Singularities in universes with negative cosmological constant

It is well known that many universes with negative cosmological constant contain singularities. We shall generalize this result by proving that all closed universes with negative cosmological constant are both future and past timelike geodesically incomplete if the strong energy condition holds. No global causality conditions or restrictions on the initial data are used in the proof. Furthermore, we shall show that all open universes with a Cauchy surface and a negative cosmological constant are singular if the strong energy condition holds