Abstract
We study the geodesic distance induced by right-invariant metrics on the group \({\text {Diff}}_\text {c}(\mathcal {M})\) of compactly supported diffeomorphisms of a manifold \(\mathcal {M}\) and show that it vanishes for the critical Sobolev norms \(W^{s,n/s}\), where n is the dimension of \(\mathcal {M}\) 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—(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.
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Notes
Similar observations (in the context of contactomophorisms) also appear in [10].
The right-hand side inequality in (3.8) is the reason we need \(\lambda _k\) to be so small, which is achieved by the two-stage squeezing. In the subcritical case \(sp<n\), the \(W^{s,p}\)-capacity of small balls is much smaller, and hence \(\lambda _k\) can be larger (that is, the results of Lemma 3.1 hold for larger values of \(\lambda _k\)), and then the one-stage squeezing used in [8] suffices.
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Acknowledgements
We are grateful to Martin Bauer, Philipp Harms and Stephen Preston for introducing us their paper and the notion of displacement energy. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada under operating Grant 261955.
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Jerrard, R.L., Maor, C. Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents. Ann Glob Anal Geom 56, 351–360 (2019). https://doi.org/10.1007/s10455-019-09670-z
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DOI: https://doi.org/10.1007/s10455-019-09670-z