The Half-Volume Spectrum of a Manifold

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.