Sergiu Klainerman

• Princeton University

This is a short survey on the black hole stability problem written in honor of Yvonne Choquet Bruhat’s 100th birthday, subject very dear to her and to which she has contributed greatly, in particular her foundational local existence result [1] and the maximal globally hyperbolic development of a given initial data set. The main focus is on the rece...

Kerr stability for small angular momentum has been proved in the series of works by Klainerman-Szeftel, Giorgi-Klainerman-Szeftel and Shen. Some of the most basic conclusions of the result, concerning various physical quantities on the future null infinity are derived in the work of Klainerman-Szeftel. Further important conclusions were later deriv...

We revisit the classical results of the formation of trapped surfaces for the Einstein vacuum equation relying on the geodesic foliation, rather than the double null foliation used in all previous results, starting with the seminal work of Christodoulou \cite{Chr1} and continued in \cite{KRodn}, \cite{An}, \cite{AnLuk}, \cite{KLR}, \cite{An1}. The...

The goal of the paper is to show that the event horizons of the spacetimes constructed in \cite{KS}, see also \cite{KS-Schw}, in the proof of the nonlinear stability of slowly rotating Kerr spacetimes $\mathcal{K}(a_0,m_0)$, are necessarily smooth null hypersurfaces. Moreover we show that the result remains true for the entire range of $|a_0|/m_0$...

This a brief introduction to the sequence of works \cite{KS:Kerr}, \cite{GKS-2022}, \cite{KS-GCM1}, \cite{KS-GCM2} and \cite{Shen} which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author woul...

This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the m...

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for |a|≪m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d...

This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e $|a|/m\ll 1$, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide complete proofs for Theorems M1 and M2 as well the curvature estimates of Theorem M8, which were stated witho...

This is our main paper in a series in which we prove the full, unconditional, nonlinear stability of the Kerr family $Kerr(a, m)$ for small angular momentum, i.e. $|a|/m\ll 1$, in the context of asymptotically flat solutions of the Einstein vacuum equations (EVE). Three papers in the series, \cite{KS-GCM1} and \cite{KS-GCM2} and \cite{GKS1} have al...

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp -weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without comm...

This chapter examines the proof for Theorem M1, deriving decay estimates for the quantity q for k ≤ k small + 20 derivatives. To this end, it uses the wave equation satisfied by q. The spacetime M is decomposed as M = (int)M u (ext)M and that u is an outgoing optical function on (ext)M while u is an ingoing optical function. The chapter relies on t...

This chapter presents the main theorem, its main conclusions, as well as a full strategy of its proof, divided in nine supporting intermediate results, Theorems M0–M8. The chapter specifies the closeness to Schwarzschild of the initial data in the context of the Characteristic Cauchy problem. The conclusions of the main theorem can be immediately e...

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To...

This chapter discusses the proof for Theorem M0, together with other first consequences of the bootstrap assumptions. The only bootstrap assumption used in the proof of Theorem M0 is the bootstrap assumption BA-D on decay for k = 0, 1 derivatives. The chapter then relies on (4.1.5) and the assumptions (4.1.1) on the initial data layer. This observa...

This chapter evaluates the proof for Theorems M4 and M5. It relies on the decay of q, α‎ and α‎ to prove the decay estimates for all the other quantities. More precisely, the chapter relies on the results of Theorems M1, M2, and M3 to prove Theorems M4 and M5. The detailed proof of Theorem M4 provides the main decay estimates in (ext)M . The proof...

One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. This book takes an important step toward solving the fundamental black hole stability problem in general...

This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in general null frames, in the context of axially symmetric polarized spacetimes. The chapter translates the null structure and null Bianchi identities as...

This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions ( u , s ) such that: on R , ( u , s ) defines an outgoing geodesic foliation as in section 2.2.4. The chapter then outlines the elliptic Hodge lemma. It also looks at the deforma...

This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first improves the bootstrap assumptions on decay estimates. The chapter then improves the bootstrap assumptions on energies and weighted energies for R and Γ‎ r...

The goal of this paper is to provide a geometric framework for analyzing the uniform decay properties of solutions to the Teukolsky equation in the fully nonlinear setting of perturbations of Kerr. It contains the first nonlinear version of the Chandrasekhar transformation introduced in the linearized setting in \cite{D-H-R-Kerr} and \cite{Ma} with...

This is a follow-up of our paper \cite{KS-Kerr1} on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of $\ell=1$ modes on a $2$-sphere embedded in a $1+3$...

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for $|a|\ll m$. The paper builds on the strategy laid out in \cite{KS} in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of \cite{KS} was the introd...

We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field), in the exterior of the domain of influence of a compact set. This complements the previous well known results, restricted to compactly supported initial conditions, based on the so called hyperboloidal method. That method takes advanta...

We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field), in the exterior of the domain of influence of a compact set. This complements the previous well known results, restricted to compactly supported initial conditions, based on the so called hyperboloidal method. That method takes advanta...

