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Global nonlinear stability of Schwarzschild spacetime under polarized perturbations:

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Bibliographic Details
Main Authors:	Klainerman, Sergiu 1950- (Author), Szeftel, Jérémie 1977- (Author)
Format:	Book
Language:	English
Published:	Princeton ; Oxford Princeton University Press [2020]
Series:	Annals of mathematics studies Number 210
Subjects:	Hochschulschrift
Online Access:	Inhaltsverzeichnis
Physical Description:	xi, 840 Seiten Illustrationen
ISBN:	9780691212432 9780691212425

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adam_text	Contents List of Figures xiii Acknowledgments xv 1 Introduction 1 1 1 Basic notions in general relativity 1 111 Spacetime and causality 1 112 The initial value formulation for Einstein equations 2 113 Special solutions 3 114 Stability of Minkowski space 10 115 Cosmic censorship 11 1 2 Stability of Kerr conjecture 13 121 Formal mode analysis 15 122 Vectorﬁeld method 16 1 3 Nonlinear stability of Schwarzschild under polarized perturbations 17 131 Bare-bones version of our theorem 17 132 Linear stability of the Schwarzschild spacetime 17 133 Main ideas in the proof of Theorem 1 6 18 134 Beyond polarization 21 135 Note added in proof 22 1 4 Organization 22 2 Preliminaries 24 2 1 Axially symmetric polarized spacetimes 24 211 Axial symmetry 24 212 Z-frames 25 213 Axis of symmetry 26 214 Z—polarized S—surfaces 28 215 Invariant S—foliations 45 216 Schwarzschild spacetime ‚ 50 2 2 Main equations 51 221 Main equations for general S—foliations 51 222 Null Bianchi identities 54 223 Hawking mass 56 224 Outgoing geodesic foliations 57 225 Additional equations 70 226 Ingoing geodesic foliation 71 227 Adapted coordinates systems 71 2 3 Perturbations of Schwarzschild and invariant quantities 78 231 Null frame transformations 78 232 Schematic notation F9 and I}, 8‘2 233 The invariant quantity q 234 Several identities for q 2 4 Invariant wave equations ‚ ___lI ‘ 84 241 Preliminaries _,_ : - 84 242 Wave equations for a, g, and q___Ui :5 - 7 Main Theorem 3 1 General covariant modulated admissible spacetimes _ ‚ 89 3 1 1Initialdatalayer :H 89 312 Maindeﬁnition _: 89 313 Renormalized curvature components and Ricci coefficients 3 2 Main norms ‚ ___ 96 321 Main norms in (“0M _ ‚ 96 322 Main norms in “MM ‚ ‚ 99 323 Combined norms 100 324 Initial layer norm _ 100 3 3 Main theorem _ 101 331 Smallness constants 101 332 Statement of the main theorem 102 3 4 Bootstrap assumptions and ﬁrst consequences 105 341 Main bootstrap assumptions 105 342 Control of the initial data 105 343 Control of averages and of the Hawking mass 106 344 Control of coordinates system 107 345 Pointwise bounds for higher order derivatives 109 346 Construction of a second frame in (“0 44 109 3 5 Global null frames 111 351 Extension of frames 111 352 Construction of the first global frame 112 353 Construction of the second global frame 113 3 6 Proof of the main theorem 11-1 361 Main intermediate results 114 362 End of the proof of the main theorem 115 363 Conclusions 116 3 7 The general covariant modulation procedure 125 371 Spacetime assumptions for the GCM procedure 125 372 Deformations of surfaces 128 373 Adapted