arXiv subjects

J. Basilio

Publications and source records attributed to J. Basilio.

2 recordsLinked to original sources

arXiv2019

Sequences of three dimensional manifolds with positive scalar curvature

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.

math.MG↗

arXiv2017

Sewing Riemannian Manifolds with Positive Scalar Curvature

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.

math.DG↗