Galloway–Ling conjecture on CMC Cauchy surfaces

Let $(M,g)$ be a globally hyperbolic, future timelike geodesically complete spacetime with compact Cauchy surfaces satisfying the strong energy condition. A CMC Cauchy surface is a Cauchy surface whose mean curvature is constant.

Galloway–Ling conjecture. Then $M$ contains a CMC Cauchy surface.

The conjecture concerns the existence of constant-mean-curvature Cauchy surfaces in cosmological spacetimes. It is motivated by results establishing such a surface under the additional assumption of nonpositive timelike sectional curvature; the status of the stated general version is not resolved in the supplied text.