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Total variation flow of curves in Riemannian manifolds

Lorenzo Giacomelli, Michał Łasica, Salvador Moll

TL;DR

This work develops a PDE-analytic framework for the total variation flow of parametrized curves taking values in a Riemannian submanifold. It introduces a robust notion of strong solution for the $L^2$-gradient flow of the constrained TV functional $TV_I^{\mathcal N}$ and proves global existence for initial data in $BV_{\rm rad}(I,\mathcal N)$ via a two-step Lipschitz-then-BV approximation, accompanied by a detailed identification of the flux $\boldsymbol z$ outside and on the jump set. Uniqueness is established in the nonpositive curvature setting, through a variational equality/inequality framework, and the flow is shown to converge to a constant in finite time. The paper also clarifies the relationship with existing 1-harmonic map theories (notably in the case $\mathcal N=\mathbb S^{N-1}_+$), analyzes the lack of geodesic convexity in general, and presents explicit examples that illustrate jump dynamics and extinction phenomena. These results advance the well-posedness theory for BV-valued manifold-valued flows and provide PDE-based insights into gradient-flow formulations of constrained total variation on manifolds.

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.

Total variation flow of curves in Riemannian manifolds

TL;DR

This work develops a PDE-analytic framework for the total variation flow of parametrized curves taking values in a Riemannian submanifold. It introduces a robust notion of strong solution for the -gradient flow of the constrained TV functional and proves global existence for initial data in via a two-step Lipschitz-then-BV approximation, accompanied by a detailed identification of the flux outside and on the jump set. Uniqueness is established in the nonpositive curvature setting, through a variational equality/inequality framework, and the flow is shown to converge to a constant in finite time. The paper also clarifies the relationship with existing 1-harmonic map theories (notably in the case ), analyzes the lack of geodesic convexity in general, and presents explicit examples that illustrate jump dynamics and extinction phenomena. These results advance the well-posedness theory for BV-valued manifold-valued flows and provide PDE-based insights into gradient-flow formulations of constrained total variation on manifolds.

Abstract

We consider the functional of total variation of maps from an interval into a Riemannian submanifold of . We define a notion of strong solution to the system of equations corresponding to the -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.

Paper Structure

This paper contains 15 sections, 35 theorems, 374 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counterexamples to geodesic semiconvexity
  4. Regular solutions and the pointwise estimate on $|\boldsymbol u_x|_{\mathcal{N}}$
  5. Existence of solutions
  6. Basic facts
  7. Identifying $\boldsymbol z$ away from the jump set
  8. Identifying $\boldsymbol z$ on the jump set
  9. Identifying $\boldsymbol u_t$
  10. Strict convergence and its consequences
  11. Variational (in)equality and uniqueness
  12. Asymptotic behavior
  13. The case $\mathcal{N}=\mathbb S_+^{N-1}$
  14. Examples of solutions
  15. Appendix

Key Result

Theorem 1.3

For any $\boldsymbol u_0 \in BV_{\rm rad}(I, \mathcal{N})$ and any $T>0$, there exists a strong solution $\boldsymbol u$ to (smootheqn,smoothbc) in $[0,T[$ such that Furthermore, $\boldsymbol u$ satisfies the energy inequality and the pointwise monotonicity property

Figures (2)

  • Figure 1: Vectors $\boldsymbol T_{\boldsymbol w}^\pm$.
  • Figure 2: From left to right: $\boldsymbol u$, $\boldsymbol v^\varepsilon$, $\boldsymbol \gamma^\varepsilon_{\frac{1}{2}}$

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 56 more