-tensor theories as well as the so-called Horndeski theories. I will then discuss black hole solutions in these theories and their perturbations, illustrated by some particular cases.
Abstract. In 2014, Carlotto and Schoen constructed initial data sets that solve the vacuum Einstein constraints and that interpolate between any asymptotically-flat vacuum solution in a cone and Euclidean space outside a wider cone. Starting from a naive interpolation (g,K) of the two solutions to be glued, they corrected it to an exact solution that is asymptotic flat with a power-law decay slightly worse than that of (g,K). With Philippe G. LeFloch, we reached an optimal version of their gravitational shielding by proving estimates whose power-law decay is controlled by the accuracy with which (g,K) solves the constraints, even beyond harmonic decay (namely the decay rate of black hole metrics). At the harmonic decay rate, we encounter corrections in the kernel of asymptotic operators built from the linearized constraints. Our work allows for very slow decay of the metrics, in which case one must define the relative ADM energy and momentum of a pair of sufficiently close initial data sets.
Abstract. The Kerr-de Sitter metric in arbitrary dimension was proposed by Gibbons et al. as a generalization the four dimensional Kerr-de Sitter metric obtained by Carter. While the role of particular cases of Kerr-de Sitter (such as Kerr-Myers-Perry, Schwarzschild-Tangherlini, de Sitter or Minkowski) certainly play a pivotal role in gravity, the role of Kerr-de Sitter is far less clear. In this talk I will explore local geometric properties that characterize the Kerr-de Sitter metric in arbitrary dimension. The main tool will be to analyze its asymptotic data at null infinity. In particular, I will show that the simplest (non-trivial) asymptotic data gives rise to a class of spacetimes called Kerr-de Sitter-like and I will identify what makes Kerr-de Sitter special within this class. Based on this, I will present various characterization properties of the Kerr-de Sitter-like metrics in arbitrary dimension.
Abstract. The question we address in this talk is how can one formulate a (locally) well-posed initial value problem in modified theories of gravity. We review recent results including scalar-tensor and Einstein-Cartan theories before focusing on a spherically evolution problem in f(R) theory.
Abstract. I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.
Abstract. Both parts of this talk involve numerical simulations on compactified hyperboloidal slices reaching future null infinity. The first part, presenting joint work with Flavio Rossetti, focuses on the asymptotic decay of the linear wave equation on flat and hyperbolic FLRW spacetimes with a time-dependent scale factor. I will describe the setup, which allows us to recover decay rates obtained from evolutions on usual truncated Cauchy slices, and also consider solutions of a non-linear wave with self-interactions. The second part will summarize my approach using conformal compactification to free hyperboloidal evolutions of the Einstein equations. I will describe the main ingredients, present relevant results, and update on the current status towards 3D evolutions.
Abstract. We construct static and axially symmetric magnetically charged hairy black holes in the gravity-coupled Weinberg-Salam theory. Large black holes merge with the Reissner-Nordstr\”om (RN) family, while the small ones are extremal and support a hair in the form of a ring-shaped electroweak condensate carrying superconducting W-currents and up to 22% of the total magnetic charge. The extremal solutions are asymptotically RN, with a mass below the total charge, due to the negative Zeeman energy of the condensate interacting with the black hole magnetic field. Therefore, they cannot decay into RN black holes. As their charge increases, they show a phase transition, when the horizon symmetry changes from spherical to oblate. At this point, they have the mass typical for planetary size black holes of which about 11% are stored in the hair. Being obtained within a well-tested theory, our solutions are expected to be physically relevant.
— 11 au 13 juin 2025 : Conférence à Paris organisée par Philippe LeFloch, dans le cadre de la Conférence Lichnerowicz 2025
— Rencontres en ligne organisées tous les semestres
Alberto FARINA (Rouen)
Résultats de classification, théorèmes de rigidité et EDPs semi-linéaires sur les variétés riemanniennes : une approche par p-fonction
Résumé. Nous considérons les solutions d’équations elliptiques semi-linéaires critiques et sous-critiques sur des variétés riemanniennes complètes, non compactes et étudions leur classification ainsi que l’effet de leur présence sur la variété sous-jacente. Lorsque la courbure de Ricci est non-négative, nous prouvons à la fois la classification des solutions positives à l’équation critique et la rigidité de la variété ambiante. Les mêmes résultats sont établis pour les solutions de l’équation de Liouville sur les surfaces riemanniennes. Nos résultats sont obtenus en montrant qu’une fonction auxiliaire appropriée (P-fonction) est constante. Ceci implique la classification à la fois des solutions et de la variété sous-jacente. L’analyse effectuée sur la fonction P permet également de classifier les solutions non-négatives d’équations sous-critiques sur les variétés qui vérifient une inégalité de Sobolev et une condition d’intégrabilité sur la partie négative de la courbure de Ricci.
