English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 geodesicdistance 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 questionwe study in this paper. We show that the diameter is infinite for strong enoughnorms, when $(s-1)p\ge \text{dim}\mathcal{M}$, and that for spheres thediameter is finite when $(s-1)p<1$. In particular, this gives a fullcharacterization of the diameter of $\text{Diff}(S^1)$. In addition, we showthat for $\text{Diff}_c(\mathbb{R}^n)$, if the diameter is not zero, it isinfinite.

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