The Half-Volume Spectrum of a Manifold

TL;DR

This paper introduces the half-volume spectrum of a manifold, proves a Weyl law for it, and shows it is realized by constant mean curvature and minimal surfaces using Allen-Cahn theory.

Contribution

It defines the half-volume spectrum and establishes the Weyl law for it, extending the volume spectrum concept to include phase transition settings.

Findings

Weyl law holds for the half-volume spectrum.

Each half-volume spectrum value is achieved by specific geometric surfaces.

Uses Allen-Cahn min-max theory to connect spectra with geometric structures.

Abstract

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.