# Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry

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## Abstract

In 1996, Huisken–Yau showed that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed CMC-surfaces if it is asymptotically equal to the (spatial) Schwarzschild solution and has positive mass. Their assumptions were later weakened by Metzger, Huang, Eichmair–Metzger and others. We further generalize these existence results in dimension three by proving that it is sufficient to assume asymptotic flatness and non-vanishing mass to conclude the existence and uniqueness of the CMC-foliation and explain why this seems to be the conceptually optimal result. Furthermore, we generalize the characterization of the corresponding coordinate CMC-center of mass by the ADM-center of mass proven previously by Corvino–Wu, Huang, Eichmair–Metzger and others (under other assumptions).

## Mathematics Subject Classification

89-02 53-02 58-02## Notes

### Acknowledgments

The author wishes to express gratitude to Gerhard Huisken for suggesting this topic, many inspiring discussions and ongoing supervision. Further thanks is owed to Lan-Hsuan Huang for suggesting the use of the Bochner–Lichnerowicz formula in this setting—a central step in the argument (Lemma 4.5). Finally, this paper would not have attained its current form and clarity without the useful suggestions by Carla Cederbaum and the referee.

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