The closed-universe recollapse conjecture

It is widely believed that all expanding S3 closed universes that satisfy the standard energy conditions recollapse to a second singularity. We show that this is false even for Friedmann universes: they construct an ever-expanding S3 Friedmann universe in which the matter tensor satisfies the strong, weak and dominant energy conditions and the generic condition. We prove a general recollapse theorem for Friedmann universes: if the positive pressure criterion, dominant energy condition and matter regularity condition hold, then an S3 Friedmann universe must recollapse. We show that all known vacuum solutions with Cauchy surface topology S3 or S2×S1 recollapse, and we conjecture that this is a property of all vacuum solutions of Einstein's equations with such Cauchy surfaces. The authors consider a number of Kantowski–Sachs and Bianchi IX universes with various matter tensors, and formulate a new recollapse conjecture for matter-filled universes.