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#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Sorbonne Université, Paris

*Organizers*

#### Philippe G. LeFloch (Sorbonne, Paris)

#### Jacques Smulevici (Sorbonne, Paris)

#### Jérémie Szeftel (Sorbonne, Paris)

**Dates of the Seminar this Fall 2018**

**September 24, October 29, November 19**

#### Monday September 24, 2018

*room 15/16-309*

#### 14h Dietrich Häfner (Grenoble)

#### Scattering for Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric and the Hawking effect

Abstract. We consider Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric which describes rotating black holes. Whereas there exists a conserved L2 norm for the Dirac field, no positive conserved quantity exists for the Klein-Gordon field, which makes the analysis more difficult for the latter. We obtain asymptotic completeness results for the Dirac field on the Kerr and for the Klein-Gordon field on the De Sitter Kerr metric. We then present a rigorous result about the Hawking effect for fermions in the setting of a collapse of a rotating charged star. This effect predicts the creation of particles by black holes.

#### 15h30 Jean-Philippe Nicolas (Brest)

#### Peeling for scalar fields on the Kerr metric

Abstract. The peeling is an asymptotic behavior of massless fields along outgoing null geodesics in asymptotically flat spacetimes, initially observed by Sachs at the beginning of the 1960’s, then reformulated in very simple terms by Penrose in 1965 using conformal geometry. The question of its genericity, especially when talking about the peeling of the Weyl tensor of an Einstein spacetime, was controversial for several decades after Penrose’s paper. For Einstein’s equations, the question is now essentially settled, but given an Einstein spacetime, it is not clear whether there is a large class of Cauchy data giving rise to solutions with a good peeling. Lionel Mason and the speaker answered the question for fields of spin 0, 1/2 and 1 on Schwarzschild’s spacetime in 2009 and 2012. We extended recently the results to linear and non linear scalar fields on the Kerr geometry in a joint work with Pham Truong Xuan. We shall recall the history of the subject, describe the principles of the approach developed with Lionel Mason and talk about the specific features of our work for Kerr metrics.

#### Monday October 29, 2018

*room 15/16-101*

#### 14h Joe Keir (Cambridge)

#### The weak null condition and the p-weighted energy method

Abstract. The Einstein equations in wave coordinates are an example of a system which does not obey Klainerman’s “null condition”. Their failure to satisfy this condition leads to many difficulties, both in Lindblad-Rodnianski’s proof of global existence and in any attempt to apply other techniques to these equations. One such technique is the “p-weighted energy method” of Dafermos- Rodnianski, which is a very powerful and robust method that can easily be adapted to understand the behavior of waves in many interesting situations, including black holes. In this talk I will explain how to modify this method to systems which only obey the “weak null condition”, including the Einstein equations. This involves adapting the p-weighted energy method, and combining it with the many of the geometric methods used by Christodoulou and Klainerman. Among other things, this allows us to enlarge the class of wave equations which are known to admit small-data global solutions, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by these systems: it is due to the formation of “shocks at infinity”.

#### 15h30 Volker Schlue (Paris)

#### Scattering from infinity for semi-linear wave equations

Abstract. I will discuss the construction of global solutions from scattering data (at null infinity) for various semi-linear wave equations on Minkowski space satisfying the (weak) null condition. I will elaborate on the proof which relies, i) on a fractional Morawetz estimate, and (ii) on the construction of suitable approximate solutions from the scattering data. Finally I will outline the application of these results to Einstein’s equations in harmonic coordinates. This is joint work with Hans Lindblad.

#### Monday November 19, 2018

*room 15/16-309*

#### 14h Adam Layne (Stockholm)

#### Stability within T2-symmetric expanding spacetimes

Abstract. We present a recently completed, non-polarized analogue of the asymptotic characterization of T2-symmetric Einstein flow solutions by P. LeFloch and J. Smulevici. We impose a far weaker condition, but obtain similar rates of decay for the normalized energy and associated quantities. Critical to this work have been novel numerical simulations which indicate that there is locally attractive behavior for those T2-symmetric solutions not subject to this weakened condition. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarized asymptotics are on one hand stable within a larger class than merely polarized solutions, but unstable within all T2-symmetric solutions.

#### 15h30 Grigorios Fournodavlos (Sorbonne)

#### Dynamics of the Einstein vacuum equations about the Schwarzschild black hole interior

Abstract. We will talk about the dynamical behavior of the Schwarzschild black hole singularity, in the context of the Einstein equations in vacuum, from the point of view of the Cauchy problem in general relativity. As it is well known, the Schwarzschild singularity is highly unstable under arbitrarily small perturbations, which makes the study of its dynamics in full generality a difficult problem. We will begin by giving an overview of the current status of the near-Schwarzschild-black hole interior problem and we will compare it to the dynamics observed near other singularity models, in black hole interiors or Big Bangs. Then we will discuss linear and non-linear partial results in the near-Schwarzschild case, both backwards and forwards in time, with and without symmetries.

