Sobolev metrics on spaces of manifold valued curves
The paper studies metric and geodesic completeness of reparametrization-invariant Sobolev metrics on the spaces of open and closed immersed curves in a Riemannian manifold. For \(D = [0,2\pi]\) or \(D=S^1\) and \((\mathcal N,g)\) a complete Riemannian manifold with bounded geometry, the space of Sobolev immersions\N\[\N\mathcal I^n(D,\mathcal N) = \{c \in H^n(D,\mathcal N) : c'(\theta)\ne 0 \text{ for all } \theta \in D\}\N\]\Nis endowed with a certain class of reparametrization-invariant Sobolev metrics \(G\) of order \(n\ge 2\) (so that \(H^n(D,\mathcal N) \subset C^1(D,\mathcal N)\)). It is proved that \((\mathcal I^n(D,\mathcal N),\operatorname{dist}^G)\) is a complete metric space, \((\mathcal I^n(D,\mathcal N),G)\) is geodesically complete, and any two elements in the same connected component of \((\mathcal I^n(D,\mathcal N),G)\) can be joined by a minimizing geodesic. The results extend previous work of [\textit{M. Bruveris}, J. Geom. Mech. 7, 125--150 (2015; Zbl 1328.58007)] for closed immersed curves in Euclidean space. The more complicated structure of \(\mathcal I^n(D,\mathcal N)\) and the effects of the curvature of \(\mathcal N\) are the main differences to the Euclidean situation. For open curves, an additional challenge is posed by the presence of boundary terms in the geodesic equation.\N\NFurther related results are obtained. For instance, it was known [\textit{M. Bauer} et al., ESAIM Control Optim. Calc. Var. 25, Paper No. 72, 24 p. (2019; Zbl 07194611)] that the space of open immersed curves with respect to constant-coefficient Sobolev metrics is metrically incomplete. Here it is shown that the only way a path of immersed curves can leave this space in finite time is vanishing of the entire curve.\N\NUsing a no-loss-no-gain result of Ebin-Marsden-type, some of the results are extended to the space of \(C^\infty\) immersed curves; for open curves, regularity is only obtained in the interior.
scientific article; zbMATH DE number 7851376
scientific article