Volume Above Distance Below

TL;DR

This paper proves that under certain volume and metric conditions, a sequence of Riemannian manifolds converges in the intrinsic flat sense, offering new tools for stability analysis in geometric rigidity theorems.

Contribution

It establishes volume-preserving intrinsic flat convergence for manifolds with metrics bounded below, and introduces a novel method for estimating intrinsic flat distances.

Findings

Convergence in volume implies intrinsic flat convergence under specified conditions.

Examples show convergence does not necessarily imply smooth or Gromov-Hausdorff convergence.

Provides a new technique for estimating intrinsic flat distances between manifolds.

Abstract

Given a pair of metric tensors $g_1 \ge g_0$ on a Riemannian manifold, $M$, it is well known that $\operatorname{Vol}_1(M) \ge \operatorname{Vol}_0(M)$. Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same $g_1=g_0$. Here we prove that if $g_j \ge g_0$ and $\operatorname{Vol}_1(M)\to \operatorname{Vol}_0(M)$ then $(M,g_j)$ converge to $(M,g_0)$ in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, $C^0$, Lipschitz, or even Gromov-Hausdorff convergence in this setting. Our theorem may also be applied as a tool towards proving other open conjectures concerning the geometric stability of a variety of rigidity theorems in Riemannian geometry. To complete our proof, we provide a novel way of estimating the intrinsic flat distance between Riemannian manifolds which is interesting in its own right.