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This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic.

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation-the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant-however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.

The conformal Einstein equations for a tracefree (radiation) perfect fluid are derived in terms of the Levi-Civita connection of a conformally rescaled metric. These equations are used to provide a non-linear stability result for de Sitter-like tracefree (radiation) perfect fluid Friedman-Lemaître-Robertson-Walker cosmological models. The solutions thus obtained exist globally towards the future and are future geodesically complete.The conformal Einstein field equations have proven a powerful tool to analyse the stability and the global properties of vacuum, electro-vacuum and Yang-Mills-electro-vacuum spacetimes -see e.g. [9,10,11,18,19,20]. By contrast, to the best of our knowledge, there has been no attempt to make use of conformal methods to analyse similar issues in spacetimes whose matter content is given by a perfect fluid. In this article we make a first step in this direction. We discuss the stability and the global properties of a class of cosmological spacetimes having as a source a perfect fluid with tracefree energy-momentum tensor. The solutions we construct are non-linear perturbations of a Friedman-Lemaître-Robertson-Walker (FLRW) reference spacetime.The present analysis is to be regarded as a first step in the development of conformal methods for the discussion of cosmological models whose matter content is described by a perfect fluid. Hence, we restrict our attention to the simplest case from the point of view of conformal methods: perturbations of a traceless prefect fluid cosmological model with compact spatial sections of positive constant curvature. Generalisation of our analysis to more general background solutions and equations of state will be discussed elsewhere.The problem of the non-linear stability of FLRW cosmologies and the exponential decay of perturbations is considered in [23]. In that reference, a frame formulation of the Einstein-perfect fluid system [12] is used to obtain a suitable symmetric hyperbolic evolution system for which the Kreiss-Lorenz theory can be readily applied -see [17]. The results obtained hold for a large class of equations of state, but not very stiff ones -like the pure radiation case discussed in the present article. More recently, the problem of the non-linear stability of the irrotational Euler-Einstein system for de Sitter-like spacetimes has been analysed in [24]. This analysis shows that FLRW background solutions with pressurep and densityρ related by a barotropic equations of state of the formp = γρ with 1 < γ < 4 3 are future asymptotically stable under small irrotational *

A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain: (i) a new proof of the stability of Einstein-Maxwell de Sitterlike spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein-Maxwell spacetimes.