Christodoulou's recent analysis of naked singularities in time-symmetric Tolman-Bondi collapse is simplified and generalised to a wider class of Tolman-Bondi models. The strengths ofthe naked singularities are assessed in terms of limiting focusing conditions.
A marginal 2-surface is, by definition, covered by a 2-surface admitting a nowhere-zero null normal field of zero expansion. A complete marginal 2-surface which is either compact, or non-compact and subject to certain asymptotic geometric constraints, is said to be well tempered. A well tempered marginal 2-surface admitting a nowhere-timelike variation through well tempered marginal 2-surfaces is said to be stable. In a spacetime satisfying the dominant energy condition, a stable well tempered marginal 2-surface is homeomorphic to S2, P2, R2, T2, S*R a Klein bottle or a Mobius band. Only the topologies S2, P2 and R2 may be compatible with genericity conditions. Of stable compact embedded marginal 2-surfaces which are bounding in a spacelike hypersurface, those homeomorphic to P2 occur in pairs, as do those homeomorphic to a Klein bottle. Stable compact embedded marginal 2-surfaces which are achronal and develop from data on a simply connected partial Cauchy surface are homeomorphic to S2 or T2.
According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinities J + and J>~ is asymptotically simple if it has no closed timelike curves, and all its endless null geodesies originate from J~ and terminate at J> + . The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints. The present paper deals with the general case. It is shown that + is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal region J § of =/ + is diffeomorphic to § 2 x U, and every compact connected spacelike 2-surface in/ + is contained in JQ and is a strong deformation retract of both J% and ,/ + . Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincare conjecture, are diffeomorphic to IR 3 .
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