  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
3-sphere which has scalar curvature greater than or equal to 6 and is not round
must have an embedded minimal sphere of area strictly smaller than $4\pi$ and
index at most one. If the Ricci curvature is positive we also prove sharp
estimates for the width.
