The (p;m)-width of Riemannian manifolds and its realization

TL;DR

This paper introduces the (p,m)-width concept in Riemannian manifolds, showing it can be realized as the volume of minimal hypersurfaces, advancing understanding of geometric variational problems.

Contribution

It defines the (p,m)-width as a stratification of p-width and proves its realization as minimal hypersurface volume with multiplicities.

Findings

(p,m)-width can be realized as volume of minimal hypersurfaces

Provides a stratification of p-width in Riemannian manifolds

Advances understanding of width realization in geometric analysis

Abstract

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Marques and Neves used the $p$-width to prove the existence of infinite many minimal hypersurfaces in compact manifolds with positive Ricci curvature. However, whether the $p$-width can be realized as the volume of minimal hypersurfaces is not known yet. We introduced the concept of the $(p, m)$-width which can be viewed as the stratification of the $p$-width, and proved that the $(p, m)$-width can be realized as the volume of minimal hypersurfaces with multiplicities.