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arXiv · 1407.0601

Completeness Properties of Sobolev Metrics on the Space of Curves

Martins Bruveris
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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 $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ 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.

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Martins Bruveris. 2015-04-08. Completeness Properties of Sobolev Metrics on the Space of Curves. https://arxiv.org/abs/1407.0601

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