A sub-additive inequality for the volume spectrum

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 ${\omega }_{k+m+1}\le {\omega }_k+{\omega }_m+W$ for all $k,m\ge 0$, 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 $\epsilon$-phase-transition spectrum $\{c_{\epsilon }(p){\}}_{p=1}^{\infty }$ using the Allen-Cahn approach.

References in corpus (2)

• A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory
• Existence of multiple closed CMC hypersurfaces with small mean curvature