# Contracting axially symmetric hypersurfaces by powers of the $\sigma_k$-curvature

## TL;DR

This paper studies the contraction of convex, axially symmetric hypersurfaces under a curvature flow driven by powers of the _k-curvature, proving convergence to a sphere without initial pinching conditions.

## Contribution

It establishes convergence results for curvature flows of convex hypersurfaces driven by _k^, including explicit pinching estimates and exponential convergence to spheres.

## Key findings

- Hypersurfaces converge exponentially to the sphere under specified curvature flows.
- Pinching estimates relate the maximum and minimum principal curvatures during flow.
- Results hold for a range of _k^ with _k^ in [1/k, c(n,k)].

## Abstract

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$ by $\sigma_k^\alpha$, where $\sigma_k$ is the $k$-th elementary symmetric function of the principal curvatures and $\alpha\ge 1/k$. We prove that for any $n\geq3$ and any fixed $k$ with $1\leq k\leq n$, there exists a constant $c(n,k)>1/k$ such that that if $\alpha$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $1<k\le n \le k^2$, we can choose $c(n,k)=\frac{1}{k-1}$. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.