The Tetrahedral Property and a new Gromov-Hausdorff Compactness Theorem

TL;DR

This paper introduces the Tetrahedral Compactness Theorem, establishing conditions under which sequences of Riemannian manifolds converge in the Gromov-Hausdorff sense to rectifiable metric spaces, using a novel tetrahedral property.

Contribution

It presents a new compactness theorem based on the tetrahedral property and introduces the sliced filling volume concept, advancing understanding of manifold convergence.

Findings

Sequences with bounded volume and diameter satisfying the tetrahedral property converge Gromov-Hausdorffly.

The tetrahedral property provides a volume lower bound based on distances in spheres.

The proof utilizes intrinsic flat convergence and the sliced filling volume concept.

Abstract

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.