Curve shortening in a Riemannian manifold

Abstract

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and, then for every n > 0. Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S1, if γ0 is a ramp, then we have a global flow which converges to a closed geodesic in C∞ norm. As an application, we prove the theorem of Lyusternik and Fet. © Springer-Verlag 2006.