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My turn of century musings about the subject of PDE'sIn collaboration with S. Alexakis and A. Ionescu we prove that the domain of outer communications of a regular stationary vacuum is isometric to the domain of outer communications of a Kerr solution, provided that the stationary Killing vector-field T is small (depending only on suitable regularity properties of the black hole) on the bifurcation sphere. No other global restrictions are necessary.In collaboration with I. Rodnianski and J. Luk we extend considerably the result of Christodoulou on the dynamic formation of trapped surfaces in vacuum by allowing characteristic initial data which are strongly concentrated along a short outgoing null geodesic.This is the main paper in a sequence in which, in collaboration with I. Rodnianski and J. Szeftel, we give a complete proof of the conjecture according to which the time of existence of solutions to the IVP for the Einstein vacuum equations should depend only on L^2 bounds for the curvature of the initial data set and some low level conditions on its geometry.This is an attempt to compress the over 900 pages of the complete proof of the conjecture to less than 150 pages.In this paper, written in collaboration with A. Ionescu, we revisit the extension problem for Killing vector-fields in smooth Ricci flat Lorentzian manifolds and its relevance to the black hole rigidity problem. We prove two main new results, one concerning an extension criterion, extending that of Alexakis-Ionescu-Klainerman, and a second which provides a counterexample to the Hawking rigidity theorem for smooth metrics.In this paper, written in collaboration with I. Rodnianski, we provide a considerable extension of our previous result on pre-scared surfaces to allow for the formation of a surface with multiple pre-scared angular regions which, together, can cover an arbitrarily large portion of the surface. In a forthcoming paper we plan to show that once a significant part of the surface is pre-scared, it can be additionally deformed to produce a bona-fide trapped surface. The paper has appeared in Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1007-1031In collaboration with I. Rodnianski we give a simpler proof of the recent breakthrough result of D. Christodoulou concerning the formation of trapoped surfaces in vacuum We also enlarge the admissible set of initial conditions and reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments. We prove in fact another result, concerning the formation of pre-scarred surfaces, i.e surfaces whose outgoing expansion is negative in an open angular sector. The paper will appear in Acta Mathematica.In collaboration with S. Alexakis and A. Ionescu we show that smooth, asymptotically flat, non-degenerate solutions of the Einstein vacuum equations, which are suuficiently close to a Kerr solution, in a specific geometric sense, must be Kerr. We rely heavily on the methods introduced in our previous works on the subject. The paper has appeared in Comm. Math. Phys. 299 (2010), 89–127.In collaboration with S. Alexakis and A. Ionescu we prove the existence of a Hawking Killing vector-field K in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. We also prove that, if the space-time possesses an additional Killing vectorfield T, tangent to the horizon and not vanishing identically on the bifurcation sphere, then there must exist a local rotational Killing field commuting with T. Thus the space-time must be locally axially symmetric. The existence of a Hawking vector-field K, and the above mentioned axial symmetry, plays a fundamental role in the classification theory of stationary black holes. The paper has appeared in Geom. Funct. Anal. 20 (2010), 845–869.In collaboration with I. Rodnianski we give a geometric criterion for the breakdown of an Einstein vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. We make no restrictions on the size of the initial data. Has appeared in J. Amer. Math. Soc. 23 (2010) 345-382.In collaboration with A. Ionescu we prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski space-times, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern ill-posed Cauchy problems on bifurcate, characteristic hypersurfaces. In the case of the Kerr space-time, the hypersurface is precisely the event horizon of the black hole. The uniqueness theorem in this case, based on two Carleman estimates, is intimately connected to our strategy to prove uniqueness of the Kerr black holes among smooth, stationary solutions of the Einstein-vacuum equations, as formulated in our precedent paper. Has appeared in Commun. Math. Phys. 285 (2009), 873–900.

