Total variation flow of curves in Riemannian manifolds

TL;DR

This paper studies the evolution of curves in Riemannian manifolds driven by total variation flow, establishing existence, uniqueness, and convergence results for solutions with bounded variation.

Contribution

It introduces a notion of strong solutions for total variation flow in Riemannian manifolds and proves their global existence, uniqueness under certain curvature conditions, and finite-time convergence.

Findings

Global existence of strong solutions for BV initial data

Uniqueness in non-positive sectional curvature

Finite-time convergence to constant maps

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.