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.

  • $\begingroup$ I guess you mean that $u_0$ is not homotopically trivial, right? $\endgroup$ Oct 3 '18 at 9:12
  • $\begingroup$ @BenoîtKloeckner If it's not, then it could shrink to a point. However, it is easy to construct an example where $u_0$ is homotopically nontrivial but converges to a geodesic with positive length. (Think of a dumbbell.) $\endgroup$
    – Ryan Unger
    Oct 3 '18 at 13:07
  • $\begingroup$ I did not meant that $u_0$ not being homotopically trivial was a necessary condition 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 make some assumption on $u_0$. $\endgroup$ Oct 3 '18 at 14:47
  • $\begingroup$ According to Råde (in his Yang-Mills heat equation paper), when $(M,g)$ is real analytic this 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/… $\endgroup$ Oct 4 '18 at 14:27

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".

  • $\begingroup$ Cool, Topping's example seems to work. For the record, I asked this because Jost claims one gets full convergence in his Riemannian geometry book but offers no proof -- this had been bugging me for some time. $\endgroup$
    – Ryan Unger
    Oct 4 '18 at 18:04

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