Flow by powers of the Gauss curvature in space forms

TL;DR

This paper studies the evolution of convex hypersurfaces under a curvature flow in space forms, proving they contract to a point and converge to geodesic spheres, extending Euclidean results to curved spaces.

Contribution

It extends curvature flow results from Euclidean space to space forms, showing contraction and convergence to geodesic spheres for convex hypersurfaces.

Findings

Convex hypersurfaces contract to a point in finite time.

Flow by power  of Gauss curvature leads to convergence to geodesic spheres.

Results generalize known Euclidean space theorems to 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.