Convexity estimates for flows by powers of the mean curvature

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 $k\ge 1$ 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.

[1] B. Andrews, Contraction of convex hypersurfaces in Euclidian space, Calc. Var. Partial Differential Equations 2 (1994), 151-171. | MR 1385524 | Zbl 0805.35048

[2] B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), 151-161. | MR 1714339 | Zbl 0936.35080

[3] B. Andrews, Moving surfaces by non-concave curvature functions, 2004, arXiv:math.DG/0402273. | MR 2729317 | Zbl 1203.53062

[4] B. Chow, Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117-138. | MR 826427 | Zbl 0589.53005

[5] B. Chow, Deforming hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63-82. | MR 862712 | Zbl 0608.53005

[6] E. Dibenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. | MR 783531 | Zbl 0549.35061

[7] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266. | MR 772132 | Zbl 0556.53001

[8] O. C. Schnürer, Surfaces contracting with speed ${\left|A\right|}^2$, J. Differential Geom. 71 (2005), 347-363. | MR 2198805 | Zbl 1101.53002

[9] F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721-733. | MR 2190140 | Zbl 1087.53062