Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken–Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard $[t=0]$-timeslice of the Schwarzschild solution. In this paper, we generalize his result to asymptotically flat initial data sets and weaken additional smallness assumptions made by Metzger. Furthermore, we prove that the CE-surfaces are in a well-defined sense (asymptotically) independent of time if the linear momentum vanishes.
"Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry." J. Differential Geom. 109 (2) 257 - 289, June 2018. https://doi.org/10.4310/jdg/1527040873