Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory

Let (Mⁿ⁺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 Ω ↦ Area(∂ Ω) - ∫_Ω h + f(Vol(Ω)).