Developments in Lorentzian Geometry: GeLoCor 2021, Cordoba, Spain, February 1-5

Description

This proceedings volume gathers selected, revised papers presented at the X International Meeting on Lorentzian Geometry (GeLoCor 2021), virtually held at the University of Córdoba, Spain, on February 1-5, 2021. It includes surveys describing the state-of-the-art in specific areas, and a selection of the most relevant results presented at the conference. Taken together, the papers offer an invaluable introduction to key topics discussed at the conference and an overview of the main techniques in use today. This volume also gathers extended revisions of key studies in this field. Bringing new results and examples, these unique contributions offer new perspectives to the original problems and, in most cases, extend and reinforce the robustness of previous findings. Hosted every two years since 2001, the International Meeting on Lorentzian Geometry has become one of the main events bringing together the leading experts on Lorentzian geometry. In this volume, the reader will find studies on spatial and null hypersurfaces, low regularity in general relativity, conformal structures, Lorentz-Finsler spacetimes, and more. Given its scope, the book will be of interest to both young and experienced mathematicians and physicists whose research involves general relativity and semi-Riemannian geometry.  Organization Preface Contents Semi-Riemannian Cones with Parallel Null Planes 1 Introduction 2 The Induced Structure on the Base 3 Consequences of the Fundamental Equations 4 The Local Form of the Metric on the Base References Nilpotent Structures of Neutral 4-Manifolds and Light-Like Surfaces 1 Introduction 2 Complex Structures and Paracomplex Structures of 4-Dimensional Neutral Vector Spaces 3 Nilpotent Structures of 4-Dimensional Neutral Vector Spaces 4 Almost Complex Structures and Almost Paracomplex Structures of Neutral 4-Manifolds 5 Almost Nilpotent Structures of Neutral 4-Manifolds 6 Light-Like Surfaces in Neutral 4-Manifolds References Positive Energy Theorems in Fourth-Order Gravity 1 Introduction 2 Preliminaries 3 Conservation Principles and Fourth Order Energy 4 Positive Energy Theorem for Einstein Metrics 5 Positive Energy Theorem for Stationary Solutions 6 The Q-Curvature Positive Mass Theorem References Curvature and Killing Vector Fields on Lorentzian 3-Manifolds 1 Introduction 2 The Newman-Penrose Formalism for Lorentzian 3-Manifolds 3 The Newman-Penrose Formalism and Global Obstructions 3.1 Evolution Equations for Divergence, Twist, and Shear 4 The Newman-Penrose Formalism and Local Classifications 4.1 The Riemannian Case 4.2 Local Coordinates 4.3 The Local Classification 4.4 The Lorentzian Setting References Bochner-Flat Para-Kähler Surfaces 1 Introduction 2 Walker Structures 2.1 Self-Dual Walker Manifolds 3 Bochner-Flat Para-Kähler Surfaces 3.1 Bochner-Flat Para-Kähler Surfaces of Constant Scalar Curvature 3.2 Some Examples of Bochner-Flat Para-Kähler Structures of Non-constant Scalar Curvature References Remarks on the Existence of CMC Cauchy Surfaces 1 Introduction 2 Some CMC Existence Results 2.1 CMC Existence Result from a Spacetime Curvature Condition 2.2 CMC Existence Result Related to a Conjecture of Dilts and Holst 3 Remarks on the Conformal Structure of Cosmological Spacetimes References Lorentzian Area and Volume Estimates for Integral Mean Curvature Bounds 1 Introduction 2 Background 2.1 Our Setting 2.2 Comparison Spaces 2.3 The Cosmological Time Function and Its Properties 3 Area and Volume Estimates 3.1 Basic Area and Volume Estimates Using Integral Mean Curvature Bounds 3.2 Proof of Theorem 2 4 Generalized Area Estimates for MathID486Σt 5 Extending Theorem 2 to Subsets and Non-compact MathID519Σ with Finite Area 6 Example: For p less than np