This is the main paper in a series of three in which we prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, i.e. solutions of the Einstein vacuum equations for asymptotically flat $1+3$ dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield with cl...

This paper reports on the recent proof of the bounded L 2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L 2-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $\Phi$ defined from a fixed Kerr solution $\KK(M,a)$, $0\le a < M $, with values in the two dimensional hyperbolic space $\HHH^2$. A part...

This is the main paper in a sequence in which we give a complete proof of the bounded \(L^2\) curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the \(L^2\)-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set....

This paper reports on the recent proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

We prove a black hole rigidity result for slowly rotating stationary solutions of the Einstein vacuum equations. More precisely, 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 (dependin...

We present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat everywhere else) whose future development contains a trapped surface. This extends considerably the previous result of Christodo...

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We...

This memoir contains an overview of the proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set. We note that though the resu...

We revisit the extension problem for Killing vector-fields in smooth Ricci flat manifolds, and its relevance to the black hole rigidity problem. We prove both a stronger version of the main local extension result established earlier, as well as two types of results concerning non-extendibility. In particular, we show that one can find local, statio...

Given that one of the goals of the conference is to address the issue of the unity of Mathematics, I feel emboldened to talk about a question which has kept bothering me all through my scientific career: Is there really a unified subject of Mathematics which one can call PDE? At first glance this seems easy: we may define PDE as the subject which i...

We follow up our work [On the formation of trapped surfaces, preprint (2009); arXiv:0912.5097] concerning the formation of trapped surfaces. We provide a considerable extension of our result there 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 p...

In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a t...

This is a timely book in general relativity (GR) written by one of the founders of what one might call mathematical relativity. Yvonne Choquet-Bruhat was one of the first analysts in the world, and indeed of a very select group of mathematicians, to concentrate their attention to the great mathematical challenges of the general theory of relativity...

The goal of the paper is to prove a perturbative result, concerning the uniqueness of Kerr solutions, a result which we believe will be useful in the proof of their nonlinear stability. Following the program started in A. D. Ionescu and S. Klainerman [Invent. Math. 175, No. 1, 35–102 (2009; Zbl 1182.83005)], we attempt to remove the analyticity ass...

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop.B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, th...

We provide $L^1$ estimates for a class of transport equations containing singular integral operators. 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.

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 bounde...

We prove two uniqueness theorems concerning linear wave equations; the first theorem is set 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 Kerr space-time, the hypersurface is precisely...

A fundamental conjecture in general relativity asserts that the domain of outer communication of 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 [17], Carter [4] and R...

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 paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear waves. We prove a uniform lower bound on the radius of injectivity of these null boundaries in terms of the Rieman...

We develop a geometric invariant Littlewood–Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove som...

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative settin...

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 we introduce a notion of primitive parametrix for the homogeneous wave equation for which we can prove, under very br...

This is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein-vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. Following [Kl-Ro]...

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this pap...

We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove som...

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood -Paley theory, in a noncommutative setti...

One of the central difficulties of settling the L 2-bounded curvature conjecture for the Einstein-Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be co...

We show that, under stronger asymptotic decay and regularity properties than those used in Christodoulou D and Klainerman S (1993 The Global Non Linear Stability of the Minkowski Space (Princeton Mathematical Series vol 41) (Princeton, NJ: Princeton University Press)) and Klainerman S and Nicolò F (2003 The Evolution Problem in General Relativity (...

Available from http://www.aps.org/meet/APR03/baps/abs/S4040004.html

We improve recent results of H. Bahouri and J.-Y. Chemin and of D. Tataru concerning local well-posedness theory for quasilinear wave equations. Our approach is based on the proof of the Strichartz estimates using a combination of geometric methods and harmonic analysis. The geometric component relies on and takes advantage of the nonlinear structu...

Preface * Introduction * Analytic methods in the initial value problem * Definitions and results * Estimates for the connection coefficients * Estimates for the curvature tensor * The error estimates * The initial hypersurface and the last slice * Conclusions * Bibliography * Index

We develop a geometric invariant Littlewood-Paley theory for ar- bitrary tensors of a compact dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove som...

Let the function w(p) define a foliation on ∑0. Its leaves are $$S_0 (\nu ) = \left. {\left\{ {p \in \Sigma _0 \left| {w(p) = \nu } \right.} \right.} \right\}

We consider Hodge systems of equations defined on a compact two-dimensional Riemann surface. We recall Definition 3.1.4 of Chapter 3. Definition 3.1.4 Given the 1-form ξ on S we define its Hodge dual 1$${}^*\xi _a = \in _{ab} \xi ^b .$$ Clearly *(*ξ) = -ξ. If ξis a symmetric traceless 2-tensor, we define the following left, *ξ and right, ξ*, Hodge...

Let S be a closed 2-dimensional surface embedded in a 3+1-dimensional spacetime (M, g). We assume that S has a compact filling by which we mean that there exists a Cauchy hypersurface ∑ containing S such that S is the boundary of a compact region of ∑. Let γ be the induced metric on S, $$\gamma (x,y) = g(x,y)$$ (3.1.1) for all X,Y ∈ T S, the tangen...