frame transformations - 128 374 GCM results - 129 375 Main ideas 131 3 8 Overview of the proof of Theorems MO—M8 133 381 Discussion of Theorem MO 133 382 Discussion of Theorem Ml - - 134 383 Discussion of Theorem M2 - - 135 384 Discussion of Theorem M3 136 385 Discussion of Theorem M4 - 137 386 Discussion of Theorem Ms 138 387 Discussion of Theorem Me 138 388 Discussion of Theorem M7 - - 139 389 Discussion of Theorem MS 140 3 9 Structure of the rest of the book Consequences of the Bootstrap Assumptions 4 1 Proof of Theorem M0 4 2 Control of averages and of the Hawking mass 421 Proof of Lemma 3 15 422 Proof of Lemma 3 16 Control of coordinates systems Pointwise bounds for higher order derivatives Proof of Proposition 3 20 Existence and control of the global frames 461 Proof of Proposition 3 23 462 Proof of Lemma 4 16 463 Proof of Proposition 3 26 Decay Estimates for q (Theorem M1) 5 1 Preliminaries 511 The foliation of M by r 512 Assumptions for Ricci coefﬁcients and curvature 513 Structure of nonlinear terms 514 Main quantities Proof of Theorem M1 521 Flux decay estimates for q 522 Proof of Theorem M1 523 Proof of Proposition 5 10 5 3 Improved weighted estimates 531 Basic and higher weighted estimates for wave equations 532 Proof of Theorem 5 14 533 Proof of Theorem 5 15 5 4 Decay estimates 541 First ﬂux decay estimates 542 Flux decay estimates for q 543 Proof of Theorem 5 9 544 Proof of Proposition 5 12 545 Proof of Proposition 5 13 Decay Estimates for a and g (Theorems M2, M3) 6 1 Proof of Theorem M2 611A renormalized frame on (em/Vt 612A transport equation for a 613 Estimates for transport equations in eg 614 Decay estimates for a 615 End of the proof of Theorem M2 6 2 Proof of Theorem M3 621 Estimate for g in (WM 622 Estimate for g on 2, 623 Proof of Proposition 6 10 624 Proof of Lemma 6 12 625 Proof of Proposition 6 14 626 Proof of Lemma 6 16 7 Decay Estimates (Theorems M4, M5) _ _ 295 711 Geometric structure of 2,___ I ‘ ‘ 295 712 Mainassumptions ,,_: 295 713 Basic lemmas ‚ __ I 296 714 Main equations ‚ „ _I 299 715 Equationsinvolvingq 301 716 Additionalequations ‚ __H 302 7 2 Structure of the proof of Theorem M4 ‚ _ „ I :05 7 3 Decay estimates on the last slice 2„ ‚ ‚ _ 3(1)? 731 Preliminaries _ _ 311 732 Differential identities involving GCM conditions on 2* 314 733 Control of the ﬂux of some quantities on 2, _ _ „ _ 315 734 Estimates for some 4=1 modes on 2 _ 322 735 Decay of Ricci and curvature components on 2 _ 332 7 4 Control in (“Ü/M, Part I _ 335 741 Preliminaries 335 742 Proposition 7 33 338 743 Estimates for Ft, [1 in (“0M 339 744 Estimates for the E=1 modes in (“QM 340 745 Completion of the proof of Proposition 7 33 343 7 5 Control in (em/Vt, Part II 346 751 Estimate for n 347 752 Crucial lemmas 347 753 Proof of Proposition 7 35, Part I 355 754 Proof of Proposition 7 35, Part II 359 7 6 Conclusion of the proof of Theorem M4 362 7 7 Proof of Theorem M5 366 8 Initialization and Extension (Theorems N16, 1M7, IVIS) 372 8 1 Proof of Theorem M6 37? 