Flow by powers of the Gauss curvature in space forms

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