Regular 1-harmonic flow

TL;DR

This paper studies the 1-harmonic flow of maps into submanifolds, proving existence, uniqueness, and long-term behavior of solutions, including finite-time convergence to constant maps under certain conditions.

Contribution

It establishes the first results on existence, uniqueness, and finite-time convergence for the 1-harmonic flow into submanifolds, extending previous harmonic map theories.

Findings

Unique solutions for Lipschitz initial data

Global existence and finite-time convergence under curvature conditions

Solutions become constant in finite time for certain target manifolds

Abstract

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some assumptions.