Sequences of three dimensional manifolds with positive scalar curvature

TL;DR

This paper introduces two innovative methods for constructing sequences of 3D manifolds with positive scalar curvature that converge to limit spaces with pulled regions, impacting Gromov's conjectures.

Contribution

The authors develop two new techniques extending sewing along curves to produce limit spaces with pulled regions, advancing understanding of scalar curvature limits.

Findings

Constructed sequences converging to pulled string spaces

Extended sewing methods to create manifolds with specific limit behaviors

Influenced Gromov's conjectures on positive scalar curvature

Abstract

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with "pulled regions". The examples created rigorously within using these methods were announced by us a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of "sewing along a curve" developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.