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

摘要

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 W-s,W-n/s, where n is the dimension of M and s is an element of (0, 1). This completes the proof that the geodesic distance induced by W-s,W-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.