Question

Let $n\in\mathbb{N}$ Is the volume functional continuous on the set of isometry classes of compact riemannian $n$-manifolds with volume $\geq \varepsilon$\ (with respect to Gromov--Hausdorff distance)?

Without the volume bound, a collapsing torus gives a counterexample. But it seems that this is the only singularity. There are a number of weird results on the semi-continuity of the volume functional. For instance, every metric on $S^3$ is the limit of a metric with volume converging to $0$. I don't know what happens to the curvature of these metrics.

Answer 1

I assume you talk about continuity in Gromov-Hausdorff topology.

In general, volume is not continuous, as easy examples show (see e.g. Colding's paper "Large manifolds with positive Ricci curvature", examples 1-2 which have a lower volume bound).

On the other hand, for manifolds with a lower Ricci curvature bound, Colding shows the for each $r$ the volume of $r$-balls is continuous in Gromov-Hausdorff topology; see his paper "Ricci curvature and volume convergence".