Flow by powers of the Gauss curvature in space forms

Abstract

In this paper, we prove that convex hypersurfaces under the flow by powers $\alpha>0$ of the Gauss curvature in space forms $\mathbb{N}^{n+1}(\kappa)$ of constant sectional curvature $\kappa$ $(\kappa=\pm 1)$ contract to a point in finite time $T^*$. Moreover, convex hypersurfaces under the flow by power $\alpha>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in $\mathbb{N}^{n+1}(\kappa)$. This extends the known results in Euclidean space to space forms.