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Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves

Published 29 Dec 2023 in math.DG and math.AP | (2312.17497v2)

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 $Hq$-immersed curves with the distance induced by the Riemannian metric.

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