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Let (M^n+1, g) be a closed Riemannian manifold of dimension 3 n+1 5. We show that, if the metric g is generic, then M contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of M. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form Area () - _ h + f (Vol () ).
Mazurowski et al. (Wed,) studied this question.