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

TL;DR

This paper investigates the conditions under which the geodesic distance induced by right-invariant Sobolev metrics on diffeomorphism groups vanishes or remains positive, revealing a sharp threshold at $s=1$ for the Riemannian case.

Contribution

It establishes the precise Sobolev regularity threshold for vanishing versus positive geodesic distance on diffeomorphism groups, correcting previous conjectures.

Findings

Geodesic distance vanishes for $s<rac{n}{p}$

Geodesic distance is positive for $s>rac{n}{p}$ or $s extgreater{}1$

In dimension $n extgreater{}1$, the distance vanishes if and only if $s<1$ for $p=2$

Abstract

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ 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 $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. 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. [BBHM13].