Two Rigidity Theorems for Gravity

In this paper we prove two theorems on initial data for general relativity. The results present “rigidity phenomena” for the extrinsic curvature, caused by the negativity of the scalar curvature. More precisely, Theorem 1 states that in the case of non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. Theorem 4 states a similar but sharper fact for vacuum initial data, namely if the scalar curvature is somewhere negative then a vacuum initial data set cannot be asymptotically flat over a “generic” Riemannian three-manifold (defined below). Since positivity, negativity of the scalar curvature is strongly influenced by the topology of the underlying three-manifold, we observe a subtle link between the topology of the space-like submanifold and the asymptotic behaviour of the initial data set in a globally hyperbolic space-time.