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 . 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 . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if , whereas if then finite-time blowup may occur. The geodesic completeness for is obtained by proving metric completeness of the space of -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.
© The Author(s) 2024.