On Long-time Evolution in General Relativity and Geometrization of 3-manifolds

In this paper, we describe certain relations between the vacuum Einstein evolution equations in general relativity and the geometrization of 3-manifolds. In its simplest terms, these relations arise by analysing the long-time asymptotic behavior of natural space-like hypersurfaces Στ , diffeomorphic to a fixed Σ, in a vacuum space-time. In the best circumstances, the induced asymptotic geometry on Στ would implement the geometrization of the 3-manifold Σ. We present several results illustrating this relationship, some of which require however rather strong hypotheses. Thus, besides proving these results, a second purpose of this paper is to clarify and make precise what some of the major difficulties are in carrying this program out to a deeper level. In this setting, we thus discuss a number of open problems and conjectures, some of which are well-known, which are of interest both in general relativity and in 3-manifold geometry. The idea of geometrizing 3-manifolds, and its proof in many important cases, is due to Thurston. We refer to [38] for an introduction from the point of view of hyperbolic geometry. A survey of geometrization from the point of view of general Riemannian geometries is given in [1], where some early relations with issues in general relativity were also presented. We also refer to [6] and [34] for surveys on topics in general relativity related to this paper. Let (M, g) be a vacuum space-time, i.e. a 4-manifold M with smooth (C) Lorentz metric g, of signature (−,+,+,+), satisfying the Einstein vacuum equations