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 $T^{\ast }$. 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.