Sobolev metrics on spaces of manifold valued curves
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866913214022811648 |
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| author | Bauer, Martin Maor, Cy Michor, Peter W. |
| author_facet | Bauer, Martin Maor, Cy Michor, Peter W. |
| contents | We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_13315 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Sobolev metrics on spaces of manifold valued curves Bauer, Martin Maor, Cy Michor, Peter W. Differential Geometry Analysis of PDEs 58B20, 58D10, 35G55, 35G60 We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics). |
| title | Sobolev metrics on spaces of manifold valued curves |
| topic | Differential Geometry Analysis of PDEs 58B20, 58D10, 35G55, 35G60 |
| url | https://arxiv.org/abs/2007.13315 |