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 $$\underset{t\to \infty }{\lim }L({\gamma }_t)>0,$$ then for every $n>0$, $$\underset{t\to \infty }{\lim }\sup (|\frac{D^{n}T}{\partial s^n}|)=0.$$ 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.

23 pages

References in corpus (1)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds