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Cover -- Title -- Copyright -- Contents -- List of Figures -- Acknowledgments -- 1 Introduction -- 1.1 Basic notions in general relativity -- 1.1.1 Spacetime and causality -- 1.1.2 The initial value formulation for Einstein equations -- 1.1.3 Special solutions -- 1.1.4 Stability of Minkowski space -- 1.1.5 Cosmic censorship -- 1.2 Stability of Kerr conjecture -- 1.2.1 Formal mode analysis -- 1.2.2 Vectorfield method -- 1.3 Nonlinear stability of Schwarzschild under polarized perturbations -- 1.3.1 Bare-bones version of our theorem -- 1.3.2 Linear stability of the Schwarzschild spacetime

1.3.3 Main ideas in the proof of Theorem 1.6 -- 1.3.4 Beyond polarization -- 1.3.5 Note added in proof -- 1.4 Organization -- 2 Preliminaries -- 2.1 Axially symmetric polarized spacetimes -- 2.1.1 Axial symmetry -- 2.1.2 Z-frames -- 2.1.3 Axis of symmetry -- 2.1.4 Z-polarized S-surfaces -- 2.1.5 Invariant S-foliations -- 2.1.6 Schwarzschild spacetime -- 2.2 Main equations -- 2.2.1 Main equations for general S-foliations -- 2.2.2 Null Bianchi identities -- 2.2.3 Hawking mass -- 2.2.4 Outgoing geodesic foliations -- 2.2.5 Additional equations -- 2.2.6 Ingoing geodesic foliation

3.2.2 Main norms in ^(int)M -- 3.2.3 Combined norms -- 3.2.4 Initial layer norm -- 3.3 Main theorem -- 3.3.1 Smallness constants -- 3.3.2 Statement of the main theorem -- 3.4 Bootstrap assumptions and first consequences -- 3.4.1 Main bootstrap assumptions -- 3.4.2 Control of the initial data -- 3.4.3 Control of averages and of the Hawking mass -- 3.4.4 Control of coordinates system -- 3.4.5 Pointwise bounds for higher order derivatives -- 3.4.6 Construction of a second frame in ^(ext)M -- 3.5 Global null frames -- 3.5.1 Extension of frames -- 3.5.2 Construction of the first global frame

3.5.3 Construction of the second global frame -- 3.6 Proof of the main theorem -- 3.6.1 Main intermediate results -- 3.6.2 End of the proof of the main theorem -- 3.6.3 Conclusions -- 3.7 The general covariant modulation procedure -- 3.7.1 Spacetime assumptions for the GCM procedure -- 3.7.2 Deformations of surfaces -- 3.7.3 Adapted frame transformations -- 3.7.4 GCM results -- 3.7.5 Main ideas -- 3.8 Overview of the proof of Theorems M0-M8 -- 3.8.1 Discussion of Theorem M0 -- 3.8.2 Discussion of Theorem M1 -- 3.8.3 Discussion of Theorem M2 -- 3.8.4 Discussion of Theorem M3

Essential mathematical insights into one of the most important and challenging open problems in general relativity—the stability of black holes 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. In this book, Sergiu Klainerman and J�er�emie Szeftel take a first important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes—or Schwarzschild spacetimes—under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, Klainerman and Szeftel introduce a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, this book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.

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