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:
Q. 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.