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**International Conference**

#### MATHEMATICAL GENERAL RELATIVITY

#### Monday May 28 to Friday June 1rst 2018

#### Institut Henri Poincaré

11 rue Pierre et Marie Curie, Paris

**Invited Speakers**

**Spyros Alexakis**(Univ. of Toronto)**Xinliang An**(Univ. Toronto)**Lars Andersson**(Einstein Inst., Potsdam)**Stefanos Aretakis**(Princeton)**Grigorios Fournodavlos**(Univ. of Cambridge)**Dejan Gajic**(Cambridge University)**Peter Hintz**(Univ. of California, Berkeley)**Gustav Holzegel**(Imperial College, London)**Cécile Huneau**(Ecole Polytechnique, Palaiseau)**Jérémie Joudioux**(Univ. of Vienna)**Jonathan Luk**(Stanford Univ.)**Sun-jin Oh**(Korea Inst. Advanced Study)**Andrzej Rostworowski**(Univ. Krakow)**Jan Sbierski**(Oxford Univ.)**Yakov Shlapentokh-Rothman**(Princeton)**Volker Schlue**(Sorbonne Univ.)**Jérémie Szeftel**(Sorbonne Univ.)

*Schedule of the conference*

Schedule for MONDAY

- 9:00am coffee
- 9:30am
**L. Andersson** - 10:30am coffee break.
- 11:00am
**S. Alexakis** - 2:30pm
**P. Hintz** - 3:30pm coffee break
- 4:00pm
**V. Schlue**

Schedule for TUESDAY

- 9:00am: coffee
- 9:30am
**J Joudioux** - 10:30am coffee break
- 11:00am
**G. Fournodavlos** - 2:30pm
**Sun-jin Oh** - 3:30pm coffee break
- 4:00pm
**D Gajic**

Schedule for WEDNESDAY

- 9:00am coffee
- 9:30am
**G Holzegel** - 10:30am coffee break
- 11:00am
**Y. Shlapentokh-Rothman** - noon:
**C. Huneau** - 7:00pm
**Reception at Sorbonne Univ.**Central Tower, Jussieu campus, 4 Place Jussieu

Schedule for THURSDAY

- 9:00am coffee
- 9:30am
**S. Aretakis** - 10:30am coffee break
- 11:00am
**J. Sbierski** - 2:30pm
**A. Rostworowski** - 3:30pm coffee break
- 4:00pm
**X. An**

Schedule for FRIDAY

- 9:00am: coffee
- 9:30am
**J. Szeftel** - 10:30am coffee break
- 11:00am
**J Luk**

*Titles and attracts of the lectures*

**Spyros Alexakis**(Univ. of Toronto): Singularity formation in black hole interiors: Polarized perturbations of SchwarzschildAbstract: We consider the stability of the Schwarzschild singularity in vacuum under polarized and axially symmetric perturbations. We find that the space-like singularities persist under such perturbations, but their dynamics exhibit a great richness, consistent with the asymptotically velocity term dominated behavior. The result relies crucially on a new approach for the Einstein equations in axial symmetry. Joint work with G. Fournodavlos.**Xinliang An**(Univ. Toronto): On apparent horizon formation.Abstract:Combining both hyperbolic and elliptic techniques, we study the formation of a marginally outer trapped tube (apparent horizon) in gravitational collapse. Analytic and geometric properties of this apparent horizon will also be discussed.

**Lars Andersson**(Einstein Inst., Potsdam): Linear stability for the Kerr spacetime.Abstract: The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. Recent work of Ma, and of Dafermos, Holzegel and Rodnianski provides energy, Morawetz, and pointwise decay estimates for solutions of the Teukolsky equation. In this talk I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linearized stability for the Kerr spacetime, including energy, Morawetz, and pointwise estimates for the linearized metric. The proof relies on using a radiation gauge. This is based on ongoing joint work with Thomas Bäckdahl, Pieter Blue, and Siyuan Ma.

