Convexity estimates for flows by powers of the mean curvature

We study the evolution of a closed, convex hypersurface in ${ℝ}^{n+1}$ in direction of its normal vector, where the speed equals a power $k\ge 1$ of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to $1$, depending only on $k$ and $n$, then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.