We define the half-volume spectrum \(\{{\tilde{\omega }_p\}_{p\in \mathbb {N}}}\) of a closed manifold \((M^{n+1},g)\) . This is analogous to the usual volume spectrum of M, except that we restrict to p-sweepouts whose slices each enclose half the volume of M. We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum \(\tilde{c}(p)\) in the phase transition setting. Moreover, for \(3 \le n+1 \le 7\) , we use the Allen–Cahn min-max theory to show that each \(\tilde{c}(p)\) is achieved by a constant mean curvature surface enclosing half the volume of M plus a (possibly empty) collection of minimal surfaces with even multiplicities.
