# Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

## TL;DR

This paper characterizes when the geodesic distance on diffeomorphism groups vanishes or is positive for certain Sobolev metrics, completing the understanding for critical Sobolev exponents.

## Contribution

It proves that the geodesic distance vanishes for critical Sobolev norms $W^{s,n/s}$ on diffeomorphism groups, extending previous results to the critical case.

## Key findings

- Geodesic distance vanishes for $W^{s,n/s}$ norms with $s<1$
- Distance is positive for norms outside the critical case
- Completes the classification of geodesic distance behavior for Sobolev metrics

## Abstract

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the geodesic distance induced by $W^{s,p}$ vanishes if $sp\le n$ and $s<1$, and is positive otherwise. The proof is achieved by combining the techniques of two recent papers --- [JM19] by the authors, which treated the sub-critical case, and [BHP18] of Bauer, Harms and Preston, which treated the critical 1-dimensional case.