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

Abstract

We study the geodesic distance induced by right-invariant metrics on the group ${\text {Diff}}_\text {c}({\mathcal {M}})$ 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 $s<\min \{n/p,1\}$, where n is the dimension of ${\mathcal {M}}$. We also show that previous results imply that whenever $s > n/p$ 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. (Ann Glob Anal Geom 44(1):5–21, 2013).