Convexity estimates for flows by powers of the mean curvature
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, p. 261-277
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.
@article{ASNSP_2006_5_5_2_261_0,
author = {Schulze, Felix},
title = {Convexity estimates for flows by powers of the mean curvature},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {2},
year = {2006},
pages = {261-277},
zbl = {1150.53024},
mrnumber = {2244700},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0}
}

Schulze, Felix. Convexity estimates for flows by powers of the mean curvature. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 261-277. http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/

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