Curve shortening in a Riemannian manifold

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 lim(t ->infinity)L(gamma(t))>0, then for every n > 0, lim(t ->infinity) sup(vertical bar(DT)-T-n/partial derivative s(n)vertical bar). Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M x S-1, if gamma(0) is a ramp, then we have a global flow which converges to a closed geodesic in C-infinity. As an application, we prove the theorem of Lyusternik and Fet.