On the topology and future stability of the universe

On the topology and future stability of the universe / Hans Ringström.

A general introduction to the initial value problem for Einstein's equations coupled to collisionless matter. The book contains a proof of future stability of models of the universe consistent with the current observational data and a discussion of the restrictions on the possible shapes of the...

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Bibliographic Details
Main Author: Ringström, Hans
Format: eBook
Language:English
Published: Oxford : Oxford University Press, 2013.
Series:Oxford mathematical monographs.
Subjects:
Cauchy problem.
MATHEMATICS > Differential Equations > Partial.
Cauchy problem
Universe.
Universe
Online Access:Click for online access
Table of Contents:
  • Contents
  • PART I: PROLOGUE
  • 1 Introduction
  • 1.1 General remarks on the limits of observations
  • 1.2 The standard models of the universe
  • 1.3 Approximation by matter of Vlasov type
  • 2 The Cauchy problem in general relativity
  • 2.1 The initial value problem in general relativity
  • 2.2 Spaces of initial data and associated distance concepts
  • 2.3 Minimal degree of regularity ensuring local existence
  • 2.4 On linearisations
  • 3 The topology of the universe
  • 3.1 An example of how to characterise topology by geometry
  • 3.2 Geometrisation of 3-manifolds3.3 A vacuum conjecture
  • 4 Notions of proximity to spatial homogeneity and isotropy
  • 4.1 Almost EGS theorems
  • 4.2 On the relation between solutions with small spatial variation and spatially homogeneous solutions
  • 5 Observational support for the standard model
  • 5.1 Using observations to determine the cosmological parameters
  • 5.2 Distance measurements
  • 5.3 Supernovae observations
  • 5.4 Concluding remarks
  • 6 Concluding remarks
  • 6.1 On the technical formulation of stability
  • 6.2 Notions of proximity to spatial homogeneity and isotropy6.3 Models of the universe with arbitrary closed spatial topology
  • 6.4 The cosmological principle
  • 6.5 Symmetry assumption
  • PART II: INTRODUCTORY MATERIAL
  • 7 Main results
  • 7.1 Vlasov matter
  • 7.2 Scalar field matter
  • 7.3 The equations
  • 7.4 The constraint equations
  • 7.5 Previous results
  • 7.6 Background solution and intuition
  • 7.7 Drawing global conclusions from local assumptions
  • 7.8 Stability of spatially homogeneous solutions
  • ""7.9 Limitations on the global topology imposed by local observations""""8 Outline, general theory of the Einsteinâ€?Vlasov system""; ""8.1 Main goals and issues""; ""8.2 Background""; ""8.3 Function spaces and estimates""; ""8.4 Existence, uniqueness and stability""; ""8.5 The Cauchy problem in general relativity""; ""9 Outline, main results""; ""9.1 Spatially homogeneous solutions""; ""9.2 Stability in the n-torus case""; ""9.3 Estimates for the Vlasov matter, future global existence and asymptotics""; ""9.4 Proof of the main results""; ""10 References to the literature and outlook""
  • 10.1 Local existence10.2 Generalisations
  • 10.3 Potential improvements
  • 10.4 References to the literature
  • PART III: BACKGROUND AND BASIC CONSTRUCTIONS
  • 11 Basic analysis estimates
  • 11.1 Terminology concerning differentiation and weak derivatives
  • 11.2 Weighted Sobolev spaces
  • 11.3 Sobolev spaces on the torus
  • 11.4 Sobolev spaces for distribution functions
  • 11.5 Sobolev spaces corresponding to a non-integer number of derivatives
  • 11.6 Basic analysis estimates
  • 11.7 Locally x-compact support
  • 12 Linear algebra

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