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CMC Foliations of Closed Manifolds

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Abstract

We prove that every closed, smooth \(n\)-manifold \(X\) admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of \(X\) can be chosen so that when \(n\ge 2\), the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of \(X\) that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.

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Notes

  1. Classical Reeb components of foliations of closed three-manifolds are defined at the beginning of Sect. 4. In Definition 4.5 we will describe the notion of enlarged Reeb component of a codimension-one foliation, which makes sense in all dimensions and differs for the classical definition in the case \(n=3\) as it does not contain a single compact leaf.

  2. A three-manifold \(X\) is irreducible if every embedded two-sphere in \(X\) bounds a three-ball in \(X\).

  3. This index (resp., nullity) is the number of negative eigenvalues (resp., multiplicity of zero as an eigenvalue) of the Jacobi operator of \(L\) viewed as a compact, two-sided hypersurface with CMC in \(X\).

  4. See Definition 2.2 for the definition of a limit leaf of a CMC lamination.

  5. Here maximality refers to atlases satisfying Properties 2 and 3.

  6. Here, minimal refers to the partial order given by inclusion.

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Acknowledgments

This material is based upon work for the NSF under Award No. DMS - 1309236. Joaquín Pérez would like to thank Prof. Jesús A. Alvarez López for valuable conversations about Oshikiri’s results. Joaquín Pérez was supported in part by MEC/FEDER Grant no. MTM2011-22547, and Regional J. Andalucía Grant No. P06-FQM-01642.

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Authors and Affiliations

  1. Mathematics Department, University of Massachusetts, Amherst, MA, 01003, USA

    William H. Meeks III

  2. Department of Geometry and Topology, University of Granada, Granada, Spain

    Joaquín Pérez

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  1. William H. Meeks III
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  2. Joaquín Pérez
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Correspondence to Joaquín Pérez.

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Meeks, W.H., Pérez, J. CMC Foliations of Closed Manifolds. J Geom Anal 26, 1647–1677 (2016). https://doi.org/10.1007/s12220-015-9603-9

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