8 2 Proof of Theorem M? 376 8 3 Proof of Theorem MS 387 831 Main norms 389 832 Control of the global frame 391 833 Iterative procedure - 393 834 End of the proof of Theorem MS - 396 8 4 Proof of Proposition 8 7 ‚ ‚ - 399 841A wave equation for ß ‚ - 399 842 Control of use) _ 400 843 End of the proof of Proposition 8 7 ‚ - - 405 8 5 Proof of Proposition 8 8 408 851A wave equation for a + ng --- 408 852 End of the proof of PrOposition 8 8 417 8-6 Proof of Proposition 89__ 418 861 Control of a and T20 - - 418 8-6-2 Control ofg T 420 863 End of the proof of Proposition 8 9 424 8 7 Proof of Proposition 8 10 424 871 r—weighted divergence identities for Bianchi pairs 425 872 End of the proof of Proposition 8 10 435 873 Proof of (8 3 12) 440 8 8 Proof of Proposition 8 11 442 881 Proof of Proposition 8 31 444 882 Weighted estimates for transport equations along 64 in (“QM 454 883 Several identities 460 884 Proof of Proposition 8 32 464 885 Proof of Proposition 8 33 471 8 9 Proof of Proposition 8 12 479 891 Weighted estimates for transport equations along 63 in (“WM 480 892 Proof of Proposition 8 42 482 8 10 Proof of Proposition 8 13 485 9 GCM Procedure 486 9 1 Preliminaries 486 911 Main assumptions 488 912 Elliptic Hodge lemma 489 9 2 Deformations of S surfaces 489 921 Deformations 489 922 Pullback map 490 923 Comparison of norms between deformations 492 924 Adapted frame transformations ‚ 496 9 3 Frame transformations 504 931 Main GCM equations 513 932 Equation for the average of a 518 933 Transversality conditions 519 9 4 Existence of GCM spheres 520 941 The linearized GCM system 524 942 Comparison of the Hawking mass 526 943 Iteration procedure for Theorem 9 32 527 944 Existence and boundedness of the iterates 530 945 Convergence of the iterates 535 9 5 Proof of Proposition 9 37 and of Corollary 9 38 538 951 Proof of Proposition 9 37 538 952 Proof of Corollary 9 38 542 9 6 Proof of Proposition 9 43 545 961 Pullback of the main equations 545 962 Basic lemmas 548 963 Proof of the estimates (9 6 5), (9 6 6), (9 6 7) 556 97A corollary to Theorem 9 32 559 9 8 Construction of GCM hypersurfaces 566 981 Definition of 2o 569 982 Extrinsic properties of 2o 570 983 Construction of 20 583 10 Regge-Wheeler Type Equations 600 10 1 Basic Morawetz estimates 600 10 1 1 Structure of the proof of Theorem 10 1 601 10 12A simpliﬁed set of assumptions 602 10 1 3 Functions depending on m and 1 602 10 1 4 Deformation tensors of the vectorfields R, T, X 10 1 5 Basic integral identities 10 1 6 Main Morawetz identity ‚ _ _ ‚ _ 10 17A ﬁrst estimate ‚ _ 10 1 8 Improved lower bound in (“0M 10 1 9 Cut-off correction in (“MM 10 1 10 The redshift vectorﬁeld 10 1 11 Combined estimate __ sss__I s_ s ‚ ‚ _ _ sus ‚ _ _ ‘ I s-suo_ 10 1 14 Analysis of the error term 65 10 1 15 Proof of Theorem 10 1 ‚ Dafermos—Rodnianski rp-Weighted estimates ‚ _ 10 2 1 Vectorﬁeld X = f(r)e4 10 2 2 Energy densities for X = f(r)e4 ‚ ‚ 10 2 3 Proof of Theorem 10 37 Higher weighted estimates L 10 3 1 Wave equation for t!) ‚Y 10 3 2 The rp—weighted estimates for z/J Higher derivative estimates 10 4 1 Basic assumptions 10 4 2 Strategy for recovering higher order derivatives 10 4 3 Commutation formulas with the wave equation 10 4 4 Some weighted estimates for wave equations 10 4 5 Proof of Theorem 5 17 10 4 6 Proof of Theorem 5 18 10 5 More weighted estimates for wave equations onns --- A Appendix to Chapter 2 A 1 Proof of Proposition 2 64 A 2 Proof of Proposition 2 71 A 3 Proof of Lemma 2 72 A 4 Proof