#### Philippe G. LeFloch (Paris)

MAIN THEMES OF THE PROGRAM

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts (Cauchy problem, cosmic censorship, asymptotic behavior). These developments have brought into focus the deep connections between the Einstein equation and other important geometric PDE’s, including the wave map equation, Yang-Mills equation, Yamabe problem, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthrough, concerning the uniqueness and stability of the Kerr black hole model, the formation of trapped surfaces, and the bounded L2 curvature problem. Specifically, the themes of mathematical interest that will be developed in the present Program and are currently most active include:

• The initial value problem for Einstein equation and the causal geometry of spacetimes with low regularity, formation of trapped surfaces
• Techniques of Lorentzian geometry: injectivity radius estimates, geometry of null cones; construction of parametrix
• Geometry of black hole spacetimes: uniqueness theorems, censorship principles
• Coupling of Einstein equation for self-gravitating matter models, weakly regular spacetimes with symmetry

General schedule for the Trimester available here

SCIENTIFIC ACTIVITIES during the Trimester

WORKSHOPS AND CONFERENCES

REGISTER HERE

Sept. 14 to 18, 2015 Summer School – INTRODUCTION TO MATHEMATICAL GENERAL RELATIVITY

List of speakers

Greg Galloway (Miami)

Gerhard Huisken (Tuebingen)

Hans Ringstrom (Stockholm)

Sept. 23 to 25, 2015  Workshop – RECENT ADVANCES IN MATHEMATICAL GENERAL RELATIVITY

List of speakers

Spyros Alexakis (Toronto)

Piotr Chrusciel (Vienna)

Joao Costa (Lisbon)

Semyon Dyatlov (Cambridge, USA)

Stefan Hollands (Cardiff)

Alexandru Ionescu (Princeton)

Lionel Mason (Oxford)

Vincent Moncrief (Yale)

Jean-Philippe Nicolas (Brest)

Harvey Reall (Cambridge, UK)

Hans Ringstrom (Stockholm)

Mu-Tao Wang (New York)

Sept. 28 to Oct. 1, 2015  Workshop – GEOMETRIC ASPECTS OF MATHEMATICAL RELATIVITY (Hold in Montpellier and organized by Marc Herzlich and Erwann Delay)

List of speakers

Piotr Chrusciel (Vienna)

Michael Eichmair (Zürich)

Mu-Tao Wang (New York)

Oct. 26 to 29, 2015  Workshop – DYNAMICS OF SELF-GRAVITATING MATTER

List of speakers

Hakan Andreasson (Gothenburg)

Thierry Barbot (Avignon)

Robert Beig (Vienna)

David Fajman (Vienna)

Marc Mars (Salamanca)

David Maxwell (Fairbanks)

Todd Oliynyk (Monash)

Volker Schlue (Toronto)

Bernd Schmidt (Potsdam)

Jared Speck (Cambridge, USA)

Eric Woolgar (Alberta)

List of speakers

Jean-Pierre Bourguignon (Bures-sur-Yvette)

Pieter Blue (Edinburgh)

Demetrios Christodoulou (Zürich & Athens)

Mihalis Dafermos (Princeton)

Thibault Damour (Bures-sur-Yvette)

(Cape Town)

Richard Hamilton (New York) *

Gustav Holzegel (London)

Jonathan Luk (Cambridge, UK)

(Oxford)

Igor Rodnianski (Princeton)

Richard Schoen (Stanford)

Jacques Smulevici (Orsay)

Jérémie Szeftel (Paris)

Robert Wald (Chicago)

Qian Wang (Oxford)

* (to be confirmed)

Dec. 14 to 16, 2015  International Conference-  RELATIVITY AND GEOMETRY – IN MEMORY OF A. LICHNEROWICZ  (Organized by Giuseppe Dito, Jean-Pierre Francoise, Paul Gauduchon, Richard Kerner, Yvette Kosmann-Schwarzbach et Daniel Sternheimer)

List of speakers

Robert Bryant (Durham)

Pierre Cartier (Gif-Sur-Yvette)

Thibault Damour (Gif-Sur-Yvette)

Nathalie Deruelle (Paris 7)

Simon Donaldson  (Stony Brook & London)

Michel Dubois-Violette  (Paris 11)

Edward Frenkel  (Berkeley)

Simone Gutt  (Bruxelles)

James Isenberg  (Eugene)

