Remarks on the existence of CMC Cauchy surfaces

TL;DR

This paper reviews known results and presents new findings on the existence of constant mean curvature (CMC) spacelike hypersurfaces in cosmological spacetimes, which are crucial for solving Einstein's equations.

Contribution

It revisits previous CMC existence results and introduces new existence theorems motivated by recent conjectures, enhancing understanding of spacetime conformal structures.

Findings

Reaffirmed the existence of CMC hypersurfaces under certain conditions

Proposed new existence results based on recent conjectures

Discussed implications for the conformal structure of cosmological spacetimes

Abstract

As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.