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

TL;DR

This paper establishes a method to sweep out 3-manifolds with positive Ricci curvature using surfaces and 1-cycles of controlled area and length, leveraging min-max theory to connect geometric bounds with topological sweepouts.

Contribution

It introduces a new technique to construct sweepouts of 3-manifolds by surfaces and 1-cycles with bounds related to volume, utilizing min-max minimal surface theory.

Findings

Existence of sweepouts by surfaces of genus ≤ 3 with area bounds proportional to volume^{2/3}

Construction of sweepouts by 1-cycles with length bounds proportional to volume^{1/3}

Diameter bounds for min-max surfaces under positive scalar curvature assumptions

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 $\leq 3$ 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.