Sergiu Klainerman  (Princeton)

Maxim Kontsevich  (Gif-Sur-Yvette)

Alan Weinstein  (Berkeley)

Program coordinated by the Centre Emile Borel at IHP. Financial support provided by the Institut Henri Poincaré and the ANR Project “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15/25-326

11h Emmanuel Hebey (Cergy-Pontoise) Systèmes de Kirchhoff critiques stationnaires sur des variétés compactes

14h  Lydia Bieri (Ann Arbor) Gravitational radiation and two types of memory

Abstract.  We are believed to live on the verge of detection of gravitational waves, which are predicted by General Relativity. In order to understand gravitational radiation, we have to investigate analytic and geometric properties of corresponding solutions to the Einstein equations. Gravitational waves leave a footprint in the spacetime regions they pass, changing the manifold – and therefore displacing test masses – permanently. This is known as the memory effect. It has been believed that for the Einstein equations, being nonlinear, there exists one such effect with a small linear’ and a large nonlinear’ part. In this talk, I present some of my joint work with D. Garfinkle showing that these are in fact two different effects.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Thierry Barbot (Avignon) Surfaces polygonales fuchsiennes et espace de Teichmüller décoré

Abstract. Dans l’article “Fuchsian polyhedra in Lorentzian space-forms, Mathematische Annalen 350, 2, pp. 417-453, 2011″, F. Fillastre a montré que toute métrique euclidienne avec singularités coniques d’angles > 2 pi sur une surface compacte se réalise de manière unique comme une surface de Cauchy polygonale dans un espace-temps globalement hyperbolique localement plat radial (i.e. dont le groupe d’holonomie fixe un point de l’espace de Minkowski). Dans cet exposé, j’évoquerai le travail de L. Brunswic dans son travail de thèse sous ma direction, qui vise à reprouver ce résultat et à l’étendre au cas des surfaces polygonales dans un espace-temps localement plat mais admettant des particules massives. Le but est de montrer qu’il y a encore existence et unicité une fois prescrit la masse des particules massives (le cas régulier montré par Fillastre correspondant au cas où l’angle singulier des particules massives est 2pi). Je montrerai aussi que la situation étudiée par R. Penner dans l’article “The Decorated Teichmϋller Space of Punctured Surfaces, Commun. Math. Phys. 113, 299-339 (1987)” est un cas limite de la situation étudiée par Brunswic, et correspond au cas où les particules sont d’angle conique nul. Je montrerai aussi comment répondre positivement à la question dans le cas où il n’y a qu’une singularité.

15h30 Andrea Seppi (Pavia) Convex surfaces in (2+1)-dimensional Minkowski space

Abstract.  It is known that the hyperbolic plane admits an isometric embedding into Minkowski space; in 1983 Hanu and Nomizu first observed the existence of non-equivalent isometric embeddings, thus showing a relevant difference with the Euclidean case. In this talk, I will introduce some natural properties of a convex surface in Minkowski space, concerning causality and asymptotic behavior. I will then explain some new results (jointly with Francesco Bonsante) on the classification of constant curvature surfaces with bounded principal curvatures and on the solvability of Minkowski problem in (2+1)-dimensional Minkowski space. If time permits, I will give the main ideas of the proof and especially the relation to some type of Monge-Ampere equations.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Shiwu Yang (Cambridge) Decay properties of solutions of Maxwell Klein-Gordon equations

Abstract.  I will present some recent progress on the asymptotic behavior of global solutions to Maxwell-Klein-Gordon equations. I will show that the integrated local energy and the energy flux through the outgoing null hypersurfaces decays polynomially in the retarded time in Minkowski space with data merely bounded in some gauge invariant weighted Sobolev space. This in particular includes the case with large charge. One novelty of this work is that these decay estimates precisely capture the asymptotic properties for the non-linear fields with arbitrarily large data. If in addition that the initial data for the scalar field is sufficiently small, then we show the pointwise decay of the solutions. This result improves the previous result of Lindblad and Sterbenz in which smallness is required for both the scalar field and the Maxwell field.

