# zbMATH — the first resource for mathematics

Shortening embedded curves. (English) Zbl 0686.53036
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.
Reviewer: T.Ochiai

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J35 Heat and other parabolic equation methods for PDEs on manifolds
Full Text: