Curvature and closed trapped surfaces in 4-dimensional space-times

Abstract

The way a spacelike surfaceH sits in a 4-dimensional space-timeM may be measured by the average null curvature functionφ H and the shape functionΩ H ofH. Relations between the shape functionΩ H of a spaceiike surfaceH and curvature of a 4-dimensional space-time obeying the Einstein equation are investigated. Some relations between the shape functions of compact spacelike surfaces and infinite curvature are obtained and discussed. Assuming some curvature conditions, some results concerning the evolution of closed trapped surfaces from a restricted type of marginally trapped surfaces diffeomorphic toS 2 are obtained.