Curve Shortening Flow in a Riemannian Manifold

Abstract

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\mathbf{M}$ be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and $$ \lim_{t\to\infty}L(\gamma_{t})>0, $$ then for every $n>0$, $$ \lim_{t\to \infty}\sup(|\frac{D^{n}T}{\partial s^{n}}|)=0. $$ In particular, the limiting curve exists and is a closed geodesic in $\mathbf{M}$. 2). For $\gamma_{0}$ is a ramp, we have a global flow and the flow converges to a geodesic in $C^{\infty}$ norm.