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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.