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

Robert L. Jerrard, Cy Maor*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We study the geodesic distance induced by right-invariant metrics on the group Diff c(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms Ws , n / s, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by Ws , p vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.

Original languageAmerican English
Pages (from-to)351-360
Number of pages10
JournalAnnals of Global Analysis and Geometry
Volume56
Issue number2
DOIs
StatePublished - 1 Sep 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature B.V.

Keywords

  • Diffeomorphism group
  • Fractional Sobolev spaces
  • Infinite-dimensional geometry
  • Vanishing geodesic distance

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