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 $$q\in [0,\infty )$$ 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 $$q>1/2$$ q > 1 / 2 . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if $$q>3/2$$ q > 3 / 2 , whereas if $$q<3/2$$ q < 3 / 2 then finite-time blowup may occur. The geodesic completeness for $$q>3/2$$ q > 3 / 2 is obtained by proving metric completeness of the space of $$H^q$$ H q -immersed curves with the distance induced by the Riemannian metric.