Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$. The sweepout of surfaces is generated from a min-max minimal surface. If further assuming a positive scalar curvature lower bound, we can get a diameter upper bound for the min-max surface.