15h30 Gustav Holzegel (London) Local and global dynamics in asymptotically anti de Sitter spacetimes

Abstract.  Asymptotically anti de Sitter (aAdS) spacetimes play a prominent role in theoretical physics and mathematics.  Due to the presence of a timelike hypersurface at infinity these spacetimes are not globally hyperbolic, a fact that leads to intricate initial boundary value problems when studying global dynamics of hyperbolic equations on these backgrounds. In this talk, I will present several local and global results for the massive wave equation on aAdS spacetimes (including black hole spacetimes) with emphasis on how different boundary conditions (Dirichlet, Neumann or dissipative) influence the global dynamics. In particular, I will outline a recent proof (obtained in collaboration with J. Luk, J. Smulevici and C. Warnick) of linear stability and decay for gravitational perturbations on anti de Sitter space under dissipative boundary conditions. The proof unravels an interesting trapping phenomenon near the conformal boundary which necessarily leads to a degeneration in the decay estimates. Time permitting some future applications will also be discussed.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h François Fillastre (Cergy-Pontoise) Minkowski problem in Minkowski space

Abstract. T. Barbot, F. Beguin and A. Zeghib solved a smooth Lorentzian version of the Minkowski problem in dimension (2+1). More precisely they proved that if M is a flat 3-dimensional maximal globally hyperbolic spatially compact spacetime, then there exists a unique strictly convex smooth space-like surface in M with a prescribed smooth positive Gauss curvature. We will look at this problem for any dimensions. The existence part is solved in a generalized way (a measure is prescribed rather than a function). Concerning the regularity of the solution, the 2+1 case is specific. The arguments are based on tools from the geometry of convex sets. Joint work with Francesco Bonsante (Pisa).

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Bruno Premoselli (Cergy-Pontoise) Robustness of the conformal constraints in a scalar-field setting

Abstract. The constraint equations arise in the initial-value formulation of the Einstein equations. The conformal method allows one to rewrite the constraint equations into a determined system of nonlinear, supercritical, elliptic PDE’s. In this talk, we will investigate some stability properties for this elliptic system. The notion of stability under consideration, defined as the continuous dependence of the set of solutions of the conformal constraint system with respect to its coefficients, is reformulated for the conformal method. The analysis of these stability properties involves blow-up techniques concerning defects of compactness for supercritical nonlinear elliptic equations. This is a joint work with Olivier Druet.

15h30 Christophe Bavard (Bordeaux) Points conjugués des tores lorentziens

Abstract. Les points conjugués jouent un rôle important dans l’étude des variétés riemanniennes et lorentziennes, en particulier pour l’étude du rayon d’injectivité. Dans le cadre riemannien, l’absence de points conjugués impose des contraintes assez fortes sur la topologie de la variété, et parfois même sur sa géométrie. Ainsi, un résultat de Hopf (1948), généralisé par Burago et Ivanov (1994), affirme qu’un tore riemannien sans points conjugués est nécessairement plat. Dans cet exposé, je montrerai l’existence de métriques sans points conjugués dans toute composante connexe de l’espace des métriques lorentziennes sur le tore de dimension 2 ; cela prouve en particulier l’existence de tores lorentziens sans points conjugués et non plats. Il s’agit d’un travail conjoint avec Pierre Mounoud.

#### Workshop from January 20 to 23, 2015

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts, including the Cauchy problem, cosmic censorship, and asymptotic behavior. These developments have brought into focus the deep connections between the Einstein equation and other important geometric partial differential equations, including the wave map equation, Yang-Mills equation, Yamabe equation, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthroughs concerning the uniqueness and stability of black hole models, the formation of trapped surfaces, and the bounded L2 curvature problem. The themes of mathematical interest that will be particularly developed in the present Program include the formation of trapped surfaces and the nonlinear interaction of gravitational waves. The new results are based on a vast extension of the earlier technique by Christodoulou and Klainerman establishing the nonlinear stability of the Minkowski space. This Program will be an excellent place in order to present the recent breakthrough on the bounded L2 curvature problem for the Einstein equation, which currently provides the lower regularity theory for the initial value problem, as well as the recently developed theory of weakly regular Einstein spacetimes with distributional curvature.

