On completeness of groups of diffeomorphisms

  • Martins Bruveris

    EPFL, Lausanne, Switzerland
  • François-Xavier Vialard

    Université Paris-Dauphine, Paris, France
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We study completeness properties of the Sobolev diffeomorphism groups Ds(M)\mathcal D^s(M) endowed with strong right-invariant Riemannian metrics when MM is Rd\mathbb R^d or a compact manifold without boundary. We prove that for s>s > dim (M)/2+1(M)/2+1, the group Ds(M)\mathcal D^s(M) is geodesically and metrically complete and any two diffeomorphisms in the same component can be joined by a minimal geodesic. 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.

Cite this article

Martins Bruveris, François-Xavier Vialard, On completeness of groups of diffeomorphisms. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1507–1544

DOI 10.4171/JEMS/698