Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

We study the geodesic distance induced by right-invariant metrics on the group ${\mathrm{Diff}}_{\text{c}}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min \{n\mathrm{/}p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s>n\mathrm{/}p$ or $s\ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conjecture made in Bauer et al. [BBHM13].