**Stefanos Aretakis**(Princeton): Asymptotics for the wave equation on black hole backgroundsAbstract: We will present asymptotic results for solutions to the wave equation for the full Reissner-Nostrom family of black holes. These spacetimes are spherically symmetric asymptotically flat solutions to the Einstein-Maxwell system. We will consider both the sub-extremal and the extremal cases. We will see that conservation laws on null hypersurfaces play an important role in the precise late-time asymptotics for solutions to the wave equation. For the extremal Reissner-Nordstrom the situation is more subtle given that there are two independent conservation laws (in contrast to the sub- extremal where this is only one such conservation law). We will also present a scattering theory in the extremal case which in particular allows us to construct exponentially decaying smooth solutions. This work is joint with Gajic (Cambridge) and Angelopoulos (UCLA).

**Grigorios Fournodavlos**(Univ. of Cambridge): On ‘hard stars’ in general relativityAbstract: After a brief review of the classical results on gravitational collapse in spherical symmetry, from the Oppenheimer-Snyder model (1939) to Christodoulou’s two-phase model (1995), I will discuss one possible end state in the latter model: hard stars. These are idealized models of neutron stars. I will present a variational characterization and discuss its relevance to the orbital stability problem in spherical symmetry. Various obstacles to a global in time result are outlined, in particular the absence of a dispersion mechanism, the trapped surface formation scenario due to reflecting boundary conditions (cf. AdS-scalar field) and the possibility of phase transitions within the two phase model to avoid Rayleigh-Taylor instabilities. This is a joint work with Volker Schlue.

**Dejan Gajic**(Cambridge University): Conservation laws and late-time tails of waves on Schwarzschild for all angular momenta

Abstract: In 1972, Price suggested that inverse polynomial tails should be present in the late-time behaviour of scalar fields on Schwarzschild black holes with fixed angular momentum and the decay rates should depend in a precise manner on the angular momentum. In the decades since, many features of these tails have been explored both numerically and heuristically. The presence of polynomial tails along event horizons has important implications for the nature of singularities inside dynamical black holes. In this talk I will discuss work done in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of these polynomial late-time tails in Schwarzschild spacetimes. I will give a sketch of how the decay rates of Price can be derived using only physical space methods and how the coefficients in the late-time asymptotics of the scalar field are related to the existence of conservation laws.**Peter Hintz**(Univ. of California, Berkeley): Global stability problemsAbstract: I will discuss the problem of proving the stability of (families of) exact spacetimes (M,g) such as Minkowski space or the family of Kerr-de Sitter (KdS) black holes as solutions of Einstein’s vacuum equation, focussing on geometric aspects of this problem: how to compactify M for the purpose of analyzing the underlying nonlinear wave equation; how to choose a gauge to break the diffeomorphism invariance of Einstein’s equation; and the role of constraint damping.

**Gustav Holzegel**(Imperial College, London): Boundedness and Decay for Solutions to the Teukolsky Equation on slowly rotating Kerr spacetimesAbstract.Abstract: I will outline a proof (joint work with M. Dafermos and I. Rodnianski) of boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equations on a Kerr exterior background with parameters satisfying |a| << M. The estimates are obtained through natural generalisations of the higher order quantities P and \underline{P} introduced in our previous work on the linear stability of the Schwarzschild metric.

**Cécile Huneau**(Ecole Polytechnique, Palaiseau): High frequency limit for Einstein equations with U(1) symmetry.Abstract: I will present the construction of a family of solutions to vacuum Einstein equations with U(1) symmetry which consist of an arbitrary number of high frequency waves travelling in different directions. In the high frequency limit, our family of solutions converges to a solution of Einstein equations coupled to null dusts. This construction is an illustration of the so called backreaction, studied by physicists (Isaacson, Burnet, Green, Wald…) : the small scale inhomogeneities have an effect on the large scale dynamics in the form of an energy impulsion tensor in the right-hand side of Einstein equations. This is a joint work with Jonathan Luk (Stanford).

**Jérémie Joudioux**(Univ. of Vienna): The vector-field method for the transport equation with application to the Einstein-Vlasov system.Abstract: The vector-field method, developed by Klainerman, was a key tool to understand the global existence of solutions to quasi-linear wave equations. In a series of work in collaboration with D. Fajman (Vienna), and J. Smulevici (Orsay), the vector- field method is extended to the relativistic transport equation where it is used to derive decay estimates for velocity averages for solutions to the relativistic Vlasov equation. An important application of this method is the proof of the stability of Minkowski space as a solution to the Einstein-Vlasov system. I will present in this talk this commutator technique for the transport equation, and describe the decay estimates for velocity averages, and sketch the key steps of the stability proof.Joint work with D. Fajman (Vienna) and J. Smulevici (Orsay).

**Jonathan Luk**(Stanford Univ.): The interior of extremal black holesAbstract: I will contrast the interior regions of subextremal and extremal black holes and present a recent result regarding the interior of dynamical extremal black holes for the Einstein-Maxwell-charged scalar field system in spherical symmetry. This is a joint work with Dejan Gajic.

