Convexity estimates for flows by powers of the mean curvature

Schulze, Felix

Schulze, Felix

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 2, p. 261-277

Abstract

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 $k \geq 1$ 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.

MR 2244700 | Zbl 1150.53024 | 2 citations in Numdam

@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, Serie 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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