0805.3880v1 The Formation of Black Holes in General Relativity.pdf

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1、TheFormationofBlackHolesinGeneralRelativityDemetriosChristodoulouMay18,2008arXiv:0805.3880v1[gr-qc]26May2008ContentsPrologueChapter1:TheOpticalStructureEquations1.1Thebasicgeometricsetup1.2Theopticalstructureequations1.3TheBianchiequations1.4CanonicalcoordinatesystemsChapter2:TheChara

2、cteristicInitialData2.1Thecharacteristicinitialdata2.2ConstructionofthesolutioninaninitialdomainChapter3:L∞EstimatesfortheConnectionCoefficients3.1Introduction3.2L∞estimatesforχ′3.3L∞estimatesforχ′3.4L∞estimatesforη,η3.5L∞estimatesforω,ω3.6ThesmallnessrequirementonδChapter4:L4(S)Estimat

3、esforthe1stDerivativesoftheConnectionCoefficients4.1Introduction4.2L4(S)estimatesfor∇/χ4.3L4(S)estimatesfor∇/χ4.4L4(S)estimatesfor∇/η,∇/η4.5L4(S)estimatesford/ω,d/ω4.6L4(S)estimatesforDω,DωChapter5:TheUniformizationTheorem5.1Introduction.AnL2(S)estimateforK−K5.2SobolevinequalitiesonS.Th

4、eisoperimetricconstant5.3Theuniformizationtheorem5.4LpelliptictheoryonSChapter6:L4(S)Estimatesforthe2ndDerivativesoftheConnectionCoefficients6.1Introduction6.2L4(S)estimatesfor∇/2χ,K−K6.3L4(S)estimatesfor∇/2χ26.4L4(S)estimatesfor∇/2η,∇/2η6.5L4(S)estimatefor∇/2ω6.6L4(S)estimatefor∇/2ωCha

5、pter7:L2Estimatesforthe3rdDerivativesoftheConnectionCoefficients7.1Introduction7.2L2estimatesfor∇/2η,∇/2η7.3L2elliptictheoryforgeneralizedHodgesystemsonS7.4L2estimatesfor∇/3χ′,d/K7.5L2estimatesfor∇/3χ′7.6L2estimatesfor∇/3η,∇/3η7.7L2estimatefor∇/3ω7.8L2estimatesfor∇/2ωand∇/3ω7.9L2estimat

6、esford/Dω,d/Dω,D2ω,D2ωChapter8:TheMultiplierFieldsandtheCommutationFields8.1Introduction8.2L∞estimatesforthedeformationtensorsofL,KandS8.3ConstructionoftherotationvectorfieldsOi8.4L∞estimatesfortheOand∇/Oii8.5L∞estimatesforthedeformationtensorsoftheOiChapter9:EstimatesfortheDerivatives

7、oftheDeformationTensorsoftheCommutationFields9.1L4(S)estimatesforthe1stderivativesofthedeformationtensorsofL,S9.2L4(S)estimatesforthe1stderivativesofthedeformationtensorsoftheOi9.3L2estimatesforthe2ndderivativesofthedeformationtensorsofL,S9.4L2estimatesforthe2ndderivativesofthedeforma

8、tiont