The curve shortening flow with parallel 1-form

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