Long-term participants

Michael Anderson (Stony Brook)

Piotr Chrusciel (Vienna)

Mihalis Dafermos (Princeton)

Cécile Huneau (Paris)

Alexandru D. Ionescu (Princeton)

James Isenberg (Eugene)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Paris)

Jared Speck (Cambridge, USA)

Jinhua Wang (Hangzhou)

Mu-Tao Wang (New York)

Qian Wang (Oxford)

Willie Wong (Lausanne)

Speakers during the Workshop

• Tuesday January 20
• Sung-Ji Oh (Berkeley) Linear instability of the Cauchy horizon in subextremal Reissner-Nordström spacetime under scalar perturbations
• Volker Schlue (Toronto) Stationarity of time-periodic vacuum spacetimes
• Alexandru D. Ionescu (Princeton) The Euler–Maxwell system for electrons: global solutions in 2D
• Joachim Krieger (Lausanne) Concentration-compactness for the critical Maxwell-Klein-Gordon equation
• Wednesday January 21
• Xianliang An (Piscataway) Two results on formation of trapped surfaces
• Tahvildar-Zadeh (Piscataway) The Dirac electron and the Kerr-Newman spacetime
• Mihalis Dafermos (Princeton)
• Jim Isenberg (Eugene) Asymptotically hyperbolic shear-free solutions of the Einstein constraint equations
• Thursday January 22
• Cécile Huneau (Paris) Stability in exponential time of Minkowski
• Jacques Smulevici (Orsay) Vector field methods for transport equations with applications to the Vlasov-Poisson system
• Mu-Tao Wang (New York) Quasi-local angular momentum and the limit at infinity
• Spyros Alexakis (Toronto) The Penrose inequality for perturbations of the Schwarzschild exterior
• Friday January 23
• Mihai Tohaneanu (Statesboro) Pointwise decay for the Maxwell system on black holes
• Qian Wang (Oxford)
• Peter Blue (Edinburgh) Revisiting decay of fields outside a Schwarzschild black hole
• Philippe G. LeFloch (Paris) Weak solutions to the Einstein equations in spherical or T2 symmetry

Attendees List

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–104

14h Qian Wang (Oxford)  A geometric approach to the sharp local well-posedness theory for quasilinear wave equations

Abstract. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces Hs with s>2+(2-\sqrt 3)/2 for general quasilinear wave equation in (1+3) spacetime by him and Rodnianski. Via this approach they obtained the local well-posedness with s>2 for (1+3) vacuum Einstein equations. A proof of the sharp H2+ local well-posedness result for general quasilinear wave equation was provided by Smith and Tataru by constructing a parametrix using wave packets. The difficulty of the problem is that one has to face the major hurdle caused by the Ricci tensor of the metric for the quasilinear wave equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in (1+3) spacetime by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

15h30 Jonathan Luk (Cambridge, UK) Stability of the Kerr Cauchy horizon

Abstract. The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25-321

14h Arick Shao (Imperial College) Unique continuation from infinity for linear waves

Abstract. We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects. This is joint work with Spyros Alexakis and Volker Schlue.

15h30 Claude Warnick (Warwick)  Dynamics in anti-de Sitter spacetimes

Abstract. When solving Einstein’s equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike hypersurface. Questions of local well-posedness and global stability are sensitive to the choices of boundary conditions. I will present recent work exploring the effects of non-trivial boundary data for the asymptotically AdS initial-boundary value problem, including a recent result in collaboration with Holzegel. I will also outline some interesting open problems in the area.

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 1525-321

14h Jan Sbierski (Cambridge, UK) A Zorn-free proof of the existence of a maximal Cauchy development for the Einstein equations

Abstract. In 1969, Choquet-Bruhat and Geroch showed that there exists a unique maximal Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In particular, their proof ensures the existence of the maximal development without actually constructing it. In this talk, we present a proof of the existence of a maximal Cauchy development that avoids the use of Zorn’s lemma and, moreover, provides an explicit construction of the maximal development.

15h15 Sergiu Klainerman (Princeton)  Remarks on the stability of the Kerr solution in axial symmetry

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-321

Abstract. We will show that some curvature operators of Ricci (or Einstein) type are locally invertible, in some weighted Sobolev spaces on Rn, near the euclidian metric. In the smooth case, we then deduce that the image of some Riemann-Christoffel type operators are smooth submanifolds in the neighborhood of the Euclidian metric.

Abstract. The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-103

###### 14h  Florian Beyer  (Dunedin)  Graceful exit from inflation for minimally coupled Bianchi A scalar field models

Abstract. We consider the dynamics of Bianchi A scalar field models which undergo inflation. The main question is under which conditions does inflation come to an end and is succeeded by a decelerated epoch. This so-called ‘graceful exit’ from inflation is an important ingredient in the standard model of cosmology, but is, at this stage, only understood for restricted classes of solutions. We present new results obtained by a combination of analytical and numerical techniques.

