Recent trends in Lorentzian geometry [electronic resource] /
Sánchez, Miguel.editor.
Ortega, Miguel.editor.
Romero, Alfonso.editor.
Traditionally, Lorentzian geometry has been used as a necessary tool to understand general relativity, as well as to explore new genuine geometric behaviors, far from classical Riemannian techniques. Recent progress has attracted a renewed interest in this theory for many researchers: long-standing global open problems have been solved, outstanding Lorentzian spaces and groups have been classified, new applications to mathematical relativity and high energy physics have been found, and further connections with other geometries have been developed.   Samples of these fresh trends are presented in this volume, based on contributions from the VI International Meeting on Lorentzian Geometry, held at the University of Granada, Spain, in September, 2011. Topics such as geodesics, maximal, trapped and constant mean curvature submanifolds, classifications of manifolds with relevant symmetries, relations between Lorentzian and Finslerian geometries, and applications to mathematical physics are included.     This book will be suitable for a broad audience of differential geometers, mathematical physicists and relativists, and researchers in the field.
Description based upon print version of record.
Recent Trends in Lorentzian Geometry; Preface; Contents; Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space; 1 Generalized CMC-1 Faces in de Sitter 3-Space; 2 Extended Hyperbolic Metrics on Riemann Surfaces; 3 Fundamental Properties of Co-orientable Extended Hyperbolic Metrics; 4 Classification of de Sitter Catenoids; 5 Hyperbolic Metrics with At Most Two Regular Singularities; Appendix A: Projective Connections; Appendix B: A Property of Subgroups in PSU(1,1); References
Calabi-Bernstein Results and Parabolicity of Maximal Surfaces in Lorentzian Product Spaces1 Introduction; 2 The Classical Calabi-Bernstein Theorem in R31; 2.1 Space-Like Graphs and the Calabi-Bernstein Theorem; 2.2 Romero's Proof Based on the Liouville Theorem for Harmonic Functions on R2; 2.3 Alías and Palmer's Proof Based on a Duality Result; 2.4 Alías and Palmer's Proof Based on a Local Integral Inequality for the Gaussian Curvature; 3 Some Preliminaries on Lorentzian Product Spaces; 4 A Parametric Version of a Calabi-Bernstein Result; 5 A Nonparametric Version of a Calabi-Bernstein Result
6 Some Nontrivial Entire Maximal Graphs in H2R16.1 Duality Between Minimal and Maximal Graphs; 6.2 More Examples; 7 Relative Parabolicity of Maximal Surfaces; 7.1 Relative Parabolicity and Entire Maximal Graphs; 8 A Local Estimate for Maximal Surfaces in a Lorentzian Product Space; References; Umbilical-Type Surfaces in SpaceTime; 1 Introduction; 2 Basic Concepts and Notation; 2.1 Extrinsic Geometry: Second Fundamental Forms and Weingarten Operators; 2.2 Special Bases on x(S); 2.3 The Mean Curvature Vector Field H and Its Causal Character; 2.4 The Extrinsic Vector Field G
2.5 The Normal Connection One-Form s2.6 Curvatures: Gauss and Ricci Equations; 3 Umbilical-Type, Pseudo-umbilical, and Related Surfaces; 4 Proof of the Main Theorems; 5 Some Important Corollaries and Consequences; 6 Final Considerations; References; Stability of Marginally Outer Trapped Surfaces and Applications; 1 Introduction; 2 Definition of Marginally Outer Trapped Surface; 2.1 Geometry of Spacelike Surfaces; 2.2 Marginally Outer Trapped Surfaces; 3 Stability of MOTS; 3.1 Principal Eigenvalue of the Stability Operator; 3.2 Dependence of the Stability Properties on the Direction
4 Barrier Properties of MOTS4.1 MOTS and Symmetries; 5 MOTS and Killing Horizons; 5.1 Killing Horizons; 5.2 Stability Operator of MOTS in Killing Horizons; 6 Axially Symmetric MOTS and Angular Momentum; References; Area Inequalities for Stable Marginally Trapped Surfaces; 1 Introduction; 2 Geometric and Physical Elements; 2.1 Geometry of 2-Surfaces; 2.1.1 Axisymmetry; 2.2 Electromagnetic Field; 2.2.1 Yang-Mills Fields; 3 Stability of Marginally Outer Trapped Surfaces; 3.1 Basic Definitions; 3.2 Integral-Inequality Characterizations of MOTS Stability
3.3 Variants to the Stably Outermost Condition
Includes bibliographical references.
Traditionally, Lorentzian geometry has been used as a necessary tool to understand general relativity, as well as to explore new genuine geometric behaviors, far from classical Riemannian techniques. Recent progress has attracted a renewed interest in this theory for many researchers: long-standing global open problems have been solved, outstanding Lorentzian spaces and groups have been classified, new applications to mathematical relativity and high energy physics have been found, and further connections with other geometries have been developed.   Samples of these fresh trends are presented in this volume, based on contributions from the VI International Meeting on Lorentzian Geometry, held at the University of Granada, Spain, in September, 2011. Topics such as geodesics, maximal, trapped and constant mean curvature submanifolds, classifications of manifolds with relevant symmetries, relations between Lorentzian and Finslerian geometries, and applications to mathematical physics are included.     This book will be suitable for a broad audience of differential geometers, mathematical physicists and relativists, and researchers in the field.
Convex geometry .
Discrete geometry.
Hyperbolic geometry.
Differential geometry.
Convex and Discrete Geometry.
Hyperbolic Geometry.
Differential Geometry.