  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.
