On the topology and future stability of the universe.

*(English)*Zbl 1270.83005
Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 978-0-19-968029-0/hbk). xiv, 718 p. (2013).

The 34 chapters of this book deal with several aspects of the local and global dynamics of solutions of the Einstein field equation. The first chapters present standard text-book material like the Cauchy problem, the topology of the isotropic universe, and approximations to homogeneous world models.

The next part contains subjects not usually found in text-books, like Vlasov matter and the Einstein-Vlasov equation. Then there are several chapters on methods to solve non-linear partial differential equations locally and globally including the introduction of distributions.

The final most advanced topics include the future global nonlinear stability and cosmological models with arbitrary closed spatial topology.

The appendices present several details of the main part which have mostly also an interest in their own right. Examples: A counterexample to local existence for a special non-linear wave equation, the metric of isotropic world models in several coordinates, and the curvature of left-invariant metrics.

The Acknowledgement says that Alan Rendall played an important role in the initial stages of the project which resulted in this book.

The book ends with 159 items reference list and four pages subject index. For another book on a similar topic written by the same author see [The Cauchy problem in general relativity. Zürich: European Mathematical Society (EMS) (2009; Zbl 1169.83003)].

The next part contains subjects not usually found in text-books, like Vlasov matter and the Einstein-Vlasov equation. Then there are several chapters on methods to solve non-linear partial differential equations locally and globally including the introduction of distributions.

The final most advanced topics include the future global nonlinear stability and cosmological models with arbitrary closed spatial topology.

The appendices present several details of the main part which have mostly also an interest in their own right. Examples: A counterexample to local existence for a special non-linear wave equation, the metric of isotropic world models in several coordinates, and the curvature of left-invariant metrics.

The Acknowledgement says that Alan Rendall played an important role in the initial stages of the project which resulted in this book.

The book ends with 159 items reference list and four pages subject index. For another book on a similar topic written by the same author see [The Cauchy problem in general relativity. Zürich: European Mathematical Society (EMS) (2009; Zbl 1169.83003)].

Reviewer: Hans-Jürgen Schmidt (Potsdam)

##### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83F05 | Cosmology |

85A40 | Cosmology |

54F65 | Topological characterizations of particular spaces |

00A79 | Physics (Use more specific entries from Sections 70-XX through 86-XX when possible) |

83C55 | Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.) |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |

35L05 | Wave equation |

35Q83 | Vlasov equations |