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On Completeness of Groups of Diffeomorphisms

Bruveris, Martins; Vialard, François-Xavier (2017), On Completeness of Groups of Diffeomorphisms, Journal of the European Mathematical Society, 19, 5, p. 1507–1544. 10.4171/JEMS/698

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Journal of the European Mathematical Society
Volume
19
Number
5
Publisher
Springer
Pages
1507–1544
Publication identifier
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Author(s)
Bruveris, Martins
Department of Mathematics [Imperial College London]
Vialard, François-Xavier
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
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.
Subjects / Keywords
Strong Riemannian metric; Sobolev metrics; Completeness; Diffeomorphism groups; Minimizing geodesic