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

Abstract

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 $\le 3$ and areas bounded by $Cvol(M^3,g{)}^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $Cvol(M^3,g{)}^{1/3}$. 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.

19 pages, 2 figures; Theorem 1.2 replaced with (weaker) Corollary 1.2; the proof of Proposition 4.3 corrected following referee's suggestion

References in corpus (2)

• Diastolic and isoperimetric inequalities on surfaces
• New applications of Min-max Theory

Cited by in corpus (1)

• Volume growth and positive scalar curvature