# Sewing Riemannian Manifolds with Positive Scalar Curvature

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## 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.

## Keywords

Scalar curvature Gromov-Hausdorff Intrinsic flat## Mathematics Subject Classification

53C23## Notes

### Acknowledgements

J. Basilio was partially supported as a doctoral student by NSF DMS 1006059. C. Sormani was partially supported by NSF DMS 1006059.

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