Convexity estimates for flows by powers of the mean curvature
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 5 (2006) no. 2, p. 261-277
We study the evolution of a closed, convex hypersurface in n+1 in direction of its normal vector, where the speed equals a power k1 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, Volume 5 (2006) no. 2, pp. 261-277. http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/

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