Estimates for the Volume of a Lorentzian Manifold

We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume. 0. Introduction Let N be a (n + 1)-dimensional Lorentzian manifold and suppose that N can be decomposed in the form (0.1) N = N0 ∪N− ∪N+, where N0 has finite volume and N− resp. N+ represent the critical past resp. future Cauchy developments with not necessarily a priori bounded volume. We assume that N+ is the future Cauchy development of a Cauchy hypersurface M1, and N− the past Cauchy development of a hypersurface M2, or, more precisely, we assume the existence of a time function x, such that (0.2) N+ = x 0−1([t1, T+)), M1 = {x = t1}, N− = x 0−1((T−, t2]), M2 = {x = t2}, and that the Lorentz metric can be expressed as (0.3) ds̄ = e2ψ{−dx0 + σij(x, x)dxdx}, where x = (x) are local coordinates for the space-like hypersurface M1 if N+ is considered resp. M2 in case of N−. The coordinate system (x)0≤α≤n is supposed to be future directed, i.e. the past directed unit normal (ν) of the level sets (0.4) M(t) = {x = t} Received by the editors April 18, 2002. 2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.