Total variation flow of curves in Riemannian manifolds

Lorenzo Giacomelli; Michał Łasica; Salvador Moll

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arXiv · 2511.08459

Total variation flow of curves in Riemannian manifolds

Lorenzo Giacomelli·Michał Łasica·Salvador Moll

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Abstract

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.

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Lorenzo Giacomelli, Michał Łasica, Salvador Moll. 2025-11-11. Total variation flow of curves in Riemannian manifolds. https://arxiv.org/abs/2511.08459

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