###### 15h30  Cécile Huneau (ENS, Paris)Vacuum constraint equations for asymptotically flat space-times with a translational Killing field

Abstract. In the presence of a space-like translational Killing field, vacuum Einstein equations in 3+1 dimensions reduces to 2+1 Einstein equations with a scalar field. Minkowski space-time is a trivial solution of vacuum Einstein equation with a translational Killing field. A natural question is therefore the nonlinear stability of Minkowski solution in this setting. A first step in addressing this problem is the study of the constraint equations. The main difficulty in that case is due to the delicate inversion of the Laplacian. In particular, we have to work in the non constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.

### of Liquid-Vapor Flows

#### Thursday February 27, 2014 at 1:00pm

____________________________

### Subway station: Jussieu

#### Schedule and abstracts here !

____________________________

INVITED SPEAKERS

#### ____________________________

CONTRIBUTING SPEAKERS

#### ____________________________

Main organizer

Philippe G. LeFloch (Paris)

Co-organizers

Benjamin Boutin (Rennes)

Frédéric Coquel (Palaiseau)

#### ____________________________

PRACTICAL INFORMATIONS

How to come to the Laboratoire Jacques-Louis Lions

Hotels near the University Pierre et Marie Curie

#### ____________________________

EARLIER WORKSHOPS “Micro-Macro Modeling and Simulation of Liquid-Vapor Flows”

Eight Workshop, Berlin, February 2013

Seventh Workshop, Paris, February 2012

Sixth Workshop, Stuttgart, January 2011

Fifth Workshop, Strasbourg, April 2010

Fourth Workshop, Aachen, February 2009

Third Workshop, Strasbourg, January 2008

Second Workshop, Bordeaux, November 2007

Opening Workshop, Kirchzarten, November 2005

## Nonlinear Wave Equations at IHP

### Institut Henri Poincaré, Paris

Schedule available here

Further informations available here

Poster of the conference  here

INVITED SPEAKERS

Stefanos Aretakis (Princeton)

Nicolas Burq (Paris-Sud)

Pieter Blue (Edinburgh)

Mihalis Dafermos (Princeton)

Jean Marc Delort (Paris-Nord)

Gustav Holzegel (London)

Alexandru Ionescu (Princeton)

Joachim Krieger (EPFL)

Jonathan Luk (UPenn)

Franck Merle (Cergy & IHES)

Sung-Jin Oh (Princeton)

Fabrice Planchon (Nice)

Pierre Raphael (Nice)

Igor Rodnianski (MIT)

Chung-Tse Arick Shao (Toronto)

Jacques Smulevici (Paris-Sud)

Jacob Sterbenz (San Diego)

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-103

###### 14h  Florian Beyer  (Dunedin)  Asymptotics and conformal structures of solutions to Einstein’s field equations

Abstract. Roger Penrose’s idea that the essential information about the asymptotics of solutions of the Einstein’s field equations is contained in the conformal structure and the associated conformal boundary has led to astonishing successes. In his original work, he provided several examples which made the importance of his idea evident. However, the question whether general solutions of Einstein’s field equations are compatible with this proposal remained unanswered. Motived by this, Helmut Friedrich has initiated a research programme to tackle this problem based on his so-called conformal field equations. In this talk I report on the status of this work and some of Friedrich’s results, but also on joint work with  collaborators at the University of Otago.

###### 15h30  Julien Cortier (IHES, Bures-sur-Yvette)  On the mass of asymptotically hyperbolic manifolds

Abstract. By analogy with the ADM mass of asymptotically Euclidean manifolds, a set of global charges can be defined for asymptotically hyperbolic manifolds. We will review their various definitions and , in particular, focus on the notion of “mass aspect” tensor, which gives rise to the  energy-momentum vector and arises  in the hyperbolic formulation of the positive mass theorem. We will compute these quantities for examples such that the Schwarzschild-anti de Sitter metrics, and we will present a family of counter-examples with “non-positive” mass when completeness is not assumed.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

#### .

###### 11h (Room  15-25- 104)  Sergiu Klainerman (Princeton)   On  the formation of trapped surfaces

Abstract. I will talk about a new result obtained in collaboration with J. Luk and I. Rodnianski in which we relax significantly Christodoulou’s main condition for the formation of trapped surfaces in vacuum.

