On the waist and width inequality in complete 3-manifolds with positive scalar curvature

TL;DR

This paper demonstrates that complete non-compact 3-manifolds with positive scalar curvature can be foliated by surfaces with bounded area and diameter, revealing geometric constraints related to curvature.

Contribution

It introduces a singular foliation structure with controlled geometric properties in 3-manifolds with positive scalar curvature, advancing understanding of their geometry.

Findings

Existence of singular foliation with bounded area and diameter

Geometric constraints imposed by positive scalar curvature

Controlled geometric structure in non-compact 3-manifolds

Abstract

We show that a complete non-compact 3-manifold with scalar curvature bounded below by a positive constant admits a singular foliation by surfaces of controlled area and diameter.