A fundamental conjecture in General Relativity asserts that the domain of outer communicationof a regular, stationary, four dimensional, vacuum black hole solution is isometrically diffeomorphic to the domain of outer communication of a Kerr black hole. So far the conjecture has been resolved, by combining results of Hawking

Carter and Robinson , under the additional hypothesis of non-degenerate horizons and \textit{real analyticity} of the space-time. In collaboration with A. Ionescu we develop a new strategy to bypass analyticitybased on a tensorial characterization of the Kerr solutions, due to Mars\cite{Ma1}, and new geometric Carleman estimates. We prove, under a technical assumption (an identity relating the Ernst potential and the Killing scalar) on the bifurcate sphere of the event horizon, that the domain of outer communication of a smooth, regular, stationary Einstein vacuum spacetime of dimension $4$ is locally isometric to the domain of outer communication of a Kerr spacetimehttp. The paper has appeared in Invent. Math. 175 (2009), 35–102.The purpose of this note, in collaboration with M. Machedon, is to give a new proof of uniqueness of the Gross- Pitaevskii hierarchy, first established in a paper by L. Erdos- B. Schlein and H.T. Yau, in a different space,based on space-time estimates similar in spirit to those of the fist generation of bilinear estimates introduced by the authors almost 15 years ago in connection to improved well posedness for nonlinear wave equation verifying the null condition. The paper has appeared in Communications in Mathematical Physics, Springer, vol. 279, no. 1, pp. 169-185, 2008 .In collaboration with I. Rodnianski provide $L^1$ estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff-Sobolev parametrices in a Lorentzian space-time verifying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest. Has appeared in J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 477–505.Togethere with I Rodnianski we investigate the regularity of past boundaries of points in regular Einstein-vacuum spacetimes. We provide conditions, compatible with bounded L^2 curvature, which are sufficient to ensure a lower bound on the radius of injectivity. Such lower bounds are essential in understanding the causal structure of spacetimes in GR. They are particularly important in the construction of an effective Kirchoff-Sobolev parametrix for solutions of wave equations on such space-times, see the paper below, which play an essential role in proving a large data break-down criterion for solutions to the Einstein-vacuum equations. The paper has appeared in J. Amer. Math. Soc. 21 (2008) 775-795Together with I. Rordnianski we construct a first order, physical space, parametrix for solutions to covariant, tensorial, wave equations on a general Lorentzian manifold. The construction is entirely geometric; that is both the parametrix and the error terms generated by it have a purely geometric interpretation. In particular, when the background Lorentzian metric satisfies the Einstein vacuum equations, the error terms, generated at some point p of the space-time, depend, roughly, only on the flux of curvature passing through the boundary of the past causal domain of p. The virtues of our specific geometric construction becomes apparent in applications to realistic problems. Though our main application is to General Relativity, which we discuss in a subsequent paper, see `` A breakdown criterion.....'' , another simpler application shown here is a gauge invariant proof of the classical regularity result of Eardley-Moncrief for the Yang-Mills equations in R1+3. The paper has appeared in Journ Hyperb. Diff. Eqts 4, Nr 3 (2007), 401-433.This is an introductory essay on partial differential equations commisioned by Princeton Compendium's to Mathematics. I beg forgiveness from the informed reader with the predictable excuse that I was assigned only 20 pages of the Compendium. I was finally able to squeeze 20 pages more from the editors. The enclosed essay is an even longer version, yet it clearly falls short of giving a realistic picture of of the depth of the subject.To settle the $L^2$ bonded curvature conjecture for the Einstein-vaccum equations one needs to prove bilinear type estimates for solutions of the homogeneous wave equation on a fixed background with $H^2$ local regularity.In this paper, in collaboration with I. Rodninaski, we introduce a notion of primitive parametrix, based on plane waves, for the homogeneous wave equation for which we can prove, under very broad assumptions, the desired bilinear estimates. Even in flat space the proof is new and`` half geometric'' in that only one of the two solutions needs to be decomposed in plane waves. Has appeared in Jounal of Hyperbolic Diff. Eqts. vol 2, Nr 2 (2205), 279-291.This is the third paper in the series with I. Rodnianski on null