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. 



Does harmonic map heat flow of a curve always fully converge to a geodesic?
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*](https://www.jstor.org/stable/2006981?seq=1#metadata_info_tab_contents) implies the desired convergence. 


However, using Topping's construction in [Section 5 of *Rigidity in the Harmonic Map Heat Flow*](https://projecteuclid.org/euclid.jdg/1214459844) (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](https://www.semanticscholar.org/paper/Convergence-of-the-heat-flow-for-closed-geodesics-Choi-Parker/ad7aa8b3fd6ed790d693a195fb7317bbbe8771d4), it was however shown that convergence holds for a "generic" set of target manifolds, which they call manifolds with "bumpy metrics". 


