This book discusses an approach to the study of global properties of solutions to the equations of general relativity, the Einstein field equations, in which the notion of conformal transformation plays a central role. The use of conformal transformations in differential geometry dates back, at least, to the work of Hermann Weyl in the 1920s. Their application to global questions in general relativity, as presented in this book, stems from the seminal work of Roger Penrose in the 1960s in which the close connection between the global causal structure of the solutions to the equations of general relativity and conformal geometry was established. Penrose's key insights are that the close relation between the propagation of the gravitational field and the structure of light cones which holds locally in a spacetime is also preserved in the case of large scales and that the asymptotic behaviour of the gravitational field can be conveniently analysed in terms of conformal extensions of the spacetime. In the following decade Penrose's ideas were polished, extended and absorbed into the mainstream research of general relativity by a considerable number of researchers – finally leading to the influential notion of asymptotic simplicity. The subject reached its maturity when this formal theory was combined with the methods of the theory of partial differential equations (PDEs). This breakthrough is mainly due to the work of Helmut Friedrich in the early 1980s, who – through the conformal Einstein field equations – showed that ideas of conformal geometry can be used to establish the existence of large classes of solutions to the Einstein field equations satisfying Penrose's notion of asymptotic simplicity. As a result of this work it is now clear that Penrose's original insights hold for large classes of spacetimes and not only for special explicitly known solutions.

This book develops the theory of the conformal Einstein field equations from the ground up and discusses their applications to the study of asymptotically simple spacetimes. Special attention is paid to results concerning the existence and stability of de Sitter-like spacetimes, the semiglobal existence and stability of Minkowski-like spacetimes using hyperboloidal Cauchy problems and the construction of anti-de Sitter-like spacetimes from initial boundary value problems. These results belong to the canon of modern mathematical relativity.