Total variation flow of curves in Riemannian manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908644754325504 |
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| author | Giacomelli, Lorenzo Łasica, Michał Moll, Salvador |
| author_facet | Giacomelli, Lorenzo Łasica, Michał Moll, Salvador |
| contents | We consider the functional of total variation of maps from an interval into a Riemannian submanifold of $\mathbb R^N$. We define a notion of strong solution to the system of equations corresponding to the $L^2$-gradient flow of this functional. We prove global existence of strong solutions for initial data of bounded variation. We show that the solutions satisfy a variational equality, and deduce uniqueness in the case of non-positive sectional curvature. We prove convergence of strong solutions to a constant map in finite time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_08459 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Total variation flow of curves in Riemannian manifolds Giacomelli, Lorenzo Łasica, Michał Moll, Salvador Analysis of PDEs Differential Geometry 35K51 (Primary) 35A01, 35A02, 35B40, 35D35, 35K92, 35R01, 53C21, 68U10 (Secondary) We consider the functional of total variation of maps from an interval into a Riemannian submanifold of $\mathbb R^N$. We define a notion of strong solution to the system of equations corresponding to the $L^2$-gradient flow of this functional. We prove global existence of strong solutions for initial data of bounded variation. We show that the solutions satisfy a variational equality, and deduce uniqueness in the case of non-positive sectional curvature. We prove convergence of strong solutions to a constant map in finite time. |
| title | Total variation flow of curves in Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry 35K51 (Primary) 35A01, 35A02, 35B40, 35D35, 35K92, 35R01, 53C21, 68U10 (Secondary) |
| url | https://arxiv.org/abs/2511.08459 |