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Large outlying stable constant mean curvature spheres in initial data sets

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Abstract

We give examples of asymptotically flat three-manifolds \((M,g)\) which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by Huisken and Yau (Invent Math 124:281–311, 1996). On the other hand, we show that such surfaces cannot exist when \((M,g)\) has nonnegative scalar curvature. This result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces.

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Notes

  1. Note that if the left-hand side were negative for some \(\xi >1\), then we could turn the argument around and conclude the existence of closed constant mean curvature surfaces in the Schwarzschild manifold that are disjoint from the horizon. This would contradict results in [5].

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Acknowledgments

The first-named author was supported in part by the U.S. National Science Foundation under Grant DMS-1201924. He acknowledges the hospitality of the Department of Mathematics and Mathematical Statistics at Cambridge University, where part of this work was carried out. The second-named author was supported by the Swiss National Science Foundation under Grant SNF 200021-140467. We are grateful to Professor Gerhard Huisken and Professor Jan Metzger for their interest and encouragement. Finally, we thank Otis Chodosh and the referee for their helpful remarks.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Simon Brendle

  2. Departement Mathematik, ETH Zürich, 8092 , Zürich, Switzerland

    Michael Eichmair

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  1. Simon Brendle
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  2. Michael Eichmair
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Correspondence to Michael Eichmair.

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Brendle, S., Eichmair, M. Large outlying stable constant mean curvature spheres in initial data sets. Invent. math. 197, 663–682 (2014). https://doi.org/10.1007/s00222-013-0494-8

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  • DOI: https://doi.org/10.1007/s00222-013-0494-8

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