Uniqueness and nonuniqueness of limits of Teichmueller harmonic map flow

TL;DR

This paper investigates the convergence properties of the Teichmueller harmonic map flow, focusing on whether the flow converges uniformly and whether diffeomorphisms are necessary for convergence.

Contribution

It analyzes the conditions under which the flow converges to a minimal immersion without the need for diffeomorphism pullbacks and explores the uniqueness of the flow's limits.

Findings

Convergence may require diffeomorphism pullbacks.

Flow limits can be non-unique.

Uniform convergence over time is not guaranteed.

Abstract

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as time tends to infinity.