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.

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},
     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, Serie 5, Volume 5 (2006) no. 2, pp. 261-277. http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/

[1] B. Andrews, Contraction of convex hypersurfaces in Euclidian space, Calc. Var. Partial Differential Equations 2 (1994), 151-171. | MR | Zbl

[2] B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), 151-161. | MR | Zbl

[3] B. Andrews, Moving surfaces by non-concave curvature functions, 2004, arXiv:math.DG/0402273. | MR | Zbl

[4] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117-138. | MR | Zbl

[5] B. Chow, Deforming hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63-82. | MR | Zbl

[6] E. Dibenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. | MR | Zbl

[7] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266. | MR | Zbl

[8] O. C. Schnürer, Surfaces contracting with speed |A| 2 , J. Differential Geom. 71 (2005), 347-363. | MR | Zbl

[9] F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721-733. | MR | Zbl