$\alpha$-Gauss Curvature flows

Abstract

In this paper, we study the deformation of the n-dimensional strictly convex hypersurface in ${\mathbb{R}}^{n+1}$ whose speed at a point on the hypersurface is proportional to $\alpha$-power of positive part of Gauss Curvature. For $\frac{1}{n}<\alpha \le 1$, we prove that there exist the strictly convex smooth solutions if the initial surface is strictly convex and smooth and the solution hypersurfaces converge to a point. We also show the asymptotic behavior of the rescaled hypersurfaces, in other words, the rescaled manifold converges to a strictly convex smooth manifold. Moreover, there exists a subsequence whose the limit satisfies a certain equation.