# Sewing Riemannian Manifolds with Positive Scalar Curvature

## TL;DR

This paper introduces a sewing construction for three-dimensional manifolds with positive scalar curvature, analyzes the limits of such manifolds under geometric convergence, and proposes a combined curvature and minimal surface area condition to better preserve scalar curvature properties.

## Contribution

The paper presents a novel sewing method for 3D manifolds with positive scalar curvature and examines how geometric limits can fail to retain scalar curvature properties, proposing a new combined curvature and minimal surface area condition.

## Key findings

- Sewing preserves positive scalar curvature in constructed manifolds.
- Sequences of sewn manifolds can converge to spaces lacking nonnegative scalar curvature.
- Combining scalar curvature bounds with minimal surface area bounds helps maintain curvature properties.

## Abstract

We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.