of Proposition 2 73 A 5 Proof of Proposition 2 74 - - - A 6 Proof of Proposition 2 90 - - A 7 Proof of Lemma 2 92 - - A 8 Proof of Corollary 2 93 - ~ A 9 Proof of Lemma 2 91 - - A 10 Proof of Proposition 2 99 - A 11 Proof of Proposition 2 100 - - - A 12 Proof of the Teukolsky-Starobinsky identity - - - A 13 Proof of Proposition 2 107 - - A 14 Proof of Theorem 2 108 ~ - A 14 1 The Teukolsky equation for a-~- A 14 2 Commutation lemmas _ „ ‚ _ _ „ _ _ ‚ „ _ _ „ _ _ A 14 3 Main commutation - - - A 14 4 Proof of Theorem 2 108 - - B Appendix to Chapter 8 8 1 Proof of Proposition 8 14 ---- C Appendix to Chapter 9 C 1 Proof of Lemma 9 11 D Appendix to Chapter 10 D 1 Horizontal S—tensors D11 Mixed tensors D12 Invariant Lagrangian D13 Comparison of the Lagrangians D14 Energy-momentum tensor D 2 Standard calculation D 3 Vectorfield X f D 4 Proof of Proposition 10 47 Bibliography
adam_txt	Contents List of Figures xiii Acknowledgments xv 1 Introduction 1 1 1 Basic notions in general relativity 1 111 Spacetime and causality 1 112 The initial value formulation for Einstein equations 2 113 Special solutions 3 114 Stability of Minkowski space 10 115 Cosmic censorship 11 1 2 Stability of Kerr conjecture 13 121 Formal mode analysis 15 122 Vectorﬁeld method 16 1 3 Nonlinear stability of Schwarzschild under polarized perturbations 17 131 Bare-bones version of our theorem 17 132 Linear stability of the Schwarzschild spacetime 17 133 Main ideas in the proof of Theorem 1 6 18 134 Beyond polarization 21 135 Note added in proof 22 1 4 Organization 22 2 Preliminaries 24 2 1 Axially symmetric polarized spacetimes 24 211 Axial symmetry 24 212 Z-frames 25 213 Axis of symmetry 26 214 Z—polarized S—surfaces 28 215 Invariant S—foliations 45 216 Schwarzschild spacetime ‚ 50 2 2 Main equations 51 221 Main equations for general S—foliations 51 222 Null Bianchi identities 54 223 Hawking mass 56 224 Outgoing geodesic foliations 57 225 Additional equations 70 226 Ingoing geodesic foliation 71 227 Adapted coordinates systems 71 2 3 Perturbations of Schwarzschild and invariant quantities 78 231 Null frame transformations 78 232 Schematic notation F9 and I}, 8‘2 233 The invariant quantity q 234 Several identities for q 2 4 Invariant wave equations ‚ _lI ‘ ' ' 84 241 Preliminaries _,_ :'-' 84 242 Wave equations for a, g, and q_Ui' :5 - 7 Main Theorem 3 1 General covariant modulated admissible spacetimes _ ‚ 89 3 1 1Initialdatalayer :H 89 312 Maindeﬁnition _: 89 313 Renormalized curvature components and Ricci coefficients 3 2 Main norms ‚ _ 96 321 Main norms in (“0M _ ‚ 96 322 Main norms in “MM ‚ ‚ 99 323 Combined norms 100 324 Initial layer norm _ 100 3 3 Main theorem _ 101 331 Smallness constants 101 332 Statement of the main theorem 102 3 4 Bootstrap assumptions and ﬁrst consequences 105 341 Main bootstrap assumptions 105 342 Control of the initial data 105 343 Control of averages and of the Hawking mass 106 344 Control of coordinates system 107 345 Pointwise bounds for higher order derivatives 109 346 Construction of a second frame in (“0 44 109 3 5 Global null frames 111 351 Extension of frames 111 352 Construction of the first global frame 112 353 Construction of the second global frame 113 3 6 Proof of the main theorem 11-1 361 Main