Recent trends in Lorentzian geometry

✍ Scribed by Miguel Sánchez (eds.)

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 357
• Series: Springer proceedings in mathematics & statistics, 26
• Category: Library

✦ Synopsis

Hyperbolic metrics on Riemann surfaces and spacelike CMC-1 surfaces in de Sitter 3-space, Shoichi Fujimori, Yu Kawakami, Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara and Kotaro Yamada. - Bernstein results and parabolicity of maximal surfaces in Lorentzian product spaces, Alma L. Albujer and Luis J. Alias, Calabi. - Umbilical-Type Surfaces in Spacetime, Jose M. M. Senovilla. - Stability of marginally outer trapped surfaces and applications, Marc Mars. - Area inequalities for stable marginally trapped surfaces, Jose Luis Jaramillo. - Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold, Erasmo Caponio. - Global geodesic properties of Godel type spacetimes, R. Bartolo, A.M. Candela and J.L. Flores.- The geometry of collapsing isotropic fluids, Roberto Giamb'o and Giulio Magli. - Conformally standard stationary spacetimes and Fermat metrics, Miguel Angel Javaloyes. - Can we make a Finsler metric complete by a trivial projective change? Vladimir S. Matveev. - The c-boundary construction of spacetimes: application to stationary Kerr spacetime, J.L. Flores and J. Herrera. - On the isometry group of Lorentz manifolds, Leandro A. Lichtenfelz, Paolo Piccione, and Abdelghani Zeghib. - Conformally flat homogeneous Lorentzian Manifolds, Kyoko Honda and Kazumi Tsukada. - Polar actions on symmetric spaces, Jose Carlos Diaz-Ramos. - (para)-Kahler Weyl structures, P. Gilkey and S. Nikcevic