Romain GICQUAUD (Tours)
Sur le “mass aspect” des variétés asymptotiquement hyperboliques
Abstract. Il existe deux définitions de la masse pour les variétés asymptotiquement hyperboliques. La première due à P. Chrusciel et M. Herzlich est un analogue de la définition ADM classique, i.e. une intégrale sur une sphère de rayon infini d’une certaine quantité construite à partir de la métrique et de ses dérivées premières. Mais il existe une autre définition due à M.-T. Wang qui décrit la masse comme l’intégrale du premier terme non nul dans l’expansion asymptotique de la métrique (communément appelé mass aspect). Je montrerai que ce mass aspect admet une définition dans un cadre de régularité faible analogue à celui de Chrusciel-Herzlich et qu’il possède de bonne propriétés de covariance sous les changements de carte à l’infini. Ce travail est en collaboration avec Anna Sakovich (Université d’Uppsala).
Klaus KRÖNCKE (Stockholm)
Dynamical stability and instability of Poincaré Einstein manifolds
Abstract. We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and I established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics. This is joint work with Louis Yudowitz.
Benjamin MECO (Uppsala)
The generalized Jang equation in the asymptotically anti-de Sitter setting and possible applications
Abstract. The generalized Jang equation was introduced by Bray and Khuri in an attempt to prove the Penrose inequality in the setting of asymptotically Euclidean initial data sets for the Einstein equations. Since then it has appeared in a number of arguments allowing to prove geometric inequalities for initial data sets by reducing them to known inequalities for Riemannian manifolds provided that a certain geometrically motivated system of equations can be solved. We will present a novel argument along these lines that could potentially lead to a proof of the positive mass theorem for asymptotically hyperbolic initial data sets modeling constant time slices of asymptotically anti-de Sitter spacetimes. Furthermore, we will show how to construct a geometric solution of the generalized Jang equation in this setting, in the case when the dimension is less than 8 and for very general asymptotics, using methods from geometric measure theory.
Pieralberto SICBALDI (Granada)
A Schiffer-type problem in annuli and applications to Euler flows
Abstract. If on a smooth bounded domain of the plane there is a (non-constant) Neumann Laplace eigenfunction that is locally constant on the boundary, must the domain be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture. In fact, here the eigenfunction is allowed to take a different constant value on each connected component of the boundary. Many of the known rigidity properties of the original Schiffer problem are essentially preserved. In this talk I will show that the answer to such question is negative by constructing a family of nontrivial doubly connected domains with the above property. Furthermore, I will show that this implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. This talk is based on a joint work with A. Enciso, A. J. Fernández and D. Ruiz.
Caterina VÂLCU (Villetaneuse)
Solving initial data for Kaluza-Klein spacetimes
Abstract. We study the constraint equations for Einstein equations on manifolds of the form Rn+1×Tm, where Tm is a flat m-dimensional torus. Spacetimes with compact directions were introduced almost a century ago by Theodor Kaluza and Oskar Klein as an early attempt of unifying electromagnetism and general relativity in a simple, elegant way. The aim of this article is to construct initial data for the Einstein equations on manifolds of the form Rn+1× Tm, which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the conformal method to reduce the constraint equations to a system of elliptic equation and work in the near CMC (constant mean curvature) regime. The main new feature of the proof is the introduction of new weighted Sobolev spaces, adapted to the inversion of the Laplacian on product manifolds. Classical linear elliptic results need to rigorously proved in this new setting. This is joint work with Cécile Huneau.
MAIN OBJECTIVES OF THIS PROJECT
Parametrization of initial data sets for the Einstein equations. The project focuses on the global geometry of Riemannian manifolds satisfying Einstein constraints arising in general relativity. In other words, we are interested in the geometric and analytic properties of initial data sets, consisting of a spacelike hypersurface (representing a slice of “present” time) in a spacetime satisfying Einstein’s field equations (possibly coupled to matter fields). Among our main objectives, we will seek a parametrization of “all” such hypersurfaces and describe their global geometric and asymptotic properties, for instance their behavior at spacelike infinity or in the vicinity of gravitational singularities. We intend to encompass a broad variety of geometric setups, including compact as well as non-compact manifolds with, for instance, asymptotically Euclidian or asymptotically hyperbolic ends. Interestingly, many central concepts of geometry and general relativity play a role.
Seek a unification of techniques and results. Our project takes its roots in our past, namely Lichnerowicz’s pioneering work on the so-called Conformal Method, later generalized by many researchers (see below). The Project also builds upon present developments on the so-called Variational Method, first proposed by Corvino and Schoen. We also intend to contribute to shape the future in this field and seek a unification of the results and methods currently available on Einstein’s constraint equations. Despite these old and new advances on the subject, including significant contributions by members of this Project, the literature on Einstein constraints remains a collection of dispersed results and ad-hoc techniques, and still offers many outstanding open problems: definition of asymptotic invariants, rigidity properties, asymptotic behaviors, etc.
Existing and new directions. To the opinion of the members of this Project, in past years this field has not received sufficient attention (nor funding), and given all recent developments (on the evolution problem for the Einstein equations, on the study of curvature invariants in Riemann geometry, and in numerical relativity) the time is now ripe to fill this gap and push the research in geometric analysis in the proposed directions. This research Project should also open up new directions on modified theories of gravity (such as the f(R,T) theory and Kaluza-Klein theory) and numerical relativity.
RECENT CONTRIBUTIONS by the members of the project
(To be completed)
Philippe LeFloch -- CNRS DIRECTOR OF RESEARCH -- Email: contact at philippelefloch dot org
Mathematical general relativity, compressible fluids, and more 