intermediate results 114 362 End of the proof of the main theorem 115 363 Conclusions 116 3 7 The general covariant modulation procedure 125 371 Spacetime assumptions for the GCM procedure 125 372 Deformations of surfaces 128 373 Adapted frame transformations - 128 374 GCM results - 129 375 Main ideas 131 3 8 Overview of the proof of Theorems MO—M8 133 381 Discussion of Theorem MO 133 382 Discussion of Theorem Ml - - 134 383 Discussion of Theorem M2 - - 135 384 Discussion of Theorem M3 136 385 Discussion of Theorem M4 - 137 386 Discussion of Theorem Ms 138 387 Discussion of Theorem Me 138 388 Discussion of Theorem M7 - - 139 389 Discussion of Theorem MS 140 3 9 Structure of the rest of the book Consequences of the Bootstrap Assumptions 4 1 Proof of Theorem M0 4 2 Control of averages and of the Hawking mass 421 Proof of Lemma 3 15 422 Proof of Lemma 3 16 Control of coordinates systems Pointwise bounds for higher order derivatives Proof of Proposition 3 20 Existence and control of the global frames 461 Proof of Proposition 3 23 462 Proof of Lemma 4 16 463 Proof of Proposition 3 26 Decay Estimates for q (Theorem M1) 5 1 Preliminaries 511 The foliation of M by 'r 512 Assumptions for Ricci coefﬁcients and curvature 513 Structure of nonlinear terms 514 Main quantities Proof of Theorem M1 521 Flux decay estimates for q 522 Proof of Theorem M1 523 Proof of Proposition 5 10 5 3 Improved weighted estimates 531 Basic and higher weighted estimates for wave equations 532 Proof of Theorem 5 14 533 Proof of Theorem 5 15 5 4 Decay estimates 541 First ﬂux decay estimates 542 Flux decay estimates for q 543 Proof of Theorem 5 9 544 Proof of Proposition 5 12 545 Proof of Proposition 5 13 Decay Estimates for a and g (Theorems M2, M3) 6 1 Proof of Theorem M2 611A renormalized frame on (em/Vt 612A transport equation for a 613 Estimates for transport equations in eg 614 Decay estimates for a 615 End of the proof of Theorem M2 6 2 Proof of Theorem M3 621 Estimate for g in (WM 622 Estimate for g on 2, 623 Proof of Proposition 6 10 624 Proof of Lemma 6 12 625 Proof of Proposition 6 14 626 Proof of Lemma 6 16 7 Decay Estimates (Theorems M4, M5) _ _ 295 711 Geometric structure of 2,_'I ‘ ' ' ‘ 295 712 Mainassumptions ,,_: '295 713 Basic lemmas ‚ _'I'' 296 714 Main equations ‚ „ _I'' 299 715 Equationsinvolvingq 301 716 Additionalequations ‚ _H 302 7 2 Structure of the proof of Theorem M4 ‚ _ „ I :05 7 3 Decay estimates on the last slice 2„ ‚ ‚ _ 3(1)? 731 Preliminaries _ _ 311 732 Differential identities involving GCM conditions on 2* 314 733 Control of the ﬂux of some quantities on 2, _ _ „ _ 315 734 Estimates for some 4=1 modes on 2 _ 322 735 Decay of Ricci and curvature components on 2 _ 332 7 4 Control in (“Ü/M, Part I _ 335 741 Preliminaries 335 742 Proposition 7 33 338 743 Estimates for Ft, [1 in (“0M 339 744 Estimates for the E=1 modes in (“QM 340 745 Completion of the proof of Proposition 7 33 343 7 5 Control in (em/Vt, Part II 346 751 Estimate for n 347 752 Crucial lemmas 347 753 Proof of Proposition 7 35, Part I 355 754 Proof of Proposition 7 35, Part II 359 7 6 Conclusion of the proof of Theorem M4 362 7 7 Proof of Theorem M5 366 8 Initialization and Extension (Theorems N16, 1M7, IVIS) 372 8 1 Proof of Theorem M6 37? 