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|Title:||Completeness properties of Sobolev metrics on the space of curves|
|Keywords:||math.DG;math.DG;58D10 (primary), 58D20, 53A04, 35A01 (secondary)|
|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.|
|Appears in Collections:||Dept of Mathematics Research Papers|
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