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}, zbl = {1150.53024}, mrnumber = {2244700}, 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 DA - 2006/// SP - 261 EP - 277 VL - Ser. 5, 5 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2006_5_5_2_261_0/ UR - https://zbmath.org/?q=an%3A1150.53024 UR - https://www.ams.org/mathscinet-getitem?mr=2244700 LA - en ID - ASNSP_2006_5_5_2_261_0 ER -
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/
[1] Contraction of convex hypersurfaces in Euclidian space, Calc. Var. Partial Differential Equations 2 (1994), 151-171. | MR 1385524 | Zbl 0805.35048
,[2] Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), 151-161. | MR 1714339 | Zbl 0936.35080
,[3] Moving surfaces by non-concave curvature functions, 2004, arXiv:math.DG/0402273. | MR 2729317 | Zbl 1203.53062
,[4] Deforming convex hypersurfaces by the th root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117-138. | MR 826427 | Zbl 0589.53005
,[5] Deforming hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63-82. | MR 862712 | Zbl 0608.53005
,[6] Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. | MR 783531 | Zbl 0549.35061
and ,[7] Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266. | MR 772132 | Zbl 0556.53001
,[8] Surfaces contracting with speed , J. Differential Geom. 71 (2005), 347-363. | MR 2198805 | Zbl 1101.53002
,[9] Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721-733. | MR 2190140 | Zbl 1087.53062
,