8 2 Proof of Theorem M? 376 8 3 Proof of Theorem MS 387 831 Main norms 389 832 Control of the global frame 391 833 Iterative procedure - 393 834 End of the proof of Theorem MS - 396 8 4 Proof of Proposition 8 7 ‚ ‚ - 399 841A wave equation for ß ‚ - 399 842 Control of use) _ 400 843 End of the proof of Proposition 8 7 ‚ - - 405 8 5 Proof of Proposition 8 8 408 851A wave equation for a + ng --- 408 852 End of the proof of PrOposition 8 8 417 8-6 Proof of Proposition 89_ 418 861 Control of a and T20 - - 418 8-6-2 Control ofg T 420 863 End of the proof of Proposition 8 9 424 8 7 Proof of Proposition 8 10 424 871 r—weighted divergence identities for Bianchi pairs 425 872 End of the proof of Proposition 8 10 435 873 Proof of (8 3 12) 440 8 8 Proof of Proposition 8 11 442 881 Proof of Proposition 8 31 444 882 Weighted estimates for transport equations along 64 in (“QM 454 883 Several identities 460 884 Proof of Proposition 8 32 464 885 Proof of Proposition 8 33 471 8 9 Proof of Proposition 8 12 479 891 Weighted estimates for transport equations along 63 in (“WM 480 892 Proof of Proposition 8 42 482 8 10 Proof of Proposition 8 13 485 9 GCM Procedure 486 9 1 Preliminaries 486 911 Main assumptions 488 912 Elliptic Hodge lemma 489 9 2 Deformations of S surfaces 489 921 Deformations 489 922 Pullback map 490 923 Comparison of norms between deformations 492 924 Adapted frame transformations ‚ 496 9 3 Frame transformations 504 931 Main GCM equations 513 932 Equation for the average of a 518 933 Transversality conditions 519 9 4 Existence of GCM spheres 520 941 The linearized GCM system 524 942 Comparison of the Hawking mass 526 943 Iteration procedure for Theorem 9 32 527 944 Existence and boundedness of the iterates 530 945 Convergence of the iterates 535 9 5 Proof of Proposition 9 37 and of Corollary 9 38 538 951 Proof of Proposition 9 37 538 952 Proof of Corollary 9 38 542 9 6 Proof of Proposition 9 43 545 961 Pullback of the main equations 545 962 Basic lemmas 548 963 Proof of the estimates (9 6 5), (9 6 6), (9 6 7) 556 97A corollary to Theorem 9 32 559 9 8 Construction of GCM hypersurfaces 566 981 Definition of 2o 569 982 Extrinsic properties of 2o 570 983 Construction of 20 583 10 Regge-Wheeler Type Equations 600 10 1 Basic Morawetz estimates 600 10 1 1 Structure of the proof of Theorem 10 1 601 10 12A simpliﬁed set of assumptions 602 10 1 3 Functions depending on m and 1' 602 10 1 4 Deformation tensors of the vectorfields R, T, X 10 1 5 Basic integral identities 10 1 6 Main Morawetz identity ‚ _ _ ‚ _ 10 17A ﬁrst estimate ‚ _ 10 1 8 Improved lower bound in (“0M 10 1 9 Cut-off correction in (“MM 10 1 10 The redshift vectorﬁeld 10 1 11 Combined estimate _ sss_I s_ s ‚ ‚ _ _ sus ‚ _ _ ‘ I s-suo_ 10 1 14 Analysis of the error term 65 10 1 15 Proof of Theorem 10 1 ‚ Dafermos—Rodnianski rp-Weighted estimates ‚ _ 10 2 1 Vectorﬁeld X = f(r)e4 10 2 2 Energy densities for X = f(r)e4 ‚ ‚ 10 2 3 Proof of Theorem 10 37 Higher weighted estimates L 10 3 1 Wave equation for t!) ‚Y 10 3 2 The rp—weighted estimates for z/J Higher derivative estimates 10 4 1 Basic assumptions 10 4 2 Strategy for recovering higher order derivatives 10 4 3 Commutation formulas with the wave equation 10 4 4 Some weighted estimates for wave