In this chapter we assume the spacetime K is foliated by a double null canonical foliation that satisfies the assumptions $$O \leqslant \epsilon_0 ,\,D \leqslant \epsilon_0 ,$$ (6.0.1) and we make use of the inequality proved in Theorem M7 $$R \leqslant cQ_K^{\frac{1} {2}} .$$ (6.0.2)

The goal of this chapter is to introduce the reader to the global analytic methods that play a fundamental role in the remaining chapters of the book. We start with a discussion of local and global existence results for systems of nonlinear wave equations. As we have pointed out in the previous sections, the Einstein vacuum equations can be reduced...

This chapter is devoted to the proof of Theorem M7 in terms of the fundamental quantities Q, \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} \). These quantities can be expressed, according to (3.5.1), as weighted integrals of the null components of \(\hat L_0 R,\hat L_0 R,\hat L^2 _0 R,\hat L_0 \hat L_T R and \hat L_S \hat L_T R\)a...

In the first part of this chapter we briefly present the main notions of differential geometry that we are going to use systematically throughout the book. We also describe, in some detail, results connected to the symmetry properties of Einstein spacetimes. The second part of the chapter is devoted to the introduction of the initial value problem...

It should not come as a surprise that General Relativity, the most mathematical of all our physical theories, presents an extremely fertile ground for mathematicians. From its inception General Relativity has attracted the attention of geometers and analysts; yet, due to its manifold difficulties, it has failed to produce that constant stream of re...

We review recent results, obtained in (Kl-Ro1)-(Kl-Ro3), concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. O...

This is the third and last in our series of papers concerning rough solutions of the Einstein vacuum equations expressed relative to wave coordinates. In this paper we prove an important result concerning Ricci defects of microlocalized solutions, stated and used in the proof of the crucial Asymptotics Theorem in our second paper "The causal struct...

This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques Of energy estimates and Sobolev inequalities. Following our prev...

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this pap...

Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or with connections with non-zero curvature). In this note we give some techniques to prove these e...

We give a new proof based on Fourier Transform of the classical Glassey and Strauss [6] global existence result for the 3D relativistic Vlasov-Maxwell system, under the assumption of compactly supported particle densities. Though our proof is not substantially shorter than that of [6], we believe it adds a new perspective to the problem. In particu...

We undertake a systematic review of results proved in [26, 27, 30, 31, 32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs...

this paper is to review the estimates proved in [3] and extend them to all dimensions, in particular to the harder case of space dimension 2. As in [3], the main application we have in view is to equations of Wave Maps type, namely systems of equations of the form ## I + # I JK (#)Q 0 (# J , # K ) = 0. (1) Here, # = -# 2 t +# denotes the standard D...

We undertake a systematic review of some results concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we provide a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The pap...

We extend the recent result of T. Tao [6] to wave maps defined from the Minkowski space Rn+1, n ≥ 5, to a target manifold N which possesses a “bounded parallelizable” structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces HN. General compact Riemannian manifolds can be imbedded as totally geodesic submanifold...

In this paper we review some of the recent mathematical progress concerning the initial value problem formulation of general relativity. It is not our intention, however, to give an exhaustive presentation of all recent results on this topic, but rather to discuss some of the most promising mathematical techniques, which have been advanced in conne...

The aim of the paper is to develop the Fourier Analysis techniques needed in the study of optimal well-posedness and global regularity properties of the Yang-Mills equations in Minkowski space-time R n + 1 \mathbb {R}^{n+1} , for the case of the critical dimension n = 4 n=4 . We introduce new functional spaces and prove new bilinear estimates for s...

Almost global solutions are constructed to three-dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity. © 1996 John Wiley & Sons, Inc.

Consider the Schrodinger equation iu(t) = (-DELTA + V)u, u(x, 0) = u0(x) is-an-element-of L2(R(n)), n greater-than-or-equal-to 3 The following estimates are proved: (A) If V = 0 then for any 0 less-than-or-equal-to alpha < 1/2, integral-Rn+1 (\x\2alpha-2\\D(x)\(alpha)u(t,x)\2 dxdt less-than-or-equal-to C\\u0\\2, and for alpha = 1/2, s > 1/2, integr...

Consider the following class of nonlinear wave equations $$\square {\text{u + u}}\,{\text{ = }}{\text{F(u,u',u''),}}$$ (N.K.G.), where □ = ∂ t2 - ∂ 12 - ∂ 22 -∂ 32 is the D’Alembertian of the 4-dimensional Minkowski space-time and F a smooth function of u = u(t,x) and its first and second partial derivates, vanishing, together with its first deriva...

The considered equations of compressible ideal fluid flow in appropriate nondimensional form are a hyperbolic system in four variables, related to density and fluid velocity. The Euler equations of incompressible fluid flow are a distinctly different system of four equations in four unknowns. Qualitative and quantitative properties of a specific li...

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magn...

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small-amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either...