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.

Keywords: Completeness; Fractional Sobolev space; Geodesic distance; Global well-posedness; Immersions; Infinite-dimensional Riemannian geometry.