2025/09/13 by Ronny Gelman, Gelman, Ronny
Engineering · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2509.10954
openalex publication_date 2025/09/13 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28
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