# Geodesics and harmonic map heat flow

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