Curvature flows in the sphere

Abstract

We consider contracting and expanding curvature flows in Sn+1. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauβ map. The contracting hypersurfaces shrink to a point x0 while the expanding hypersurfaces converge to the equator of the hemisphere H(-x0 ). After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers x0 resp. -x0 exponentially fast in C∞(Sn ).