Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves

Abstract

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional order $q\in [0,\infty )$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if $q>1/2$. Our second main result shows that the metric is geodesically-complete (i.e., the geodesic equation is globally well-posed) if $q>3/2$, whereas if $q<3/2$ then finite-time blowup may occur. The geodesic-completeness for $q>3/2$ is obtained by proving metric-completeness of the space of $H^q$-immersed curves with the distance induced by the Riemannian metric.

version 2: corrections of some typos

References in corpus (1)

• A speed preserving Hilbert gradient flow for generalized integral Menger curvature

Cited by in corpus (1)

• The Michor-Mumford conjecture in Hilbertian H-type groups