On Completeness of Groups of Diffeomorphisms

Martins Bruveris (1, 2) , François-Xavier Vialard (3, 4)

1 Department of Mathematics [Imperial College London]
2 EPFL - Ecole Polytechnique Fédérale de Lausanne
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

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 dim M/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.

Mots clés: Strong Riemannian metric; Sobolev metrics; Completeness; Diffeomorphism groups; Minimizing geodesic

Domaines: Mathématiques générales [math.GM]; Géométrie différentielle [math.DG]

Martins Bruveris, François-Xavier Vialard. On Completeness of Groups of Diffeomorphisms. 2016. ⟨hal-01253261⟩