**Sun-jin Oh**(Korea Inst. Advanced Study): Strong cosmic censorship and generic mass inflation for charged black holes in spherical symmetry.Abstract: I will first review a recent joint work with J. Luk, in which the C2-formulation of the strong cosmic censorship is proved for the Einstein-Maxwell-(real)-Scalar Field system in spherical symmetry for two-ended asymptotically flat data. More precisely, it was shown that a “generic” class of data for this model gives rise to maximal future developments which are future C2-inextendible. In the second part of the talk, I will present a new, complementary theorem (also joint with J. Luk) that for a further “generic” subclass of such data, the Hawking mass blows up identically along the Cauchy horizon. This result confirms, rigorously and unconditionally, the mass inflation scenario of Poisson-Israel and Dafermos for the model at hand.

**Andrzej Rostworowski**(Univ. Krakow): New insights into nonlinear perturbations of vacuum spacetimes.Abstract: I will present a systematic and robust approach to nonlinear gravitational perturbations of maximally symmetric black holes. In particular, I will show that at each order of perturbation expansion, the system of perturbative Einstein equations can be reduced to two (for each gravitational mode in 3+1 dimensions on which the study will be focused) scalar wave equations, and the metric perturbations can be explicitly obtained, once the solutions to these scalar wave equations are known. That is, this approach extends the field of gravitational

perturbations of black holes, initiated in the seminal Regge & Wheeler 57′ paper beyond linear order. The talk will be partially based on a recent work Phys. Rev. D96, 124026 (2017).**Jan Sbierski**(Oxford Univ.): On the unique evolution of solutions to wave equationsAbstract: The well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, the time-evolution of globally hyperbolic solutions is unique. This talk investigates whether the same results hold for quasilinear wave equations defined on a fixed background. We first present an example of a quasilinear wave equation for which unique evolution of smooth globally hyperbolic solutions in fact fails and contrast this case with the Einstein equations. We then proceed by presenting conditions which guarantee unique evolution. This talk is based on joint work with Felicity Eperon and Harvey Reall.

**Yakov Shlapentokh-Rothman**(Princeton Univ.): The asymptotically self-similar regime for the Einstein vacuum equations.Abstract: We will dynamically construct singular solutions to the Einstein vacuum equations which are asymptotically self-similar in that successive rescalings around the singularity converge to a self-similar solution. Connections both to Christodoulou’s bounded variation solutions of the spherically symmetric Einstein-scalar field system and to the ambient metric construction of Fefferman and Graham will be elaborated on. This is joint work with Igor Rodnianski.

**Volker Schlue**(Sorbonne Univ.): Expanding black hole space-times: Towards the stability of the cosmological regionAbstract: The Schwarzschild de Sitter space-time is the simplest model of a black hole in the expanding universe. I will discuss the challenges arising in the stability problem for this solution of the Einstein vacuum equations with positive cosmological constant, and focus in particular on the evolution in the expanding region beyond the cosmological horizon of the black hole. I will present a result for the decay of the conformal Weyl curvature, and discuss its relation to the existence of asymptotic degrees of freedom in this problem. Moreover, I will discuss the behavior of solutions to the Eikonal equation in de Sitter, and questions related to the definition of asymptotic quantities.

**Jérémie Szeftel**(Sorbonne Univ.): The nonlinear stability of SchwarzschildAbstract: I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

*Organizers *

Philippe G. LeFloch (Paris), Jacques Smulevici (Orsay), Jérémie Szeftel (Paris)

#### Funding

**GEOWAKI**

*“The analysis of geometric non-linear wave and kinetic equations”*

Principal investigator: Jacques Smulevici

**ERC Starting Grant 2016**

**EPGR**

*“The Evolution Problem in General Relativity”*

Principal investigator: Jérémie Szeftel

**ERC Consolidator Grant 2016**

- Hotel de la Tour, 19 boulevard Edgar Quinet www.hoteldelatourparis.fr
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- Hotel des Canettes, 17 rue des Canettes www.parishotelcanettes.com
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- Hotel du Brésil, 10 rue Le Goff www.bresil-paris-hotel.com
- Hotel des 3 Collèges, 16 rue Cujas www.3colleges.com

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**_________________________________________**

#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Université Pierre et Marie Curie, Paris

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jacques Smulevici (Orsay)

#### Jérémie Szeftel (Paris)

**Dates of the Seminar for this Academic year: **

**October 9, November 20, December 4, February 12, March 12**

**+ Conference from May 28 to June 1rst**

#### Monday March 12, 2018

*room 15-16 309*

#### 14h Carla Cederbaum (Tubingen)

**On foliations related to the center of mass in general relativity**

Abstract. We will discuss new developments in the analysis of asymptotic foliations by prescribed curvature in relativistic initial data sets with prescribed asymptotic decay, generalizing results by Huisken and Yau. We will relate these foliations to the definition of the center of mass of the initial data sets under consideration. The results presented are joint work with Cortier–Sakovich and with Nerz.

