A sub-additive inequality for the volume spectrum

TL;DR

This paper establishes a sub-additive inequality for the volume spectrum of closed Riemannian manifolds and extends similar results to the $ ext{Almgren-Pitts}$ width and $ ext{Allen-Cahn}$ spectrum, revealing new spectral relationships.

Contribution

It introduces a novel sub-additive inequality for the volume spectrum and extends this to the $ ext{Allen-Cahn}$ spectrum, connecting geometric and variational spectral concepts.

Findings

Proves $ ext{volume spectrum}$ satisfies a sub-additive inequality involving the width $W$.

Establishes a similar inequality for the $ ext{Allen-Cahn}$ spectrum.

Provides new bounds and relationships between different spectral invariants.

Abstract

Let $(M,g)$ be a closed Riemannian manifold and $\{\omega_p\}_{p=1}^{\infty}$ be the volume spectrum of $(M,g)$. We will show that $\omega_{k+m+1}\leq \omega_k+\omega_m+W$ for all $k,m\geq 0$, where $\omega_0=0$ and $W$ is the one-parameter Almgren-Pitts width of $(M,g)$. We will also prove the similar inequality for the $\varepsilon$-phase-transition spectrum $\{c_{\varepsilon}(p)\}_{p=1}^{\infty}$ using the Allen-Cahn approach.