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On Completeness of Groups of Diffeomorphisms

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1403.2089v4 (1).pdf (494.2Kb)
Date
2017
Dewey
Probabilités et mathématiques appliquées
Sujet
Strong Riemannian metric; Sobolev metrics; Completeness; Diffeomorphism groups; Minimizing geodesic
Journal issue
Journal of the European Mathematical Society
Volume
19
Number
5
Publication date
2017
Article pages
1507–1544
Publisher
Springer
DOI
http://dx.doi.org/10.4171/JEMS/698
URI
https://basepub.dauphine.fr/handle/123456789/17247
Collections
  • CEREMADE : Publications
Metadata
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Author
Bruveris, Martins
4553 Department of Mathematics [Imperial College London]
Vialard, François-Xavier
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is ℝd or compact without boundary. The main result is that for dimM/2 + 1, the group Ds (M) is geodesically and metrically complete with a surjective exponential map. We also extend the result to its closed subgroups, in particular the group of volume preserving diffeomorphisms and the group of symplectomorphisms. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.

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