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

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.