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5 References 379 [Le] [Mu] J. Leray, Lectures on Hyperbolic Equations, Institute for Advanced Study, Notes, H. Muller Zum Hagen, Characteristic initial value problem for hyperbolic systems of second order differential equations, Ann. Inst. H. Poincare 53, n.2, (1990), [Ne-Pel] E. T. Newman, R.Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962), [Ne-Pe2] [Ni] [Pel] [Pe2] E. T. Newman, R. Penrose, New conservation laws for zero rest-mass fields in asymptotically fiat space-time, Proc. Roy. Soc. Lond. A305 (1968), F. Nicolo, Canonical foliation on a null hypersurface, To appear. R. Penrose, Conformal Treatment ofinfinity, Relativity, Groups, and Topology, B. dewitt and C. dewitt, eds., Gordon and Breach, R. Penrose, Zero rest mass fields including gravitation: asymptotic behavior, Proc. Roy. Soc. Lond. A284 (1962), [Pe3] R. Penrose, Gravitational collapse and spacetime singularities, Phys. Rev. Lett. 14 (1965), [Ren] A. D. Rendall, Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. Roy. Soc. Lond. A427 (1990), [Se] H. Seifert, Kausal Lorentzraume, Doctoral thesis, [Sc-Yaul] [Sc-Yau2] [Sp] [Ta] R. Schoen, S. T. Yau, Proof of the positive mass theorem I, Comm. Math. Phys. 65 (1979), R. Schoen, S. T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1981), M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc., Wilmington, D. Tataru Strichartz estimates for operators with nonsmooth coefficients, to appear in Journ. of A.M.S. [Wall R. Wald, Gravitational collapse and cosmic censorship. (1997), grqc/ [Wa2] R. Wald, General Relativity, University of Chicago Press, [W] T. Wolff, Recent work connected with the Kakeya problem, in Prospects in Mathematics, H. Rossi, ed. AMS, 1998.

6 Index (S, y) Riemannian manifold, 116 C null hypersurfaces, 94 ]1(X, W) Weyl current, 245 ]2(X, W) Weyl current, 245 ]3(X, W) Weyl current, 245 k-distribution,4 S null pair, 56 S-foliation, 62, 64 [Correction] term, 210, 212, 217 [Q] 1,2 integral norms, 204 [Qh,2().., v) integral norms, 204 null hypersurfaces, 94. canonical foliation, 81, 304 null hypersurface, 320 qirl, 116 D~'s (X) norms, 97 D~'s (L)(X) norms, 98 <fiv, 116 y induced metric, 55 p,115 (xlp,249 (X)q, 249 IL mass aspect function, 366 Nand N null hypersurfaces, 58 V covariant derivative, 295 Y1 induced connection, 55 O'~] norm, 305 O'[m norm, 305 O~] norm, 94 O~] norm, 94 0[1] norm, 94 0[1](.) norm, 96 0[1](1:0) norm, 96 0[2] norm, 94 0[2](.) norm, 96 0[2](1:0) norm, 96 0[3] norm, 93, 94 0[1] norm, 305 0[2] norm, norm, 306 0[1] norm, 305 0[2] norm, 305 0[3] norm, 305 0'3 (1:o\K) norms, 301 0'3(1:0\K) norms, ( *) norm, 96 O[l] norm, 94 0[1] (.) norm, 96 0[1](1:0) norm, 96 0[2] norm, 94 0[2](L) norm, 96 0[2](1:0) norm, 96 0[3] norm, 93 O't S (1:o\K) (X) norms, 301

