# Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process of creating a midpoint piecewise geodesic polygon converges on a surface $S \subset \mathbb{R}^3$. $S$ may be assumed smooth, closed, and oriented; or it may be assumed a Riemannian manifold. Let $P$ be a closed geodesic polygon each of whose finite number of edges is a shortest path connecting its endpoint vertices $v_i$. Moreover, the edges of $P$ are "short" in a sense specified below. $P$ need not be simple, i.e., it may intersect and cross itself. Define the midpoint polygon $M(P)$ for $P$ to have vertices $u_i$, each the midpoint of the geodesic segment $v_i v_{i+1}$ of $P$, where $u_i$ is connected to $u_{i+1}$ by a shortest path. I would like the segment $u_i u_{i+1}$ to be the unique shortest path between those points. So assume that $|u_i u_{i+1}|$ is smaller than the injectivity radius of the manifold. (This might require adding new vertices to ensure this condition holds for the next iteration.)

In the plane, iterating this midpoint polygon construction is well studied. On a surface (or Riemannian manifold) it plays a role in curve shortening, in particular Birkhoff curve shortening, which he used to prove the existence of a simple closed geodesic [B27]. Subsequently the technique has many uses. For example, Christopher Croke used Birkhoff shortening to find the length of a shortest closed geodesic on a sphere [C88].

The uses I have seen of Birkhoff shortening are on the sphere, not on more general surfaces. Which brings me to the question:

Does Birkoff shortening repeatedly applied to a geodesic polygon on a closed surface $S$ always converge (perhaps to a point)? If so, always to a closed geodesic? If not, under what conditions might it not converge?

References.
[B27] George D. Birkhoff, Dynamical Systems, AMS, 1927. p.135ff.

[C88] Christopher B. Croke, "Area and the length of the shortest closed geodesic." J. Differential Geom., Volume 27, Number 1 (1988), 1-21.

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What is Birkhoff shortening exactly? Is it iterated midpoint polygon construction or something else? It seems that there are several related constructions referred to by this name. –  Sergei Ivanov Oct 16 '10 at 10:46
@Sergei: In the sense e.g. Croke uses it, it is iterated midpoint construction with the addition of vertices if necessary to keep edges shorter than the injectivity radius. I have not seen other uses of the term, although I do not doubt there are other constructions that go by this name. –  Joseph O'Rourke Oct 16 '10 at 12:42
Another use of the term is the curvature flow for curves (which is a sort of limit as edges go to zero). Btw, you don't need to introduce new vertices to keep the edges small: if all edges are shorter than $r$, then by the triangle inequality the next iteration's edges are shorter than $r$ too. –  Sergei Ivanov Oct 16 '10 at 17:08
@Sergei: Ah, thanks, I did not understand that point about short edges! –  Joseph O'Rourke Oct 16 '10 at 17:40
It is easy to prove that partial limits form a connected family of geodesics. Moreover any two geodesics in this family must intersect. –  Anton Petrunin Oct 17 '10 at 20:31

I would advise you to have a look on section 3.7 of the paper of Bowditch, Notes on locally CAT(1) spaces, in Geometric Group Theory, R.Charney, M.Davis, and M.Shapiro eds., de Gruyter (1995). In this section Bowditch says: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds". Also he constructs in this section a 3-dimensional example where the process does not converge.

Here is the abstract of the article http://www.warwick.ac.uk/~masgak/abstracts/lco.html

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@Dmitri: Ah, this seems to be the answer (modulo Sergei's point that several constructions go by the same name): it is open for 2-manifolds. It will take me some time to get this book (not in local libraries), but that quote is unambiguous. Thanks so much! –  Joseph O'Rourke Oct 16 '10 at 13:23
Although I guess it remains possible that B.-shortening converges on embedded surfaces in $\mathbb{R}^3$ but not on all Riemannian 2-manifolds... –  Joseph O'Rourke Oct 16 '10 at 14:40

You may also have a look at the paper

www.md-net.org.uk/preprints/jacobi7.pdf

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"A convergent string method: Existence and approximation for the Hamiltonian boundary-value problem," by Hartmut Schwetlick and Johannes Zimmer. "The aim of this article is to develop a discrete framework that mimics the Birkhoff [curves shortening method]." Thanks, Tony! I would never have looked at this paper just from its title. –  Joseph O'Rourke Mar 1 '11 at 16:43