SWIF Convergence of Smocked Metric Spaces

TL;DR

This paper investigates the geometric limits of smocked metric spaces, showing that under certain conditions, their rescaled limits converge to normed spaces, with implications for Gromov's conjectures on Riemannian manifolds.

Contribution

It introduces the study of tangent cones at infinity for smocked metric spaces and establishes convergence results in Gromov-Hausdorff and Intrinsic Flat senses.

Findings

Rescaled limits of balls in smocked metric spaces converge to normed spaces.

Convergence occurs in both Gromov-Hausdorff and Intrinsic Flat senses.

Results will inform future work on Gromov's conjectures for manifolds with positive scalar curvature.

Abstract

In this paper we explore a special class of metric spaces called smocked metric spaces and study their tangent cones at infinity. We prove that under the right hypotheses, the rescaled limits of balls converge in both the Gromov-Hausdorff and Intrinsic Flat sense to normed spaces. This paper will be applied in upcoming work by Kazaras and Sormani concerning Gromov's conjectures on the properties of GH and SWIF limits of Riemannian manifolds with positive scalar curvature.