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.

necessarycondition for the convergence of a subsequence toward a geodesic, but that the later does not hold unconditionally. It seems we agree on this, so that I guess you want to makesomeassumption on $u_0$. $\endgroup$real analyticthis can be derived from Leon Simon's 83 paper jstor.org/stable/2006981?seq=1#metadata_info_tab_contents // incidentally, a related question has been asked before: mathoverflow.net/questions/134930/… $\endgroup$