Studies On The Curve Shortening Flow On Surfaces By Projection

The essay use two methods to deal with the curve shortening flow of spherical.In the first method, we project a family of closed convex spherical curves C(·,t) onto the (x,y) plane by vertical projection operator, and we obtain that the evolution equation of closed convex spherical curves is equivalent to the evolution equation of closed convex plane curvesSo, the study of the curve shortening flow of spherical curves is replaced by the study of the curve shortening flow of plane curves.In the second method, we project a family of closed convex plane curves C(·,t) onto the spherical by stereographic projection operator, and we obtain that the evolution equation of closed convex plane curves is equivalent to the evolution equation of closed convex spherical curvesSo, the study of the curve shortening flow of spherical curves is replaced by the study of the curve hortening flow of plane curves. Because a family of closed convex plane curves con-verges to a point in finite time, so a family of closed convex spherical curves curves converges to a point in finite time.