#### 15h30 Maxime Van de Moortel (Stanford)

**Stability and instability in spherical symmetry of Reissner-Nordström black holes for the Einstein-Maxwell-Klein-Gordon model**

Abstract. Penrose’s Strong Cosmic Censorship Conjecture is one of the central problems of Mathematical General Relativity. Its proof for the Einstein-Maxwell-Uncharged-Scalar-Field (EMSF) model in spherical symmetry relies on the formation of a Cauchy horizon that is C0 regular but C2 singular for generic Cauchy data. EMSF model however only admits two-ended black holes, unlike its charged analogue that allow for one-ended black holes, relevant to the study of charged gravitational collapse in spherical symmetry. In this talk I will present my work about spherically symmetric charged and massive scalar fields on black holes. This includes a study of the black hole interior, that relates the behavior of fields on the event horizon to the formation of a C0 regular and C2 singular Cauchy horizon. I will also mention my more recent work on the black hole exterior stability, for weakly charged massless scalar fields.

#### Monday February 12, 2018

*room 15-16 309*

#### 14h Shadi Tahvildar-Zadeh (Rutgers)

**General relativity at the subatomic scale**

Abstract. The idea that General Relativity (GR) may have something to say about the subatomic world is about as old as GR itself, but very few physicists have taken it seriously, and little is known rigorously about it. In this talk I use the problem of the “general- relativistic Dirac spectrum of Hydrogen” to convey the conceptual and technical issues one is up against, and survey recent results obtained in collaboration with my colleague Michael Kiessling and by some of our students and postdocs.

#### 15h30 Thomas W. Johnson (Cambridge)

Abstract. I shall discuss the linear stability of the Schwarzschild family of black holes as solutions to the Einstein vacuum equations when expressed in a generalised wave gauge, a result which complements the recent work of Dafermos, Holzegel and Rodnianski in a similar vein as the pioneering result of Lindblad and Rodnianski complemented the monumental achievement of Christodoulou and Klainerman in establishing the global nonlinear stability of the Minkowski space. The proof relies on classical insights regarding the linearised Einstein equations about the Schwarzschild family, in particular the decoupling of certain gauge-invariant scalars into the Regge—Wheeler and Zerilli equations, and recent advances for the linear wave equation on the Schwarzschild exterior, both of which shall be reviewed.

#### Monday December 4, 2017

*room 16-26 113*

#### 14h Siyuan Ma (Potsdam)

**On Maxwell field and linearized gravity in Kerr**

Abstract. I will consider both Maxwell field and linearized gravity on Kerr backgrounds, and present recent results in obtaining energy and Morawetz estimates for the extreme Newman-Penrose components.

#### 15h30 Claudio Paganini (Potsdam)

**Mode stability on the real axis**

Abstract. I will discuss a generalization of the mode stability result of Whiting (1989) for the Teukolsky equation for the case of real frequencies. The main result states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish.

#### Monday November 20, 2017

*room 16-26 113 *

#### 14h Frederico Pasqualotto (Princeton)

**Nonlinear stability for the Maxwell–Born–Infeld system on a Schwarzschild background**

Abstract. The Maxwell–Born–Infeld (MBI) theory is a hyperbolic system of PDEs which describes nonlinear electromagnetism. Due to its tensorial and quasilinear nature, this system can be seen as a nonlinear model problem to study the stability properties of solutions to the Einstein vacuum equations. In this talk, I will present a nonlinear stability result for the MBI system on a fixed Schwarzschild background, when the initial data are constrained to be small. A crucial element of the proof is the observation that some null components of the MBI field satisfy “good” Fackerell–Ipser equations, as in the linear Maxwell case. However, in the MBI case, these equations are coupled through cubic nonlinear right hand sides, which contain all components of the MBI field. In order to resolve the coupling, we prove high-order energy decay and, subsequently, pointwise decay for all the components of the MBI field. This is achieved through the application of many ideas developed in recent years, regarding the decay of linear fields.