hypersurfaces on Einstein vacuum backgrounds verifying the finite curvature flux condition.. We prove the results of the main lemma in the first paper of the series, see above. concerning sharp trace theorems in Besov type spaces. This is the most technical of the series Now available. Has appeared in GAFA, vol 16, nr 1, 164-229.Together with I. Rodnianski we develop a geometric version,of LP-theory on Riemannian manifolds by a heat flow approach. We show how to recover the usual properties of the classical LP calculus by using only very limited information concerning the reglarity of the manifold. These results are developed for application to control spacelike hypersurfaces on Einstien vacuum backgrounds which satisfy the bounded currvature flux condition of the above paper. Has appeared in GAFA, 16, nr 1, 126-163.One of the central difficulties in connection to the L^2 bounded curvature conjecture in general relativity is to be able to control the causal structure of spacetimes with this very limited regularity. This is the first, and main, in a sequence of three papers, written in collaboration with I. Rodnianski , in which we circumvent this difficulty by showing that the geometry of null hypersurfaces of Einstein vacuum spacetimes can be controlled by the total curvature flux through the hypersurface. Has appeared in Inventiones 159 (2005), 437-529.We show that under stronger asymptotic decay properties than those used in the Christodoulou -Kl. and Kl-Nicolo global stabilty results, asymptotically flat initial data sets lead to solutions of the Einstein vacuum equations which have strong peeling properties consistent with the predictions of the conformal compactification approach of Penrose. Even the existence of spacetimes with such strong peeling properties has remained open until recently. Our methods, based on Kl-Nicolo, give a systematic picture of the relationship between asymptotic propertie of the data and the peeling properties of the corresponding developments. To appear in Quantum and Classic GravityThe third paper in the series with I. Rodnianski on rough solutions to the Einstein vacuum equations. We prove an important result which was needed in our second paper.The second paper written in collaboration with I. Rodnianski on solutions with minimal regularity for the Einstein equations.. In the second paper we concentrate on the crucial new estimates for the Eikonal equation.The first in a sequence of three papers written in collaboration with I. Rodnianski on solutions with minimal regularity for the Einstein equations.. Taking advantage oif the special structure of the equations, in wave coordinates, we get the optimal regularity result , $H^{2+\epsilon}$, which can be derived by Strichartz type estimates.A new book written in collaboration with F. Nicolo on the initial value problem in General Relativity. The main goal of the book is to revisit the proof of glaobal nonlinear stability of the Minkowski space by Christodoulou-Klainerman. We provide a new self contained proof of the main part of that result concerrning the full solution of the radiation problem in vacuum, for arbitrary assymptotically flat initial data. The proof, which is a significant modification of the arguments developed in Ch-Kl is based on a double null foliation rather than the mixed null-maximal foliation used in Ch-Kl.A paper written in collaboration with I. Rodnianski and T. Tao in which we expereiment with new proofs of bilinear estimates which have the potential to generalize to quasilinear equations.This paper, in collaboration with G. Staffilani, provides a new proof an old conditional regularity result of Glassey-StraussThis paper, in collaboration with I. Rodnianski, is an extension of the recent higher dimensional critical regularity result of T. Tao to general bounded parallelizable Riemannian target manifolds.This paper, in collaboration with I. Rodnianski, improves Tataru's well known recent regularity result for quasilinear wave equations. The paper relies on a significant improvement of the geometric technique developed earlier in a commuting vectorfield approach paper.This is a survey paper in collaboration with S. Selberg concerning optimal well posedness results for nonlinear wave equations such as Wave Maps, Maxwell-Klein-Gordon and Yang Mills. The survey provides a unified treatment and simplified proofs for some old results of Klainerman-Machedon, Klainerman-Selberg, Klainerman-Tataru.This paper marks my wholehearted return to quasilinear wave equations, after a detour of more than ten years. I show that one can recover some of the recent results of Chemin-Bahouri and Tataru by reducing the crucial dispersive inequality ( the key ingredient in the Strichartz type estimates) to L^2--L^\infty decay estimates based on energy estimates and an adaptation of the commuting vectorfields method.