The curve shortening flow with parallel 1-form

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 $\Omega (T)\ge 0$ 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 $\tilde{M}$.

12 pages

References in corpus (2)

• The extension and convergence of mean curvature flow in higher codimension
• Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces