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

@article{Bauer2019CanWR,
  title={Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics},
  author={Martin Bauer and Cy Maor},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2019},
  volume={60},
  pages={1-35}
}
  • Martin BauerCy Maor
  • Published 10 October 2019
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
The group $${\text {Diff}}({\mathcal {M}})$$ Diff ( M ) of diffeomorphisms of a closed manifold $${\mathcal {M}}$$ M is naturally equipped with various right-invariant Sobolev norms $$W^{s,p}$$ W s , p . Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when $$sp\le \dim {\mathcal {M}}$$ s p ≤ dim M and $$s<1$$ s < 1 ). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite… 
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