Fractional Sobolev metrics on spaces of immersed curves

Abstract

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 $\mathrm{Imm}(S^1,{\mathbb{R}}^d)$ 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 Fréchet-manifold $\mathrm{Imm}(S^1,{\mathbb{R}}^d)$ 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 $s>\frac{3}{2}$. These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.

23 pages, 1 figure