equations 10 4 5 Proof of Theorem 5 17 10 4 6 Proof of Theorem 5 18 10 5 More weighted estimates for wave equations onns --- A Appendix to Chapter 2 A 1 Proof of Proposition 2 64 A 2 Proof of Proposition 2 71 A 3 Proof of Lemma 2 72 A 4 Proof of Proposition 2 73 A 5 Proof of Proposition 2 74 - - - A 6 Proof of Proposition 2 90 - - A 7 Proof of Lemma 2 92 - - A 8 Proof of Corollary 2 93 - ~ A 9 Proof of Lemma 2 91 - - A 10 Proof of Proposition 2 99 - A 11 Proof of Proposition 2 100 - - - A 12 Proof of the Teukolsky-Starobinsky identity - - - A 13 Proof of Proposition 2 107 - - A 14 Proof of Theorem 2 108 ~ - A 14 1 The Teukolsky equation for a-~- A 14 2 Commutation lemmas _ „ ‚ _ _ „ _ _ ‚ „ _ _ „ _ _ A 14 3 Main commutation - - - A 14 4 Proof of Theorem 2 108 - - B Appendix to Chapter 8 8 1 Proof of Proposition 8 14 ---- C Appendix to Chapter 9 C 1 Proof of Lemma 9 11 D Appendix to Chapter 10 D 1 Horizontal S—tensors D11 Mixed tensors D12 Invariant Lagrangian D13 Comparison of the Lagrangians D14 Energy-momentum tensor D 2 Standard calculation D 3 Vectorfield X f D 4 Proof of Proposition 10 47 Bibliography
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spelling	Klainerman, Sergiu 1950- Verfasser (DE-588)172678056 aut Global nonlinear stability of Schwarzschild spacetime under polarized perturbations Sergiu Klainerman, Jérémie Szeftel Princeton ; Oxford Princeton University Press [2020] xi, 840 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies Number 210 (DE-588)4113937-9 Hochschulschrift gnd-content Szeftel, Jérémie 1977- Verfasser (DE-588)1167946383 aut Erscheint auch als Online-Ausgabe 978-0-691-21852-6 Annals of mathematics studies Number 210 (DE-604)BV000000991 210 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032700459&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Klainerman, Sergiu 1950- Szeftel, Jérémie 1977- Global nonlinear stability of Schwarzschild spacetime under polarized perturbations Annals of mathematics studies
subject_GND	(DE-588)4113937-9
title	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations
title_auth	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations
title_exact_search	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations
title_exact_search_txtP	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations
title_full	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations Sergiu Klainerman, Jérémie Szeftel
title_fullStr	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations Sergiu Klainerman, Jérémie Szeftel
title_full_unstemmed	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations Sergiu Klainerman, Jérémie Szeftel
title_short	Global nonlinear stability of Schwarzschild spacetime under polarized perturbations
title_sort	global nonlinear stability of schwarzschild spacetime under polarized perturbations
topic_facet	Hochschulschrift
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032700459&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000991
work_keys_str_mv	AT klainermansergiu globalnonlinearstabilityofschwarzschildspacetimeunderpolarizedperturbations AT szefteljeremie globalnonlinearstabilityofschwarzschildspacetimeunderpolarizedperturbations

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