✦ Table of Contents

Cover......Page 1
Recent Trends in Lorentzian
Geometry......Page 4
(para)-Kähler Weyl Structures......Page 8
Appendix......Page 12
1 Generalized CMC-1 Faces in de Sitter 3-Space......Page 15
2 Extended Hyperbolic Metrics on Riemann Surfaces......Page 26
3 Fundamental Properties of Co-orientable Extended Hyperbolic Metrics......Page 37
4 Classification of de Sitter Catenoids......Page 43
5 Hyperbolic Metrics with At Most Two Regular Singularities......Page 54
Appendix A: Projective Connections......Page 56
Appendix B: A Property of Subgroups in PSU(1,1)......Page 57
References......Page 59
1 Introduction......Page 61
2 The Classical Calabi–Bernstein Theorem in R31......Page 63
2.1 Space-Like Graphs and the Calabi–Bernstein Theorem......Page 65
2.2 Romero's Proof Based on the Liouville Theorem for Harmonic Functions on R2......Page 67
2.3 Alías and Palmer's Proof Based on a Duality Result......Page 68
2.4 Alías and Palmer's Proof Based on a Local Integral Inequality for the Gaussian Curvature......Page 70
3 Some Preliminaries on Lorentzian Product Spaces......Page 72
4 A Parametric Version of a Calabi–Bernstein Result......Page 74
5 A Nonparametric Version of a Calabi–Bernstein Result......Page 77
6 Some Nontrivial Entire Maximal Graphs in H2R1......Page 79
6.1 Duality Between Minimal and Maximal Graphs......Page 80
6.2 More Examples......Page 84
7 Relative Parabolicity of Maximal Surfaces......Page 86
7.1 Relative Parabolicity and Entire Maximal Graphs......Page 91
8 A Local Estimate for Maximal Surfaces in a Lorentzian Product Space......Page 93
References......Page 95
1 Introduction......Page 98
2 Basic Concepts and Notation......Page 100
2.1 Extrinsic Geometry: Second Fundamental Forms and Weingarten Operators......Page 101
2.3 The Mean Curvature Vector Field H and Its Causal Character......Page 102
2.4 The Extrinsic Vector Field G......Page 104
2.6 Curvatures: Gauss and Ricci Equations......Page 105
3 Umbilical-Type, Pseudo-umbilical, and Related Surfaces......Page 107
4 Proof of the Main Theorems......Page 111
5 Some Important Corollaries and Consequences......Page 114
References......Page 118
1 Introduction......Page 121
2.1 Geometry of Spacelike Surfaces......Page 124
2.2 Marginally Outer Trapped Surfaces......Page 126
3 Stability of MOTS......Page 128
3.1 Principal Eigenvalue of the Stability Operator......Page 131
3.2 Dependence of the Stability Properties on the Direction......Page 134
4 Barrier Properties of MOTS......Page 136
4.1 MOTS and Symmetries......Page 139
5 MOTS and Killing Horizons......Page 141
5.1 Killing Horizons......Page 142
5.2 Stability Operator of MOTS in Killing Horizons......Page 144
6 Axially Symmetric MOTS and Angular Momentum......Page 146
References......Page 147
1 Introduction......Page 149
2.1 Geometry of 2-Surfaces......Page 150
2.1.1 Axisymmetry......Page 151
2.2 Electromagnetic Field......Page 152
2.2.1 Yang–Mills Fields......Page 153
3.1 Basic Definitions......Page 154
3.2 Integral-Inequality Characterizations of MOTS Stability......Page 155
3.3.1 On an Averaged Outermost Stably Conditions for MOTS......Page 157
3.3.2 Towards Axisymmetry Relaxation......Page 158
4 The Area-Angular Momentum Inequality......Page 159
5 The Area-Charge Inequality Generalizations......Page 161
5.2 Further Generalizations......Page 163
5.2.1 The Cosmological Constant and Stability Operator Eigenvalue......Page 164
6 The Area-Angular Momentum-Charge Inequality......Page 165
7 Discussion......Page 167
References......Page 169
1 Introduction......Page 172
2 A Review of Bishop's Proof......Page 174
3 Infinitesimal Convexity Implies Local Convexity......Page 176
4.1 Convexity with Respect to the Geodesics Having the Same Causal Character......Page 180
4.2 Geometric Convexity......Page 181
4.3 Some Applications to Geodesic Connectedness......Page 182
References......Page 185
1 Introduction......Page 187
2 The Variational Principle......Page 190
3 Geodesic Connectedness in GTS......Page 193
4 Geodesic Completeness......Page 196
References......Page 200
1 Introduction......Page 202
2 Relativistic Stars in Spherical Symmetry......Page 203
2.1 Geometrical and Physical Assumptions......Page 204
2.2 Singularity Formation and Cosmic Censor......Page 206
3 Regular Isotropic Models......Page 207
4.1 Barotropic Fluids with a Linear Equation of State......Page 209
4.2 Generalized Chaplygin Gas......Page 210
References......Page 212
1 Introduction......Page 213
2 Finsler and Randers Metrics......Page 214
3 Fermat's Principle in Conformally Standard Stationary SpaceTimes......Page 215
4 Causality and Fermat Metrics......Page 220
5 Causal Boundaries and Fermat Metrics......Page 224
6 Existence of Light-Like Geodesics......Page 227
6.1 Morse Theory for Light-Like Geodesics......Page 229
6.2 t-Periodic Light-Like Geodesics and the Closed Geodesic Problem......Page 230
6.3 Alternative Functional to Energy......Page 231
7.2 Time-Like Geodesics with Fixed Arrival Proper Time......Page 232
7.3 Conformal Maps and Almost Isometries......Page 233
References......Page 234
1 Statement of the Problem, Motivation and the Main Result......Page 237
1.1 Motivation......Page 238
1.2 Main Result......Page 240
2 Proof of Theorem 1......Page 241
References......Page 248
1 Introduction......Page 249
2 C-Boundary of SpaceTimes......Page 250
2.1 Point Set and Chronological Levels......Page 251
2.2 Topological Level......Page 252
2.3 Properties of the C-Boundary......Page 253
2.4 Relation with the Conformal Boundary......Page 255
3.2 Cauchy Completions......Page 256
3.3 Gromov Completions......Page 257
3.4 Busemann Completion......Page 259
3.4.1 Interpretation of the Operators ,......Page 260
3.4.2 Topology on the Busemann Completion......Page 261
4 C-Boundary for Stationary SpaceTimes......Page 263
5 Application to Stationary Kerr SpaceTime......Page 265
6 Appendix......Page 274
References......Page 280
1 Riemannian Versus Lorentzian Isometries......Page 282
2.1 First-Order Structure Function......Page 284
2.2 Prolongations......Page 286
3 Automorphism Groups......Page 288
3.1 The Automorphism Group of a Parallelism......Page 290
4 Groups Acting Isometrically on Compact Lorentz Manifolds......Page 293
5 Geometry of Manifolds with Non Compact Isometry Group......Page 294
5.1 When the Identity Connected Component Is Not Compact......Page 295
5.3 When the Isometry Group Has Infinitely Many Connected Components......Page 296
References......Page 298
1 Introduction......Page 299
2 Preliminaries......Page 301
3 Case 1......Page 304
4 Case 2 with One Eigenvalue......Page 311
5 Case 2 with Two Eigenvalues and -......Page 315
References......Page 318
1 Introduction......Page 319
2 Polar and Hyperpolar Actions......Page 320
3 Symmetric Spaces......Page 323
3.1 Riemannian Symmetric Spaces of Noncompact Type......Page 325
4 Classification of Polar Actions......Page 326
4.1 Euclidean Spaces......Page 327
4.2.1 Symmetric Spaces of Rank One......Page 328
4.2.2 Symmetric Spaces of Higher Rank......Page 330
4.3 Symmetric Spaces of Noncompact Type......Page 332
References......Page 337
1 Introduction......Page 339
1.1 Riemannian, Affine, and Weyl Geometry......Page 340
1.2 Kähler Geometry......Page 342
2.1 Representation Theory......Page 344
2.2 The Singer–Thorpe and the Higa Decompositions......Page 346
2.3 The Tricerri–Vanhecke Decompositions......Page 347
3.3 The Module 20,-.4 if m=4......Page 348
3.4 The Module 2-.4 if m-.46......Page 350
3.5 The Module 2-.4 if m =4......Page 351
4.1 The Representations W-.4,i for i =1,2,3......Page 353
4.2 The Representations 2-.4 and 20,-.4......Page 355
References......Page 356