Characterization of the Metric Completion of Immersed Open-Curve Spaces

The completeness properties of spaces of immersed curves equipped with reparametrization-invariant Riemannian metrics have recently been the subject of active research. This thesis studies the metric completion of spaces of immersed open curves endowed with Sobolev-type metrics and examines a previously proposed conjecture that suggests the metric completion consists of a single additional point, representing all vanishing length Cauchy sequences. We disprove the conjecture in the setting of real-valued immersed curves by demonstrating the existence of multiple distinct limit points. Furthermore, we provide a nearly complete characterization of the metric structure of the metric completion in this case. These results lead to a revised conjecture regarding the structure of the metric completion in more general settings