7 382 O%,S({2*)(X) norms, 96 O%,S (L.o)(XJ norms, 96 a asymptotic behaviour, 225 <Pt diffeomorphisms, 351 Q(A, v) integral norm, 89 Q(A, v) integral norm, 89 Q1 (A, v) integral norm, 89, 204 Q/A, v) integral norm, 89, 204 QI (A, v) integral norm, 89, 204 Q2(A, v) integral norm, 89,204 QEonlC integral norm, 90 QIC integral norm, 90, 204 :&,1,2[w] norms, 91, 208 :& norm, 91, 204, 208 RI norm, 91 R2 norm, 91, 204 R[o] norm, 91, 204 R[l] norm, 91, 204 R[2] norm, 91, 204 R O,I,2[W] norms, 91, 208 Ro norm, 91, 204, 208 RI norm, 91 R2 norm, 91, 204 R[O] norm, 91, 204 R[I] norm, 91, 204 R[2] norm, 91, 204 L.o background foliation, 298 L.o canonical foliation, 299 tre evolution equation, 79 o norms, 94 Q integral norms, 203 0* norms, 305 ADM mass, 45 o norms, 93 Rnorms, 90 Q integral norms, 88 A> annulus, 320 V norms, 101 V' norms, 98 V~ norm, 97 Vi norm, 97 V 2 norm, 98 VO,I norms, 97 feu, g) error term, 241, errorterm,242,243,259 2 error term, 242 J+ null-outgoing infinity, 349 K' 80 initial layer region, 125 KCAo, v* + 8) spacetime, 320 o norms, 94, 96, 101 0' norms, 96 0' (L.o\K) norms, 300 0* norms, 305 0% norms, 94 R norms, 96, 10 1, 204 R' norms, 96 R'1:' norm, 93 Rf norm, 93 Q integral norms, 102, 203 e evolution equation, 297 io spacelike infinity, (L.O) norm, 96 p asymptotic behavior, 223 1/I(A, v) diffeomorphism, 351 Q1EonV(A,v) integral norm, 90 Q2EonV(A,v) integral norm, 90 QI,2EonlC integral norms, 204 ji (A, v) metric, 353 C\ (f0 norm, ({Q) norm, 95 <P diffeomorphisms, 351 -'-S 0' (L.o\K) norms, 300 f4 norms, 94 R't' norm, 93 Rf norm, 93 (3) R Ricci tensor, 297 (3) R scalar curvature, 297 L.'t' space like foliation, 125 Schwarzschild cones, 349 Main Theorem, 47, 102 Index a priori estimate, 35 A. Fisher, 14 adapted null frame, 63, 85, 111 ADM mass, 52, 347, 349, 370 affine parameter, 7 angular momentum vector fields, 84 approximate conformal Killing vector field, 52

8 Index approximate Killing vector field, 52 asymptotic behavior, 101,226 asymptotically flat, 25 background foliation, 81 Bel-Robinson tensor, 42, 76 Bianchi equations, 41, 44, 75, 216, 218, 220 Bianchi identities, 8 Bondi mass, 349, 366, 367 Bondi mass formula, 368 bootstrap assumption, 162 Bootstrap assumption Bl, 106 Bootstrap assumption B2, 106 bootstrap assumptions, 106, 130, 137, 144, 146, 150, 151, , , 164, 167 canonical double null foliation, 81, 88, 96,107 canonical foliation, 81 canonical function, 82 canonical generators of the rotation group, 172 Canonical Null Foliation, 43 Canonical Null Pair, 44 Canonical Sphere Foliation, 44 Cauchy development, 13 Cauchy horizon, 25 Cauchy hypersurface, 18 Cauchy problem, 13 characteristic initial value problem, 72 Christoffel symbols, 5 chronological future, 48 Codazzi-Mainardi equation, 79 Codazzi-Mainardi equations, 296 complete future null infinity, 24 conformal compactification, 24 conformal derivatives, 57 Conformal group, 10 conformal isometry, 9 conformal Killing vector field, 9, 38,42, 43 conformal method, 37 Conformal Structure, conformally invariant, 8 connection, 5 connection coefficients, 55, 64, 65, 168, 299 constraint equations, 13 Constraint equations, 21 constraint equations, 22 contracted Codazzi-Mainardi equations, 296 contracted Gauss equation, 296 contraction mapping, 32 cosmic censorship, 23 D' Alembertian, 31 deformation tensor, 9, 75, 85, 86 deformation tensors, 174,250 development, 13 differential df, 3 double null canonicalfoliation, 127, 130, 133 double null foliation, 64, 65, 68, 78, 122 double null integrable, 63 eikonal equation, 52, 63, 82 Einstein equations, 12 Einstein vacuum equations, 70 energy-momentum tensor, 36 equivariant vector field, 65 evolution equations, 21 Evolution Lemma, 122, 135, 143 exponential map, 7 Extension theorem, 105 Fermi transported frame, 67, 297 final hypersurfaces, 96 final slice problem, 54 flux quantities, 48 Friedman spacetime, 23 Frobenius Theorem, 5 G.FR. Ellis, 23 Gauss curvature, 55, 62, 116 Gauss equation, 62, 79, 296, 299 Gaussian metric, 15 generalized energy-type norms, 38 generalized Sobolev norms, 38