#### 15h30 Volker Schlue (Paris)

**On `hard stars’ in general relativity**

Abstract. I will review the classical picture of gravitational collapse in spherical symmetry, from the Oppenheimer-Snyder model (1939) to Christodoulou’s two phase model (1995). I will then turn to the possible end states of gravitational collapse, in particular discuss non-trivial static solutions to the two-phase model, which are idealized models of neutron stars. The main results concern a variational characterization of hard stars, and I will outline their relevance for the orbital stability problem of neutron stars in spherical symmetry. I hope to conclude with a discussion of the various remaining problems that have to be overcome for a global in time result, in particular related to possible phase transitions in this model.

#### Monday October 9, 2017

*room 15/16-309*

#### 14h Daniel Monclair (Orsay)

**Attractors in spacetimes and time functions**

Abstract. A time function on a Lorentzian manifold is a continuous real valued function which is increasing along all future directed causal curves. A result of Hawking states that the existence of a time function is equivalent to stable causality, i.e. the impossibility of generating timelike loops even after small perturbations of the metric. We will discuss a construction of time functions which is quite different from Hawking’s construction, in the sense that it produces functions that still have interesting properties for non stably causal spacetimes (while Hawking’s time functions fail to be continuous without stable causality). Our approach is based on a notion of attracting sets in spacetimes, following the work of Conley on Lyapunov functions.

#### 15h30 The-Cang Nguyen (Paris)

**Global dispersion of self-gravitating massive matter in spherical symmetry**

Abstract. We study massive matter fields evolving under their own gravitational field and we generalize results established by Christodoulou for the spherically symmetric evolution of massless scalar fields governed by the Einstein equations. We encompass both Einstein’s theory and the f(R)-theory of modified gravity defined from a generalized Hilbert-Einstein functional depending on a nonlinear function f(R) of the spacetime scalar curvature R. This is a joint work with P.G. LeFloch and F. Mena.

###### _____________________________**_______________****_________________________________________****_______________**_________

**_________________________________________**

#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Université Pierre et Marie Curie, Paris

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jacques Smulevici (Orsay)

#### Jérémie Szeftel (Paris)

**Dates of the Seminar: **

**January 30, February 27, March 20, April 10, May 22, June 6, June 19, July 4**

#### Monday January 30, 2017

*room 15/25-104*

#### 14h Georgios Moschidis (Princeton, USA)

**The scalar wave equation on general asymptotically flat spacetimes. Stability and instability results**

Abstract. We will examine how certain geometric conditions on general asymptotically flat spacetimes (M,g) are related to stability or instability properties of solutions to the scalar wave equation on M. First, in the case when (M,g) possesses an event horizon with positive surface gravity and an ergo-region which is sufficiently small in terms of the near-horizon geometry, we will prove a logarithmic decay result for solutions to the wave equation, provided a uniform energy boundedness estimate holds. This result, applicable also in the absence of a horizon and an ergo-region, generalizes a result of Burq for the wave equation on the complement of an arbitrary compact obstacle in flat space. We will then apply the methods developed for the proof of this result in obtaining a rigorous proof of Friedman’s ergosphere instability for scalar waves in the case when (M,g) possesses an ergo-region and no event horizon.

#### 15h30 Xavier Lachaume (Tours)

**The constraint equations of scalar tensor and Lovelock theories**

Abstract. The ADM decomposition is the projection of the Einstein field equations on a spacelike foliation of the spacetime. It gives the constraint equations that must necessarily be satisfied by a riemannian metric and a 2-form to be the initial data of an Einstein spacetime. In this talk, we shall introduce some modified gravity theories: the scalar-tensor and Lovelock theories, and see how they behave under the ADM decomposition. We shall examine their constraint equations, and solve them in particular cases. This involves the study of whether a certain function of the elementary symmetric polynomials is concave or not.

#### Monday February 27, 2017

*room 15/16-309*

#### 14h Mokdad Mokdad (Brest)

**Conformal scattering for Maxwell fields on Reissner-Nordstrøm-de Sitter spacetimes**

Abstract. The Reissner-Nordstrøm-de Sitter spacetime models a spherically symmetric charged and non-rotating black hole in the presence of a positive cosmological constant. Depending on the parameters of the metric, this spacetime can have up tothree distinct event horizons. In the case of three horizons, we develop a scattering theory for Maxwell fields using the conformal geometric approach initiated by Penrose and Friedlander and referred to as conformal scattering. The idea is that a complete scattering theory is equivalent to the well-posedness of the Goursat problem (characteristic Cauchy problem) at the null boundary of the conformal manifold. Decay estimates obtained by geometric energy inequalities are essential tools for closing the estimates that allow the construction of the scattering operator : their role is to prove that energy cannot accumulate at timelike infinity, which can be understood as a weak form of Huygens’ principle.

