Shortening embedded curves. (English) Zbl 0686.53036

Grayson, Matthew A.

Ann. Math. (2) 129, No. 1, 71-111 (1989).

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

Keywords:

evolution of curvature curve shortening; evolution equation