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
TypeArticle accepté pour publication ou publié
Journal nameJournal of the European Mathematical Society
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Department of Mathematics [Imperial College London]
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 / KeywordsStrong Riemannian metric; Sobolev metrics; Completeness; Diffeomorphism groups; Minimizing geodesic
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