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

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Article number54
JournalCalculus of Variations and Partial Differential Equations
Issue number1
StatePublished - Feb 2021

Bibliographical note

Funding Information:
We would like to thank to Stefan Haller, Philipp Harms, Stephen Preston, Tudor Ratiu and Josef Teichman for various discussions during the work on this paper, and to Meital Maor for her help with the figures. We are in particular grateful to Kathryn Mann and Tomasz Rybicki for introducing us to the literature on fragmentation and perfectness, and to Bob Jerrard for his continuous and valuable help throughout the work on this project. This project was initiated during the BIRS workshop “Shape Analysis, Stochastic Geometric Mechanics and Applied Optimal Transport” in December 2018; we are grateful to BIRS for their hospitality. M. Bauer was partially supported by NSF-grants 1912037 and 1953244. C. Maor was partially supported by ISF-grant 1269/19.

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


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