Title: Sewing Riemannian Manifolds with Positive Scalar Curvature,
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.