Asymptotic behavior of flows by powers of the Gaussian curvature

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

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The first-named author was supported in part by the National Science Foundation under grant DMS-1649174. The third-named author was supported in part by the National Science Foundation under grant DMS-1600658.

Received 6 December 2016

Accepted 28 October 2017

Published 31 January 2018