#### 15h30 Annalaura Stingo (Paris 13)

**Global existence and asymptotic behavior for small solutions to 1D quasi-linear cubic Klein-Gordon equations**

Abstract. Let u be a solution to a quasi-linear cubic Klein-Gordon equation, with smooth, small Cauchy data. It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove that the result holds even when data are not compactly supported but only decay like 1/r at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one-term asymptotic expansion for the solutions and establish a modified scattering property.

#### Monday March 20, 2017

*room 15/16-309*

#### 14h Dominic Dold (Cambridge, UK)

**Exponentially growing mode solutions to the Klein-Gordon equation in Kerr-AdS spacetimes**

Abstract. We consider solutions to the Klein-Gordon equation in the black hole exterior of Kerr-AdS spacetimes. It is known that, if the spacetime parameters satisfy the Hawking-Reall bound, solutions (with Dirichlet boundary conditions at infinity) decay logarithmically. We shall present our recent result of the existence of exponentially growing mode solutions in the parameter range where the Hawking-Reall bound is violated. We will discuss various boundary conditions at infinity.

#### Monday April 10, 2017

*room 15/25-101*

#### 14h Bruno Premoselli (Bruxelles)

**Instability of focusing initial data sets in high dimensions**

Abstract. We will investigate blow-up properties for a class of initial data sets for the Einstein equations obtained from the conformal method in a scalar-field theory. In dimensions larger than 6, and when some stability conditions on the physics data are not satisfied, we will show that the conformal method produces blowing-up families of initial data sets. The proof of this result combines constructive variational methods with a priori asymptotic analysis blow-up techniques.

#### Monday May 22, 2017

*exceptionally taking place at IHES*

*and co-organized with S. Klainerman (Princeton)*

#### 14h Jan Sbierski (Cambridge, UK)

**The inextendibility of the Schwarzschild spacetime as a Lorentzian manifold with a continuous metric**

Abstract. The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this talk I will describe how one proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture in general relativity.

#### 15h30 Grigorios Fournodavlos (Cambridge, UK)

**Dynamics of the Einstein equations near a Schwarzschild singularity**

Abstract. We will discuss dynamical properties of the Schwarzschild interior, backwards and forwards (in time) with respect to the initial value problem for the Einstein vacuum equations.

#### Tuesday June 6, 2017

*room 15/16-309*

#### 14h Dejan Gajic (London)

**Precise asymptotics for the wave equation on stationary, asymptotically flat spacetimes**

Abstract. The late-time behaviour of solutions to the wave equation on a large class of asymptotically flat spacetimes does not conform to the strong Huygens principle. Instead, it is governed by polynomially decaying “tails”, as first discovered heuristically by Price. Their presence plays an important role in the study of singularities in black hole interiors. I will discuss a method for proving the precise leading-order asymptotics for the wave equation on these spacetimes and in the process I will introduce new energy decay estimates to obtain sharp decay rates that go beyond those obtained via traditional vector field methods. This talk is based on joint work with Yannis Angelopoulos and Stefanos Aretakis.

#### 15h30 Cécile Huneau (Grenoble)

**High frequency back reaction for the Einstein equations under polarized U(1) symmetry**

Abstract. It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition. The known exemples of this phenomena are constructed under symmetry reductions which involve two Killing fields and lead to an energy impulsion tensor which consists in at most two dust propagating in null directions. In this talk, I will explain our construction, under a symmetry reduction involving one Killing field, which lead to an energy impulsion tensor consisting in N dust fields propagating in arbitrary null directions. This is a joint work with Jonathan Luk (Stanford).

#### Monday June 19, 2017

*room 15/16-309*

#### 14h Elena Giorgi (Columbia)

**On the rigidity problem of black holes in general relativity**

Abstract. The rigidity problem in General Relativity consists in showing that an (electro)vacuum, asymptotically flat and stationary spacetime is isometric to Kerr(-Newman). The problem was solved for analytic manifolds by Hawking in the so called “no-hair theorem”. We overview the known results related to the rigidity problem for Ricci flat smooth manifolds. In the non-analytic case, Ionescu-Klainerman extended the Hawking Killing field along the horizon to the outer domain of dependence. This was done through a unique continuation procedure, relying on Carleman estimates. We generalize the result to the case of Einstein equation coupled with Maxwell equations. Finally, we summarize what is known in the case of degenerate horizons, which corresponds to the extremal Kerr.