###### 14h (Room 15-25-326)Chung-Tse Arick Shao (Toronto)   Null cones to infinity, curvature flux, and Bondi mass

Abstract. In general relativity, the Bondi mass in an asymptotically flat spacetime represents, roughly, the mass remaining in the system after some has radiated away. In this talk, we make sense of and control the Bondi mass for a single null cone in an Einstein-vacuum spacetime under minimal assumptions. In terms of regularity, we assume only small weighted curvature flux along the null cone and small data on an initial sphere of the cone. Furthermore, we make no global assumptions on the spacetime, as all our conditions deal only with the single null cone under consideration. This work is joint with S. Alexakis.

###### 15h30  (Room 15-25-326)Gustav Holzegel  (Princeton) Existence of dynamical vacuum black holes

Abstract. This is joint work with Mihalis Dafermos and Igor Rodnianski. We prove the existence of a large class of non-stationary vacuum black holes whose exterior geometry asymptotes in time to a fixed Schwarzschild or Kerr metric. The spacetimes are constructed by solving a backwards scattering problem for the vacuum Einstein equations with characteristic data prescribed on the horizon and at null infinity. The data admits the full functional degrees of freedom to specify data for the Einstein equations. An essential feature of the construction is that the solutions converge to stationarity exponentially fast with their decay rate intimately related to the surface gravity of the horizon and hence to the strength of the celebrated redshift effect which, in our backwards construction, is seen as a blueshift.

## Mathematical General Relativity

### Speaker

###### 11h15 –  José A. Font  (Valencia)  Simulations of neutron star mergers and black hole-torus systems

Abstract. Merging binary neutron stars are among the strongest sources of gravitational waves and have features compatible with the events producing short–hard gamma-ray bursts. Numerical relativity has reached a stage where a complete description of the inspiral, merger and post-merger phases of the late evolution of binary neutron star systems is possible. This talk presents an overview of numerical relativity simulations of binary neutron star mergers and the evolution of the resulting black hole–torus systems. Such numerical work is based upon a basic theoretical framework which comprises the Einstein’s equations for the gravitational field and the hydrodynamics equations for the evolution of the matter fields. The most well-established formulations for both systems of equations are briefly discussed, along with the numerical methods best suited for their numerical solution, specifically high-order finite-differencing for the case of the gravitational field equations and high-resolution shock-capturing schemes for the case of the relativistic Euler equations. A number of recent results are reviewed, namely the outcome of the merger depending on the initial total mass and equation of state of the binary, as well as the post-merger evolution phase once a black hole–torus system is produced. Such system has been shown to be subject to non-axisymmetric instabilities leading to the emission of large amplitude gravitational waves.

## Mathematical General Relativity

### Speakers

###### 10h – Miguel Sánchez Caja (Granada) Recent interrelated progress in Lorentzian, Finslerian and Riemannian geometry

Abstract.    Recently, a correspondence between the conformal structure of a class of Lorentzian manifolds (stationary spacetimes) and the geometry of a class of Finsler manifolds (Randers spaces) has been developed. This correspondence is useful in both directions. On one hand, it allows a sharp description of geometric elements on stationary spacetimes in terms of Finsler geometry. On the other hand, the geometry of spacetimes suggests, both, new geometric elements and new results, for any Finsler manifold, including the Riemannian case. Here, three levels of this correspondence will be explained: (1) Causal structure of spacetimes: properties of Finslerian distances: 0903.3501.  (2) Visibility and gravitational lensing: convexity of Finsler hypersurfaces: 1112.3892, 0911.0360. (3) Causal boundaries: Cauchy, Gromov, and Busemann boundaries in Riemannian and Finslerian settings: arXiv:1011.1154.

###### 11h30 Vladimir Matveev (Jena) Geodesic degree of mobility of Lorentzian metrics

Abstract. The degree of mobility of a metric can be defined as the dimension of the space of solutions of a certain linear PDE system of finite type whose coefficients depend on the metric, and, for a given metric, there are standard algorithms to determine it. The standard algorithms strongly depend on the metric and in most cases it is possible to find the maximal and sub-maximal values of the degree of mobility, only. I will show that the degree of mobility of a manifold is closely related to the space of parallel symmetric tensor fields on the cone over the manifold. In the case the metric is Einstein, it is essentially the tractor cone. I will use it to describe all possible values of the degree of mobility (on a simply connected manifold) for Riemannian and Lorentzian metrics. I will also consider the case when the metric is Einstein and, as a by-product, solve the classical Weyl-Petrov-Ehlers conjecture, and also show applications. Most these results are based on joint projects with  A. Fedorova and S. Rosemann.

