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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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Your terminology is far from common. Try to search for "curvature flow" and "curve shortening flow". –  Sergei Ivanov Jun 26 '13 at 21:57
    
jstor.org/stable/1971486 will be helpful for curve shortening flow in surfaces, but I don't know what happens in higher dimensions.. –  Otis Chodosh Jun 26 '13 at 21:59
    
@Kofi What is $\nabla/\partial t$? –  Andrew Jun 27 '13 at 4:02
    
The covariant derivative along $c$. –  Matthias Ludewig Jun 27 '13 at 7:51

1 Answer 1

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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Presumably, if the limit is not a point, it's a closed geodesic? –  Deane Yang Jun 30 '13 at 5:42
    
No, any geodesic is stationary w.r.t. this flow, it need not be closed. –  Matthias Ludewig Jun 30 '13 at 9:18
    
What about my question about fixing the end points? –  Matthias Ludewig Jun 30 '13 at 9:19

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