On the Structure of Solutions to the Static Vacuum Einstein Equations

These equations are the simplest equations for Ricci-flat 4-manifolds. They have been extensively studied in the physics literature on classical relativity, where the solutions represent space-times outside regions of matter which are translation and reflection invariant in the time direction t. However, with the exception of some notable instances, (c.f. Theorem 0.1 below), many of the global properties of solutions have not been rigorously examined, either from mathematical or physical points of view, c.f. [Br] for example. This paper is also motivated by the fact that solutions of the static vacuum equations arise in the study of degenerations of Yamabe metrics (or metrics of constant scalar curvature) on 3-manifolds, c.f. [A1]. Because of this and other related applications of these equations to the geometry of 3-manifolds, we are interested in general mathematical aspects of the equations and their solutions which might not be physically relevant; for example, we allow solutions with negative mass. In this paper, we will be mostly concerned with the geometry of the 3-manifold solutions (M,g, u) of (0.1), (i.e. the space-like hypersurfaces), and not with the 4-manifold metric. Thus, the choice of Riemannian or Lorentzian geometry on N in (0.2) will play no role. This considerably simplifies the discussion of singularities and boundary structure, but still allows for a large variety of behaviors; c.f. [ES] for a survey on singularities of space-times. Obviously, there are no non-flat solutions to (0.1) on closed manifolds, and so it will be assumed that M is an open, connected oriented 3-manifold. Let M̄ be the metric (or Cauchy) completion of M and ∂M the metric boundary, so that M̄ = M ∪ ∂M is complete as a metric space. In order to avoid trivial ambiguities, we will only consider maximal solutions of the equations (0.1). For example any domain Ω in R3 with the flat metric, and u a positive constant, satisfies (0.1). In this case, the metric boundary ∂Ω is artificial, and has no intrinsic relation with the geometry of the solution. The solution obviously extends to a larger domain, i.e. R3. Thus, we only consider maximal solutions (M,g, u), in the sense that (M,g, u) does not extend to a larger domain (M ′, g′) ⊃ (M,g) with u > 0 on M ′. It follows that at the metric boundary ∂M of M , either the metric or u degenerates in some way or u approaches 0 in some way, (or a combination of such).