We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha \geq \frac{1}{n+2}$. For $\alpha > \frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha=\frac{1}{n+2}$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

中文翻译：

---

通过高斯曲率的幂的流的渐近行为

我们考虑 $\mathbb{R}^{n+1}$ 中严格凸超曲面的单参数族以速度 $- K^\alpha \nu$ 移动，其中 $\nu$ 表示向外指向的单元法线向量和 $\alpha \geq \frac{1}{n+2}$。对于 $\alpha > \frac{1}{n+2}$，我们表明流在重新缩放后收敛到一个圆形球体。在仿射不变情况 $\alpha=\frac{1}{n+2}$ 中，我们的论点提供了一个替代证明，即流​​在重新缩放后收敛到椭球这一事实。