Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves
arXiv:2312.17497 · doi:10.1007/s12220-024-01652-3
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 . 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.
arXiv commentversion 2: corrections of some typos
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