The curve shortening flow with parallel 1-form

TL;DR

This paper investigates the behavior of the curve shortening flow on closed Riemannian manifolds with a parallel 1-form, proving long-time existence under certain conditions and analyzing convergence properties of the flow.

Contribution

It establishes conditions for the long-time existence of the curve shortening flow and characterizes its convergence behavior on manifolds with a parallel 1-form.

Findings

Flow exists for all time if initial tangent satisfies (T)\u2265

Flow's curvature derivatives tend to zero as time approaches infinity

Flow converges to a geodesic under specified conditions

Abstract

Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $\Omega(T)\geq 0$ on the initial curve $\co$. Here $T$ is the unit tangent vector on $\co$. The other one is about the convergence. It says that in a closed {\Rm} $\tilde{M}$, assume the curve shortening flow $\ct$ exists for all $t\in[0,\infty)$ and its length converges to a positive limit, then $ \lim\limits_{t\rightarrow\infty}max_{\ct}|\nabla^{m}A|^{2}=0$ for all $m=0,1,...$. Here $A$ denotes the second fundamental form of $\ct$ in $\tilde{M}$.