In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a three-sphere which has scalar curvature greater than or equal to and is not round must have an embedded minimal sphere of area strictly smaller than and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.
Fernando C. Marques. André Neves. "Rigidity of min-max minimal spheres in three-manifolds." Duke Math. J. 161 (14) 2725 - 2752, 1 November 2012. https://doi.org/10.1215/00127094-1813410