Volume above distance below

Abstract

$\def\Vol{\operatorname{Vol}}$ Given a pair of metric tensors $g_1 \geq g_0$ on a Riemannian manifold, $M$, it is well known that $\Vol_1(M) \geq \Vol_0 (M)$. 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 $g_j \geq g_0$ and $\Vol_j (M) \to \Vol_0 (M)$ then $(M, g_j)$ converge to $(M, g_0)$ in the volume preserving intrinsic at sense. Well known examples demonstrate that one need not obtain smooth, $C^0$, Lipschitz, or even Gromov–Hausdor 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 at distance between Riemannian manifolds which is interesting in its own right.