Fractional Sobolev metrics on spaces of immersed curves

Martin Bauer (1) , Martins Bruveris (2) , Boris Kolev (3)

1 FSU - Florida State University [Tallahassee]
2 Brunel University London [Uxbridge]
3 I2M - Institut de Mathématiques de Marseille

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\operatorname{Imm}(S^1,\mathbb{R}^d)$ and on its Sobolev completions $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$. We prove local well-posedness of the geodesic equations both on the Banach manifold $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$ and on the Fr\'{e}chet-manifold $\operatorname{Imm}(S^1,\mathbb{R}^d)$ provided the order of the metric is greater or equal to one. In addition we show that the $H^s$-metric induces a strong Riemannian metric on the Banach manifold $\mathcal{I}^{s}(S^1,\mathbb{R}^{d})$ of the same order $s$, provided $s>\frac 32$. These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.

Mots clés: Sobolev metrics of fractional order

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Martin Bauer, Martins Bruveris, Boris Kolev. Fractional Sobolev metrics on spaces of immersed curves. 2017. ⟨hal-01486557v1⟩