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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:
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.
September 14, 2015 to December 18, 2015
Trimester Program at the
Centre Emile Borel
Financial support provided by Institut Henri Poincaré
and ANR Project “Mathematical General Relativity”
Organizers
Lars Andersson (Potsdam)
Sergiu Klainerman (Princeton)
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, nonlinear stability of Minkowski space with matter
General schedule for the Trimester
SCIENTIFIC ACTIVITIES during the Trimester
- Workshops and Conferences (see below)
- On the Mathematical Theory of Black Holes, Course by S. Klainerman.
- Begins on October 13 from 2pm to 4:30pm, and the following Tuesdays.
- Video of the lectures
- An Introduction to Self-Gravitating Matter, Course by P.G. LeFloch.
- Begins on October 9 from 2pm to 4:30pm, and the following Fridays.
- LECTURE NOTES for this course
- Video 1: An introduction to self-gravitating matter
- Video 2: Modified gravity and weakly regular spacetimes
- Video 3: Weak solutions to the Einstein equations
- Video 4: Weakly regular Cauchy developments
- Video 5: Self-gravitating fluids with bounded variation
- Video 6: The geometry of weakly regular spacetimes
- Video 7: Nonlinear stability of Minkowski space for massive fields
- Weekly Seminar on Mathematical General Relativity
- organized by L. Andersson, jointly with S. Klainerman, P.G. LeFloch, J. Szeftel (Paris), and A. Zeghib (Lyon).
- Begins on October 7 from 2pm to 4:30pm, and the following Wednesdays.
- Speakers in the Seminar :
- Bruno Premoselli (Cergy-Pontoise), Martin Taylor (Cambridge, UK)
- Steffen Aksteiner (Potsdam), Lars Andersson (Potsdam)
- Jesus Oliver (San Diego)
- Anne Franzen (Utrecht), Jared Wunsch (Evanston)
- Sung-Jin Oh (Berkeley), Anna Sakovich (Vienna)
- Hans Lindblad (Baltimore), Florian Beyer (Dunedin)
- VIDEOS available (courses, main conference)
- Tea break every day 3pm-3:30pm
WORKSHOPS AND CONFERENCES
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)
Shadi Tahvildar-Zadeh (Rutgers)
Eric Woolgar (Alberta)
Nov. 16 to 20, 2015 International Conference – GENERAL RELATIVITY – A Celebration of the 100th Anniversary
List of speakers Schedule and title
Jean-Pierre Bourguignon (Bures-sur-Yvette)
Demetrios Christodoulou (Zürich & Athens)
Mihalis Dafermos (Princeton)
Thibault Damour (Bures-sur-Yvette)
Georges Ellis (Cape Town)
Richard Hamilton (New York)
Gustav Holzegel (London)
Jonathan Luk (Cambridge, UK)
Roger Penrose (Oxford)
Richard Schoen (Stanford & Irvine)
Jacques Smulevici (Orsay)
Jérémie Szeftel (Paris)
Robert Wald (Chicago)
Qian Wang (Oxford)
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
Olivier Biquard (Paris 6)
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)
Charles Francès (Strasbourg)
Edward Frenkel (Berkeley)
Christian Fronsdal (Los Angeles)
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”.
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Organizers
Michael Anderson (Stony Brook)
Sergiu Klainerman (Princeton)
Philippe G. LeFloch (Paris)
Jared Speck (Cambridge, USA)
Location: Simons Center for Geometry and Physics
Date: One-month concentration period in January 2015
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
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Organizers
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday June 17, 2015
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.