9 384 geodesic curve, 7 geodesic distance, 26, 99 geodesically complete, 7 global initial data, 98 global optical function, 349 global smallness condition, 99 global smallness initial data, 100 global Sobolev inequalities, 47, 117 global time function, 83, 370 globally hyperbolic, 19 Gronwall inequality, 122, 149, 150, 170, 178, 179, 181, 182, 199 H.Friedrich, 15 Hawking mass, 366, 367 Hodge dual, 57, 115 Hodge duals, 41, 60, 61 Hodge operator *, 45 Hodge systems, 115, 139, 141 incoming evolution equation, 123 initial data, 98 initial data constraints, 74 initial data norm, 300 initial data set, 13, 100 initial hypersurface foliations, 295 initial hypersurfaces, 96 initial layer foliation, 82, 83, 102, 125, 126, 130, 323 initial layer region, 96, 126, 130, 132 initial slice problem, 54, 79, 299 Inversion, 10, 44 isometry, 9 isoperimetric constant, 22, 121 isoperimetric inequality, Mardsen, 14 Killing vector field, 9, 38,43 lapse equation, 22 lapse function, 20 last slice connection estimates, 305 last slice foliation, 304 Levi-Civita connection, 6 Lie coefficients, 206, 207 Lie derivative, 4 Lie transported, 65 Liouville theorem, 10 local existence, 17 local Sobolev spaces Hl~c' 17 Lorentz rotations, 10 Lorentzian manifold, 1 Index Main Theorem, 101 mass aspect functions, 80, 140 maximal Cauchy development, 23 maximal foliation, 47 maximal globally hyperbolic vacuum extension, 23 maximal hypersurface, 21 maximal time, 35 Maxwell-type equations, 42 modified Lie derivative, 42, 75 Moser type estimates, 35 naked singularities, 25 normalized null pair, 64 null Codazzi equations, 62 null components, 44, 46, 347 null condition, 39, 40 null decomposition, 61 null frame, 60 null geodesic pair, 64 null geodesic vector fields, 64 null hypersurfaces, 64 null incoming integrable, 63 null outgoing hypersurfaces asymptotic behaviour, 370 null outgoing infinity limits, 359 null pair, 60, 62 null second fundamental forms, 56, 59, 62 null structure equations, 62, 68 null-incoming integrable, 63 null-outgoing integrable, 63 optical function, 52 Oscillation Lemma, 126, 127, 130, 192, ,200 outgoing evolution equation, 122

10 Index parallel transport, 6 Poincare group, 10 Poincare inequality, 211 R.Geroch, 19 R. Geroch, 23 reduced Einstein equations, 14, 15 Ricci tensor, 8 Riemann components, 101 Riemann curvature tensor, 7, 45, 60 Robertson-Walker spacetime, 23 rotation deformation tensor, 180 rotation deformation tensors, 97, 175, 177, 180,257 rotation vector fields, 44, 87, 172, 207 S.w. Hawking, 23 scalar curvature, 8 Scalings, 10, 44 second fundamental form, 13,21,83,295 shift vector, 20 signature, 223 smooth compactification, 357 Sobo1ev inequality, 35, 38 Sobolev space, 34 spacelike hypersurfaces, 36, 83 spacetime lapse function, 64 strong asymptotic flatness, 100 strong asymptotically flat, 26 strong cosmic censorship, 25 strongly asymptotically flat, structure equations, 68, 110 structure equations outgoing infinity limit, 365 T. Hughes, 14 T. Kato, 14 Theorem MO, 102, 323 Theorem MI, 102, 130 Theorem M2, 102, 177 Theorem M3, 103, 300, 302 Theorem M4, 103, 306 Theorem MS, 103,315 Theorem M6, 104,304,327 Theorem M7, 104,204 Theorem MS, 105, 241 Theorem M9, 105,320 time translations, 44 torsion, 56, 62 torsion equation, 62 Translations, 10 uniqueness, 19 wavelike coordinates, 40 weak cosmic censorship, 24 Weyl current, 244 Weyl current null components, 245 Weyl tensor field, 41, 44, 60, 75 y. Choquet-Bruhat, 14,23