$\alpha$-Gauss Curvature flows

TL;DR

This paper investigates the evolution of convex hypersurfaces in Euclidean space driven by a power of Gauss Curvature, establishing existence, convergence to a point, and asymptotic behavior of rescaled solutions.

Contribution

It proves existence and convergence results for $rac{1}{n}<eta \, ext{and} \, eta \, ext{in} \, (rac{1}{n},1]$, of smooth convex solutions to the $eta$-Gauss curvature flow.

Findings

Solutions exist and remain smooth for the specified range of $eta$.

Rescaled hypersurfaces converge to a smooth convex manifold.

A subsequence of solutions converges to a limit satisfying a specific equation.

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 $\frac{1}{n}<\alpha \leq 1$, 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.