International Congress of Mathematicians

228 B. Andrewseach t > 0, to either a great sphere (in infinite time) or to a point, with sphericallimiting shape (in finite time). If M 0 is embedded, then so is M t for each t > 0.This includes in particular Simons' result on mimimal surfaces. If we modifythe speed somewhat, then we get the following result, which gives in particular anew result for Weingarten surfaces in the 3-sphere:Theorem 2. Let 0}. Then there exists a function F which is smoothly defined on{K1K2 + 1 > 0}, and strictly monotone increasing in each argument, with sgnF =sgncf) everywhere, such that the following holds: If M 0 = XQ(S 2 ) is a smooth compactsurface in S 3 with non-negative intrinsic curvature, then the motion with speed Fdeforms M 0 through a smooth family {M t } 0