Curvature flows in the sphere

Abstract

We consider contracting and expanding curvature flows in $\mathbb{S}^{n+1}$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mathcal{H}(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ resp. $-x_0$ exponentially fast in $C^{\infty} (\mathbb{S}^n)$.