Total variation flow of curves in Riemannian manifolds

Let N be a complete Riemannian manifold. We consider the functional of total variation defined on maps from an interval I into N. This is a relaxation with respect to L2 topology on I of the length functional defined on parametrized curves. We investigate well-posedness of the steepest descent flow of this functional. I will show that, unless N is of non-positive sectional curvature (NPC), it fails critically to be semiconvex, hence Ambrosio-Gigli-Savare theory of gradient flows in metric spaces is not applicable. Then, I will introduce a notion of solution to flow equations that coincides with the one of Ambrosio-Gigli-Savare for NPC manifolds. These solutions can be shown to exist under a mild condition on the size of jumps of the initial datum. I will discuss some tools used in the proof such as a "completely local" a priori estimate and a variant of Sobolev inequality with covariant derivative. This is based on a joint project with Lorenzo Giacomelli and Salvador Moll.