The tetrahedral property and a new Gromov–Hausdorff compactness theorem

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 Hm 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.