###### 14h30 Philippe LeFloch (Paris)  Injectivity radius and canonical foliations of Einstein spacetimes

Abstract. I will discuss recent results on the local geometry of spacetimes with low regularity, when no assumption on the derivatives of the curvature tensor is made, obtained in collaboration with Bing-Long. Chen. Specifically, I will establish that, under geometric bounds on the curvature and injectivity radius, only, there exist local foliations by CMC (constant mean curvature) hypersurfaces, as well as CMC–harmonic coordinates. Importantly, these coordinates are defined in geodesic balls whose radii depend on the assumed bounds, only, and the components of the Lorentzian metric have the best possible regularity.

###### 15h00 Ghani Zeghib (Lyon) Actions on the circle and isometry groups of globally hyperbolic Lorentz surfaces (after D. Monclair)

Abstract. Let M be a globally hyperbolic spatially compact spacetime with dimension 1+1. A Cauchy surface in it is diffeomorphic to the circle and, more canonically, its family of lightlike geodesics is diffeomorphic to two copies of the circle and, under mild conditions, M embeds as an open set of the 2-torus.  The isometry group G of M acts naturally on these circles, so that G is a subgroup of Diff(S1). We will establish here that G tends to be included in PSL(2, R), the group of projective transformations of the circle S1, up to a global conjugacy by an element of the circle.

###### 15h30 Eduardo Garcia-Rio (Santiago de Compostela) Quasi-Einstein and Ricci soliton Lorentzian metrics

Abstract. Quasi-Einstein metrics are natural generalizations of Einstein metrics and gradient Ricci solitons.  Moreover, they are closely related to the existence of warped product Einstein metrics. Such metrics are defined by an overdetermined equation involving the Ricci curvature and the Hessian of a potential function. I will present some results on the geometry of Lorentzian quasi-Einstein metrics by focusing primary on those which are locally conformally flat. In this setting the Ricci curvature determines the whole curvature tensor and thus the different possibilities depend on the geometry of the level sets of the potential function: warped product metrics and pp-waves appear in a natural way.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

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###### Jean-Philippe Nicolas (Brest) Perspectives in conformal scattering

Abstract. The origins of conformal scattering are to be found in a paper by Friedlander in 1980 “Radiation fields and hyperbolic scattering theory”, in which he realized that the Lax-Phillips theory was in fact providing an interpretation of scattering theory as the well-posedness of the characteristic Cauchy problem for the conformally rescaled wave-equation on null infinity. He clearly saw that the method provided an interesting geometrical short-cut to define a scattering operator with the advantage that all the analytical structure can be recovered a posteriori. The true power of the conformal approach to scattering theory lies in its complete indifference to time dependence. This talk will review the essential features of Lax-Phillips theory and its intimate link with conformal infinity via the Radon transform and the Whittaker formula, then describe the pinciples of conformal scattering with the crucial importance of the precise resolution of the Goursat problem; we will present some results (actual scattering constructions and studies of the Goursat problem) and explain the necessary steps of the extension of the method to black hole spacetimes, which is currently under development.

###### Jérémie Szeftel (Paris) The bounded L2 curvature conjecture in general relativity

Abstract.  In order to control locally a spacetime which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need an L2 bound on the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

#### Decay for the Maxwell field outside a Kerr black hole

Abstract.  This talk will repeat some of the material from last year on the same topic (January 12, 2011) and present some new results. The goal of this talk is to prove uniform energy bounds and Morawetz (integrated decay) estimates.  In the exterior of a Kerr black hole, one of the components of the Maxwell system satisfies a wave equation with a complex potential. Trapping and the complex potential interact to provide surprisingly difficult challenges. Pseudodifferential techniques can treat a model problem with both features. However, because of the structure of the original Maxwell system, a new idea suggests classical derivatives alone should be sufficient.

#### Global structure of spherically symmetric spacetimes

Abstract. At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, an example of which is given by Einstein-Maxwell-Klein-Gordon (self-gravitating charged scalar fields). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary. In particular, for any ‘strongly tame’ Einstein-matter model, a ‘first singularity’ must emanate from a spacetime boundary to which the area-radius r extends continuously to zero.