Quasiconvex Foliations and Asymptotically Flat Metrics of Non-negative Scalar Curvature

Abstract

where R is the scalar curvature of the Riemannian metric g induced on a maximal time-slice, and k is the second fundamental form of that slice in the ambient Lorentzian 4-manifold. It is clear from (1) that R ≥ 0. The remaining Einstein vacuum equations can be seen as governing the evolution of the data consisting of the first and second fundamental form g and k. Since this evolution traces a continuous path in the space of initial data with the appropriate topology, it is natural to ask: what are the topological properties of this space of data? In particular, a question of considerable importance is whether this space is connected. It is possible to show using the conformal method that two sets of initial data (g, k), (g′, k′) are in the same pathconnected component of the space of initial data if and only the metrics g and g′ are in the same path-connected component of the space of metrics of non-negative scalar curvature [17]. A topological 2-sphere in a Riemannian manifold is said to be quasiconvex if its Gauss and mean curvatures are positive. A foliation is quasiconvex