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

Abstract

We study the geodesic distance induced by right-invariant metrics on the group ${\mathrm{Diff}}_c(M)$ 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.

arXiv commentv2: minor changes in presentation, no mathematical changes

References in corpus (4)

• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
• The Lagrangian Radon Transform and the Weil representation
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

Cited by in corpus (3)

• Vanishing distance phenomena and the geometric approach to SQG
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves