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

Martin Bauer; Cy Maor

doi:10.1007/s00526-021-01918-6

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

Martin Bauer, Cy Maor^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The group Diff (M) of diffeomorphisms of a closed manifold M is naturally equipped with various right-invariant Sobolev norms W^s^,^p. Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when sp≤ dim M and s< 1). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when (s- 1) p≥ dim M, and that for spheres the diameter is finite when (s- 1) p< 1. In particular, this gives a full characterization of the diameter of Diff (S¹). In addition, we show that for Diff _c(Rⁿ) , if the diameter is not zero, it is infinite.

Original language	English
Article number	54
Journal	Calculus of Variations and Partial Differential Equations
Volume	60
Issue number	1
DOIs	https://doi.org/10.1007/s00526-021-01918-6
State	Published - Feb 2021

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© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

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abstract = "The group Diff (M) of diffeomorphisms of a closed manifold M is naturally equipped with various right-invariant Sobolev norms Ws,p. Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when sp≤ dim M and s< 1). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when (s- 1) p≥ dim M, and that for spheres the diameter is finite when (s- 1) p< 1. In particular, this gives a full characterization of the diameter of Diff (S1). In addition, we show that for Diff c(Rn) , if the diameter is not zero, it is infinite.",

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

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