Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves
- Resource Type
- Working Paper
- Authors
- Bauer, Martin; Heslin, Patrick; Maor, Cy
- Source
- J Geom Anal 34, 214 (2024)
- Subject
- Mathematics - Differential Geometry
Mathematics - Analysis of PDEs
58B20, 58D10, 35G55, 35A01
- Language
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.
Comment: version 2: corrections of some typos