Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

Abstract

The group $\text{Diff}(\mathcal{M})$ of diffeomorphisms of a closed manifold $\mathcal{M}$ is naturally equipped with various right-invariant Sobolev norms $W^{s,p}$. Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when $sp\le \text{dim}\mathcal{M}$ and $s<1$). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when $(s-1)p\ge \text{dim}\mathcal{M}$, and that for spheres the diameter is finite when $(s-1)p<1$. In particular, this gives a full characterization of the diameter of $\text{Diff}(S^1)$. In addition, we show that for $\text{Diff}_c(\mathbb{R}^n)$, if the diameter is not zero, it is infinite.