Waist inequality for 3-manifolds with positive scalar curvature

Abstract

We construct singular foliations of compact threemanifolds (M 3 , h) with scalar curvature R h ≥ Λ 0 > 0 by surfaces of controlled area, diameter and genus. This extends Urysohn and waist inequalities of Gromov-Lawson and Marques-Neves. Minimal surfaces and decomposition of 3-manifolds2.1. Area and diameter estimates. We summarize known area and diameter bounds for two-sided minimal surfaces on manifolds of uniformly positive scalar curvature R g ≥ Λ 0 .The following area estimates are from [MN12, Proposition A.1].

Flipping Heegaard splittings and minimal surfaces

“…The requirement in Theorem 1.6 that one have an optimal foliation can likely be removed in the absence of stable minimal surfaces (when for instance, M has positive Ricci curvature). Namely, one should be able to use the mean curvature flow with surgeries to obtain an optimal foliation in any such manifold (see for instance [19] and [34])). However, in our applications the optimal foliation is readily available.…”

Macroscopic scalar curvature and codimension 2 width

“…The conjecture is only established in dimension 3 (see [CL20,LM20]), whereas in higher dimensions there are some partial results (see the discussion in the introduction of [DD20]). It is not even known if the codimension 1 width could be bounded from above.…”