Completeness Properties of Sobolev Metrics on the Space of Curves

2014/07/02 by Bruveris, Martins · 2 citations
#35A01 #53A04 #58D10 #58D20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1407.0601

Abstract

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥ 2 are metrically complete on the space \mathcal Iⁿ(S¹,\mathbb R^d) of Sobolev immersions of the same regularity and that any two curves in the same connected component can be joined by a minimizing geodesic. These results then imply that the shape space of unparametrized curves has the structure of a complete length space.

Cited by

• Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces
• Brownian motion on spaces of discrete regular curves

Related

• Completeness of Length-Weighted Sobolev Metrics on the Space of Curves
• Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces
• Sobolev Metrics on Spaces of Discrete Regular Curves
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• Sobolev metrics on shape space of surfaces