  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.

Differential Geometry;Analysis of PDEs