A CMC existence result for expanding cosmological spacetimes

TL;DR

This paper proves the existence of constant mean curvature (CMC) Cauchy surfaces in expanding cosmological spacetimes under certain conditions, using barrier constructions and mean curvature flow, and addresses related conjectures.

Contribution

It establishes new CMC existence results for cosmological spacetimes, confirming conjectures under specific geometric and energy conditions.

Findings

Existence of CMC Cauchy surfaces in expanding spacetimes

Construction of barriers in the support sense

Results extended to positive cosmological constant cases

Abstract

We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $\Lambda > 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.