#### Monday July 3, 2017

*exceptionally taking place at IHES*

*and co-organized with S. Klainerman (Princeton)*

#### 14h Steffen Aksteiner (Potsdam)

**From operator identities to symmetry operators**

Abstract. The hidden symmetry of the Kerr spacetime, encoded in its pair of conformal Killing-Yano tensors, implies hidden symmetries for various test fields on such a background. Starting from certain natural operator identities we derive two such symmetries of the linearized Einstein operator. The first one is of differential order four and the relation to the classical theory of Debye potentials as well as to the Chandrasekhar transformation will be explained. The second one is of differential order six and related to the separability of an integrability condition to the linearized Einstein equations — the Teukolsky equation. Advanced symbolic computer algebra tools for xAct were developed for this purpose and if time permits, I will give an overview on the current status.

#### 15h30 Arick Shao (London)

**Unique continuation of waves on asymptotically Anti-de Sitter spacetimes**

Abstract. In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (AdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an asymptotically AdS solution of the Einstein equations uniquely determined by its data on its conformal boundary at infinity? In this presentation, we establish a key step: we prove such a unique continuation result for wave equations on fixed asymptotically AdS spacetimes. In particular, we highlight the analytic and geometric features of AdS spacetime which enable this uniqueness result, as well as obstacles preventing such a result from holding in other cases. If time permits, we will also discuss some applications of this result toward symmetry extension and rigidity theorems.

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**_________________________________________**

#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Université Pierre et Marie Curie, Paris

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jacques Smulevici (Orsay)

#### Jérémie Szeftel (Paris)

This Fall: **October 10, November 21, and December 12, 2016**

#### Monday October 10, 2016

*room 15/25-104*

#### 14h Peter Hintz (Berkeley)

**Nonlinear stability of Kerr-de Sitter black holes**

**Abstract. **In joint work with András Vasy, we recently established the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole.

#### 15h30 Stefan Czimek (Paris)

**An extension procedure for the constraint equations**

**Abstract. **In this talk we present a new extension procedure for the maximal constraint equations of general relativity, motivated by applications to the Cauchy problem. Given a small solution on the unit ball, we can extend it to an asymptotically flat global solution. The main features are that our extension procedure does not need a gluing region, preserves regularity and works in weak regularity. For the proof, we use new methods to solve the prescribed divergence equation for the second fundamental form and the prescribed scalar curvature equation for the metric. We use the under-determinedness of the constraint equations to conserve regularity.

#### Monday November 21, 2016

*room 15/16-413*

#### 14h The-Cang Nguyen (Paris)

**Progress and recent results for the conformal equations**

**Abstract. **The presentation will be divided into two parts. First, I will introduce the conformal equations and present recent results for these equations as well as questions arising naturally. In a second part, I will talk about the “half-continuity method” and explain how to use this method for giving answers to the questions posed in the first part.

#### 15h30 Volker Schlue (Paris)

**On the nonlinear stability of expanding black hole cosmologies**

#### Monday December 12, 2016

*room 15/25-102*

#### 14h Michał Wrochna (Grenoble)

**The quantum stress-energy tensor and its intricate relationship with spacetime geometry**

**Abstract. **It is widely believed that at low energies, quantum gravity should yield an effective theory described by Einstein equations with a stress-energy tensor made of averaged fluctuations of quantum fields. The construction of that stress-energy tensor is however very problematic and its intricate dependence on spacetime geometry results in highly non-linear equations that possess no qualitative theory to date. In this talk I will review this problem as a motivation for improving the construction of linear Klein-Gordon quantum fields, and discuss recent progress that allows for a better control of the dependence on the spacetime metric (partly based on joint work with Christian Gérard).

#### 15h30 Guillaume Idelon-Riton (Regensburg)

**Some results about the scattering theory for the massive Dirac fields in the Schwarzschild-Anti-de Sitter space-time**

**Abstract. **I will first give a brief presentation of the Schwarzschild-Anti-de Sitter spacetime and of some of its geometrical properties that will concern us. Then I will present the massive Dirac equation in this space-time and first study the Cauchy problem which is not completely obvious since our spacetime is not globally hyperbolic. I will then give a result concerning the asymptotic completeness for these fields. By means of a Mourre estimate, it is possible to obtain that the minimal velocity for these fields is 1. I will then show that our dynamics behaves in asymptotic regions like a transport at unit speed in the direction of the black hole. In a third part, I will study the local energy decay for these fields. First, using the existence of exponentially accurate quasi-modes, I will show a logarithmic lower bound on the local energy decay which is in accordance with the results of G. Holzegel and J. Smulevici in the Kerr-Anti-de Sitter spacetime for the Klein-Gordon fields. In order to obtain an upper bound, I will prove the existence of resonances and give some tools in order to localize them.