Table of Contents
Preface vii
1 Introduction 1
1.1 The nonlinear stability problem for the Einstein equations 1
1.2 Statement of the main result 4
2 Overview of the Hyperboloidal Foliation Method 9
2.1 The semi-hyperboloidal frame and the hyperboloidal frame 9
2.2 Spacetime foliation and initial data set 11
2.3 Coordinate formulation of the nonlinear stability property 14
2.4 Bootstrap argument and construction of the initial data 16
2.5 Outline of the Monograph 17
3 Functional Analysis on Hyperboloids of Minkowski Spacetime 19
3.1 Energy estimate on hyperboloids 19
3.2 Sup-norm estimate based on curved characteristic integration 25
3.3 Sup-norm estimate for wave equations with source 28
3.4 Sup-norm estimate for Klein-Gordon equations 35
3.5 Weighted Hardy inequality along the hyperboloidal foliation 36
3.6 Sobolev inequality on hyperboloids 40
3.7 Hardy inequality for hyperboloids 41
3.8 Commutator estimates for admissible vector fields 43
4 Quasi-Null Structure of the Einstein-Massive Field System on Hyperboloids 45
4.1 Einstein equations in wave coordinates 45
4.2 Analysis of the support 51
4.3 A classification of nonlinearities in the Einstein-massive field system 53
4.4 Estimates based on commutators and homogeneity 58
4.5 Basic structure of the quasi-null terms 59
4.6 Metric components in the semi-hyperboloidal frame 60
4.7 Wave gauge condition in the semi-hyperboloidal frame 62
4.8 Revisiting the structure of the quasi-null terms 65
5 Initialization of the Bootstrap Argument 69
5.1 The bootstrap assumption and the basic estimates 69
5.2 Estimates based on integration along radial rays 72
6 Direct Control of Nonlinearities in the Einstein Equations 73
6.1 L∞ estimates 73
6.2 L2 estimates 74
7 Direct Consequences of the Wave Gauge Condition 77
7.1 L∞ estimates 77
7.2 L2 estimates 81
7.3 Commutator estimates 84
8 Second-Order Derivatives of the Spacetime Metric 89
8.1 Preliminary 89
8.2 L∞ estimates 91
8.3 L2 estimates 92
8.4 Conclusion for general second-order derivatives 95
8.5 Commutator estimates 95
9 Sup-Norm Estimate Based on Characteristics 99
9.1 Main statement in this section 99
9.2 Application to quasi-null terms 103
10 Low-Order Refined Energy Estimate for the Spacetime Metric 105
10.1 Preliminary 105
10.2 Main estimate established in this section 107
10.3 Application of the refined energy estimate 110
11 Low-Order Refined Sup-Norm Estimate for the Metric and Scalar Field 113
11.1 Main estimates established in this section 113
11.2 First refinement on the metric components 114
11.3 First refinement for the scalar field 116
11.4 Second refinement for the scalar field and the metric 118
11.5 A secondary bootstrap argument 120
12 High-Order Refined L2 Estimates 125
12.1 Objective of this section and prefiminary 125
12.2 Main estimates in this section 131
12.3 Applications to the derivation of refined decay estimates 136
13 High-Order Refined Sup-Norm Estimates 139
13.1 Preliminary 139
13.2 Main estimate in this section 141
14 Low-Order Refined Energy Estimate for the Scalar Field 147
Appendix A Revisiting the wave-Klein-Gordon model 151
Appendix B Sup-norm estimate for the wave equations 153
Appendix C Sup-norm estimate for the Klein-Gordon equation 159
Appendix D Commutator estimates for the hyperboloidal frame 165
Bibliography 171
