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I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes:

Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] \longrightarrow M$, consider the flow $$ \frac{\partial}{\partial \varepsilon}c(t, \varepsilon) = \frac{\nabla}{\partial t}\frac{\partial}{\partial t} c(t, \varepsilon), \quad c(t, 0) = c_0(t)$$ What is known here (short-time existence, long-time-existence etc.)? If $c$ is a closed loop, does it (maybe suppose $M$ to be compact) either converge to a closed geodesic or collapse to a point?

What I was thinking initially was that I could fix the endpoints (let's say two that are not cutpoints of each other) and hopefully get the flow to converge to the shortest geodesic between the two. However, having thought about it, it feels that the problem of fixed endpoints is ill-posed, isn't it? What other flow should I consider here?

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    $\begingroup$ Your terminology is far from common. Try to search for "curvature flow" and "curve shortening flow". $\endgroup$
    – Sergei Ivanov
    Commented Jun 26, 2013 at 21:57
  • $\begingroup$ jstor.org/stable/1971486 will be helpful for curve shortening flow in surfaces, but I don't know what happens in higher dimensions.. $\endgroup$
    – Otis Chodosh
    Commented Jun 26, 2013 at 21:59
  • $\begingroup$ @Kofi What is $\nabla/\partial t$? $\endgroup$
    – Andrew
    Commented Jun 27, 2013 at 4:02
  • $\begingroup$ The covariant derivative along $c$. $\endgroup$
    – Matthias Ludewig
    Commented Jun 27, 2013 at 7:51

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The question concerns not the curve-shortening flow but the harmonic map flow where the base manifold is a curve. In the case of closed loops mapping into compact manifolds, the arguments in the foundational paper of Eells-Sampson (Amer. J. Math. 1964) will imply long time existence and convergence (possibly to a point) of the flow. This theorem is not stated in the paper, but the crucial point, which is control over the nonlinear term in the Bochner formula for the energy density, will hold automatically in the case where the source manifold is one-dimensional. However, the example on page 155 shows that the solutions can blow up at time infinity when the target space is a complete manifold, even with nonpositive curvature.

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  • $\begingroup$ Presumably, if the limit is not a point, it's a closed geodesic? $\endgroup$
    – Deane Yang
    Commented Jun 30, 2013 at 5:42
  • $\begingroup$ No, any geodesic is stationary w.r.t. this flow, it need not be closed. $\endgroup$
    – Matthias Ludewig
    Commented Jun 30, 2013 at 9:18
  • $\begingroup$ What about my question about fixing the end points? $\endgroup$
    – Matthias Ludewig
    Commented Jun 30, 2013 at 9:19
  • $\begingroup$ The convergence claim via Eells-Sampson is in fact false; see the discussion in mathoverflow.net/questions/311915/… $\endgroup$
    – Willie Wong
    Commented May 11, 2023 at 17:43

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