Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

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∈[0,∞). 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.