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Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. As a function of $t$, let $\gamma$ evolve as if it were made of damped elastic, so that the curve tries to pull itself into a geodesic. That is, denoting $\gamma'(t,x) = \frac{\partial}{\partial x} \gamma(t,x)$, it obeys the PDE: $$ \frac{\partial}{\partial t} \gamma(t,x) = \frac{\partial}{\partial x} \gamma'(t,x) + \nabla_{\gamma'(t,x)} \gamma'(t,x)$$ where $\nabla_u v$ denotes the usual covariant derivative of $v$ with respect to $u$.

I am interested in whether $\gamma(t,\cdot)$ converges to a geodesic on $M$ at an exponential rate. I was a little surprised at how nice a proof I was able to obtain, almost as if the definition of sectional curvature was designed with this result in mind. Basically if the sectional curvature is non-positive, then it does converge with exponential speed, and if the sectional curvature is positive then the hypothesis seems to exactly exclude the case when the curve is on the sphere going from the north pole to the south pole.

My question is this: does this result already exist in the literature, and where? (Maybe it is even an exercise in some book.)

Note this PDE is a kind of heat equation. I do know about a paper by Eells, J.; Sampson, J.H. (1964), "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86: 109–160, JSTOR 2373037. But I think my problem is a little simpler than theirs. (On the other hand, I haven't looked at this paper in any real detail, so maybe it is hidden in there somewhere.)

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    $\begingroup$ I'm also very interested in this. Jost addresses the case of closed geodesics in his text "Riemann Geometry and Geometric Analysis" - however I believe there is a mistake. He uses $\gamma_{xxt}=\gamma_{xtx}$ (note the missing curvature term!) in justifying that the energy is always convex when I believe it is only so (at least his calculation only justifies it) in the case of nonpositive sectional curvature. math.msu.edu/~parker/ChoiParkerGeodesics.pdf has a supposed example of non-convergence on a compact manifold. I'm unsure how much the fixed-endpoint case differs. $\endgroup$
    – Anthony Carapetis
    Sep 17, 2013 at 13:58
  • $\begingroup$ Thank you very much for the reference. It is very helpful. I find their counterexample very plausible. This is because equation 4.5 only shows that $\int_s^t ||\dot u||^2$ is finite, whereas to prove convergence one would need $\int_s^t ||\dot u||$ to be finite. (And this was actually my goal in all of this.) $\endgroup$
    – Stephen Montgomery-Smith
    Sep 17, 2013 at 16:25
  • $\begingroup$ The only extra tool I bring to their Section 8 is Poincarés inequality, namely $\int_0^L ||u||^2 \le C^2 \int_0^L ||\dot u||^2$, where $C = \pi/2L$. This is true when the end points are fixed. You can see this, for example, using sine series. This is not true when the end points are not fixed, as the constant function is an obvious counterexample. $\endgroup$
    – Stephen Montgomery-Smith
    Sep 17, 2013 at 16:30
  • $\begingroup$ However, maybe some kind of Poincaré inequality would also work in the $S^1$ case, as the constant function is essentially the only counterexample. $\endgroup$
    – Stephen Montgomery-Smith
    Sep 17, 2013 at 16:31
  • $\begingroup$ Two comments up I meant $C=\pi/L$. $\endgroup$
    – Stephen Montgomery-Smith
    Sep 17, 2013 at 16:41

1 Answer 1

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I consider this to be the answer to my question:

http://www.math.msu.edu/~parker/ChoiParkerGeodesics.pdf

Thank you, Anthony Carapetis.

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