Shortening embedded curves (Q1262548)

Let F be a surface. We consider the evolution equation for a map C(u,t): \(S^ 1\times 1\to F\) given by \[ (*)\quad \partial C/\partial t=kN,\quad C(u,0)=c(u), \] where k is the curvature of C and N is its unit normal vector. Now suppose F is convex at infinity, namely the convex hull of every compact subset is compact. It is proved that there exists a solution for (*) on \([0,t_{\infty})\). If \(t_{\infty}\) is finite, then C converges to a point. If \(t_{\infty}\) is infinite, then the curvature of C converges to zero in the \(C^{\infty}\) norm. As an application, it is proved that a 2-sphere with a smooth Riemannian metric has at least three simple closed geodesics.