Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry
The author deals with foliations by spheres of constant expansion near infinity (CE), useful in general relativity. In [Invent. Math. 124, No. 1--3, 281--311 (1996; Zbl 0858.53071)], \textit{G. Huisken} and \textit{S.-T. Yau} proved that, near infinity, every Riemannian manifold is uniquely foliated by stable surfaces with constant mean curvature (CMC) if it is asymptotically equal to the Schwarzschild solution and has positive mass. Their decay assumptions were subsequently weakened by \textit{J. Metzger} [J. Differ. Geom. 77, No. 2, 201--236 (2007; Zbl 1140.53013)] and other authors. The paper under review extends the setting to asymptotically flat initial data sets with non-vanishing ADM mass, assumes weaker decay conditions on the metric \(g\), and partly replaces the decay condition of $k$ by an integral smallness condition. In addition, it is shown that the unique CE-surfaces constructed are asymptotically independent of time if the linear momentum vanishes. This result is motivated by earlier work of the author on the time evolution of CMC spheres.
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