Interlacing of the Volume and Half-Volume Spectra

Abstract

We give a direct proof of the weak interlacing inequalities \[ω_p\leq\widetildeω_p\leqω_{p+1}\] between the volume and half-volume spectra of a closed connected Riemannian manifold $M^{n+1}$, $n+1\geq3$. We also prove that, for a residual set of smooth metrics, the area of every smooth embedded half-volume CMC boundary lies outside the rational span of the areas of smooth closed embedded minimal hypersurfaces. For dimensions $3\le n+1\le 5$, these results yield strict interlacing for generic metrics and a lower bound of order $A^{n+1}$ for the number of distinct half-volume CMC boundaries with area at most $A$.