We study the evolution of a closed, convex hypersurface in in direction of its normal vector, where the speed equals a power 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 , depending only on and , 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}, pages = {261--277}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {2}, year = {2006}, mrnumber = {2244700}, zbl = {1150.53024}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/} }
TY - JOUR AU - Schulze, Felix TI - Convexity estimates for flows by powers of the mean curvature JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 261 EP - 277 VL - 5 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/ LA - en ID - ASNSP_2006_5_5_2_261_0 ER -
%0 Journal Article %A Schulze, Felix %T Convexity estimates for flows by powers of the mean curvature %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 261-277 %V 5 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/ %G en %F 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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