TITLE: Does harmonic map heat flow of a curve always fully converge to a geodesic?
QUESTION [7 upvotes]: Consider a smooth closed curve $u_0$ in a compact Riemannian manifold $(M,g)$. Let $u_0$ evolve by harmonic map heat flow, $\partial_tu=\nabla_{\partial_su}\partial_su$, and call the result $u(t)$. 
Since the circle is 1-dimensional, a miracle happens and we get a gradient estimate for free, so the flow exists for all time. By standard arguments, there is a sequence $t_k\to\infty$ such that $u(t_k)$ converges to a geodesic $u_\infty$ in the $C^\infty$ topology. Furthermore, one can see that $u_\infty$ is homotopic to $u_0$ because $u(t_k)$ is eventually within a small enough neighborhood of $u_\infty$. However, what's not clear to me is if the flow provides this homotopy.
That is, does $\lim_{t\to\infty}u(t)=u_\infty$ in the $C^\infty$ topology, not just up to a subsequence?
Hartman has proved this when $M$ has nonpositive sectional curvature for general harmonic maps with bounded image. I'm wondering if this is true for curves without a curvature assumption on the target.

REPLY [9 votes]: The situation is actually quite complicated, it seems. 
In the case where the target manifold is real analytic, Leon Simon's results in Asymptotics for a Class of Non-Linear Evolution Equations, with Applications to Geometric Problems implies the desired convergence. 
However, using Topping's construction in Section 5 of Rigidity in the Harmonic Map Heat Flow (note Remark 6 which states that the construction is essentially independent of the dimension of the domain), one concludes that, in general, the convergence really is only up to subsequence: the flow can fail to converge even in $C^0$. 
In a paper by Choi and Parker, it was however shown that convergence holds for a "generic" set of target manifolds, which they call manifolds with "bumpy metrics".