# 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} }

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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