A rigidity theorem for nonvacuum initial data

Abstract

In this note we prove a theorem on nonvacuum initial data for general relativity. The result presents a “rigidity phenomenon” for the extrinsic curvature, caused by the nonpositive scalar curvature. More precisely, we claim that in the case of an asymptotically flat nonvacuum initial data if the spatial metric has everywhere nonpositive scalar curvature, then the extrinsic curvature cannot be compactly supported.